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0.125.2
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\section{Lerch Zeta Function}\label{section: Lerch Zeta}
In this section, we first introduce the Lerch zeta function for totally real fields.
Then we will then define the Shintani zeta function associated to a cone $\sigma$ and a function $\phi\colon\cO_F\rightarrow\bbC$
which factors through $\cO_F/\frf$ for some nonzero ideal $\frf\subset\cO_F$.
We will then describe the generating function of its values at nonpositive integers when $\phi$ is a finite additive character.
Let $\xi\in{\Hecke}om_\bbZ(\cO_F,\bbC^\times)$ be a $\bbC$-valued character on $\cO_F$ of finite order.
As in Definition \ref{def: Lerch} of \S\ref{section: introduction},
we define the Lerch zeta function for totally real fields by the series
\[
\cL(\xi\Delta, s)\coloneqq\sum_{\alpha\in\Delta_\xi\backslash\cO_{F+}}\xi(\alpha)N(\alpha)^{-s},
\]
where $\Delta_\xi\coloneqq\{\varepsilon\in\Delta\mid \xi^\varepsilon=\xi\}$,
which may be continued analytically to the whole complex plane.
\begin{remark}
Note that we have
\[
\cL(\xi\Delta, s)=\sum_{\alpha\in\Delta\backslash\cO_{F+}}
\sum_{\varepsilon\in\Delta_\xi\backslash\Delta} \xi(\varepsilon\alpha)N(\alpha)^{-s}.
\]
Even though $\xi(\alpha)$ is not well-defined for $\alpha\in\Delta\backslash\cO_F$,
the sum $\sum_{\varepsilon\in\Delta_\xi\backslash\Delta}\xi(\varepsilon\alpha)$
is well-defined for $\alpha\in\Delta\backslash\cO_F$.
\end{remark}
The importance of $\cL(\xi\Delta, s)$ is in its relation to the Hecke $L$-functions of $F$.
Let $\frf$ be a nonzero integral ideal of $F$.
We denote by $\Cl^+_F(\frf)\coloneqq I_\frf/P^+_\frf$
the strict ray class group modulo $\frf$ of $F$, where
$I_\frf$ is the group of fractional ideals of $F$ prime to $\frf$
and $P^+_\frf\coloneqq \{ (\alpha) \mid \alpha\in F_+, \alpha\equiv 1 \operatorname{mod}^\times \frf\}$.
A finite Hecke character of $F$ of conductor $\frf$ is a character
\[
\chi\colon\Cl^+_F(\frf)\rightarrow\bbC^\times.
\]
By \cite{Neu99}*{Chapter VII (6.9) Proposition}, there exists a unique character
$
\chi_\fin\colon(\cO_F/\frf)^\times\rightarrow\bbC^\times
$
associated to $\chi$ such that $\chi((\alpha))=\chi_\fin(\alpha)$ for any $\alpha\in\cO_{F+}$ prime to $\frf$.
In particular, we have $\chi_\fin(\varepsilon)=1$ for any $\varepsilon\in\Delta$.
Extending by \textit{zero}, we regard $\chi_\fin$ as functions on $\cO_F/\frf$ and $\cO_F$ with values in $\bbC$.
In what follows, we let $\bbT[\frf]\coloneqq{\Hecke}om(\cO_F/\frf,{\ol\bbQ}^\times)\subset\bbT(\ol\bbQ)$
be the set of $\frf$-torsion points of $\bbT$.
We say that a character $\chi$, $\chi_\fin$ or $\xi\in\bbT[\frf]$ is \textit{primitive},
if it does not factor respectively through $\Cl_F^+(\frf')$, $(\cO_F/\frf')^\times$ or $\cO_F/\frf'$
for any integral ideal $\frf'\neq\frf$ such that $\frf'|\frf$.
Then we have the following.
\begin{lemma}\label{lemma: fourier}
For any $\xi\in\bbT[\frf]$, let
\[
c_\chi(\xi)\coloneqq\frac{1}{\bsN(\frf)}\sum_{\beta\in\cO_F/\frf} \chi_\fin(\beta)\xi(-\beta).
\]
Then we have
\[
\chi_\fin(\alpha)=\sum_{\xi\in\bbT[\frf]}c_\chi(\xi)\xi(\alpha).
\]
Moreover, if $\chi_\fin$ is primitive, then we have $c_\chi(\xi)=0$ for any non-primitive $\xi$.
\end{lemma}
\begin{proof}
The first statement follows from
\[
\sum_{\xi\in\bbT[\frf]}c_\chi(\xi)\xi(\alpha) = \frac{1}{\bsN(\frf)} \sum_{\beta\in\cO_F/\frf}
\chi_\fin(\beta)\biggl(\sum_{\xi\in\bbT[\frf]} \xi(\alpha-\beta)\biggr) = \chi_\fin(\alpha),
\]
where the last equality follows from the fact that $\sum_{\xi\in\bbT[\frf]} \xi(\alpha)=N(\frf)$ if $\alpha\equiv 0\pmod{\frf}$ and
$\sum_{\xi\in\bbT[\frf]} \xi(\alpha)=0$ if $\alpha\not\equiv 0\pmod{\frf}$.
Next, suppose $\chi_\fin$ is primitive, and let $\frf'\neq\frf$ be an integral ideal of $F$ such that $\frf'|\frf$ and $\xi\in\bbT[\frf ']$.
Since $\chi_\fin$ is primitive, it does not factor through $\cO_F/\frf'$, hence there exists an element $\gamma\in\cO_F$
prime to $\frf$
such that $\gamma\equiv1\pmod{\frf'}$ and $\chi_\fin(\gamma)\neq 1$. Then since $\xi\in\bbT[\frf']$, we have
$\xi(\gamma\alpha)=\xi(\alpha)$ for any $\alpha\in\cO_F$. This gives
\begin{align*}
c_\chi(\xi)=\frac{1}{\bsN(\frf)}\sum_{\beta\in\cO_F/\frf} \chi_\fin(\beta)\xi(-\beta)
&=\frac{1}{\bsN(\frf)}\sum_{\beta\in\cO_F/\frf} \chi_\fin(\beta)\xi(-\gamma\beta)\\
&=\frac{\ol\chi_\fin(\gamma)}{\bsN(\frf)}\sum_{\beta\in\cO_F/\frf} \chi_\fin(\gamma\beta)
\xi(-\gamma\beta)=\ol\chi_\fin(\gamma)c_\chi(\xi).
\end{align*}
Since $\chi_\fin(\gamma)\neq 1$, we have $c_\chi(\xi)=0$ as desired.
\end{proof}
Note that since multiplication by $\varepsilon\in\Delta$ is bijective on $\cO_F/\frf$ and since $\chi_\fin(\varepsilon)=1$,
we have $c_\chi(\xi^\varepsilon)=c_\chi(\xi)$. Then we have the following.
\begin{proposition}\label{prop: Hecke}
Assume that the narrow class number of $F$ is \textit{one},
and let $\chi\colon\Cl^+_F(\frf)\rightarrow\bbC^\times$ be a finite primitive Hecke character of $F$ of
conductor $\frf\neq(1)$.
Then for $U[\frf]\coloneqq\bbT[\frf]\setminus\{1\}$, we have
\[
L(\chi, s)=\sum_{\xi\in U[\frf]/\Delta} c_\chi(\xi)\cL(\xi\Delta, s).
\]
\end{proposition}
\begin{proof}
By definition and Lemma \ref{lemma: fourier}, we have
\begin{align*}
\sum_{\xi\in\bbT[\frf]/\Delta} c_\chi(\xi)\cL(\xi\Delta, s)
&=\sum_{\xi\in\bbT[\frf]/\Delta}\sum_{\alpha\in\Delta\backslash\cO_{F+}}
\sum_{\varepsilon\in\Delta_\xi\backslash\Delta}c_\chi(\xi)\xi(\varepsilon\alpha)\bsN(\alpha)^{-s}\\
&=\sum_{\alpha\in\Delta\backslash\cO_{F+}}\sum_{\xi\in\bbT[\frf]/\Delta}\sum_{\varepsilon\in\Delta_\xi\backslash\Delta}
c_\chi(\xi^\varepsilon)\xi^\varepsilon(\alpha)\bsN(\alpha)^{-s}\\
&=\sum_{\alpha\in\Delta\backslash\cO_{F+}}\sum_{\xi\in\bbT[\frf]}c_\chi(\xi)\xi(\alpha)\bsN(\alpha)^{-s}\\
&=\sum_{\alpha\in\Delta\backslash\cO_{F+}}
\chi_\fin(\alpha)\bsN(\alpha)^{-s} = \sum_{\fra\subset\cO_F} \chi(\fra)\bsN\!\fra^{-s}.
\end{align*}
Our assertion follows from the definition of the Hecke $L$-function and the fact that $c_\chi(\xi)=0$ for $\xi=1$.
\end{proof}
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0.125.3
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Note that since multiplication by $\varepsilon\in\Delta$ is bijective on $\cO_F/\frf$ and since $\chi_\fin(\varepsilon)=1$,
we have $c_\chi(\xi^\varepsilon)=c_\chi(\xi)$. Then we have the following.
\begin{proposition}\label{prop: Hecke}
Assume that the narrow class number of $F$ is \textit{one},
and let $\chi\colon\Cl^+_F(\frf)\rightarrow\bbC^\times$ be a finite primitive Hecke character of $F$ of
conductor $\frf\neq(1)$.
Then for $U[\frf]\coloneqq\bbT[\frf]\setminus\{1\}$, we have
\[
L(\chi, s)=\sum_{\xi\in U[\frf]/\Delta} c_\chi(\xi)\cL(\xi\Delta, s).
\]
\end{proposition}
\begin{proof}
By definition and Lemma \ref{lemma: fourier}, we have
\begin{align*}
\sum_{\xi\in\bbT[\frf]/\Delta} c_\chi(\xi)\cL(\xi\Delta, s)
&=\sum_{\xi\in\bbT[\frf]/\Delta}\sum_{\alpha\in\Delta\backslash\cO_{F+}}
\sum_{\varepsilon\in\Delta_\xi\backslash\Delta}c_\chi(\xi)\xi(\varepsilon\alpha)\bsN(\alpha)^{-s}\\
&=\sum_{\alpha\in\Delta\backslash\cO_{F+}}\sum_{\xi\in\bbT[\frf]/\Delta}\sum_{\varepsilon\in\Delta_\xi\backslash\Delta}
c_\chi(\xi^\varepsilon)\xi^\varepsilon(\alpha)\bsN(\alpha)^{-s}\\
&=\sum_{\alpha\in\Delta\backslash\cO_{F+}}\sum_{\xi\in\bbT[\frf]}c_\chi(\xi)\xi(\alpha)\bsN(\alpha)^{-s}\\
&=\sum_{\alpha\in\Delta\backslash\cO_{F+}}
\chi_\fin(\alpha)\bsN(\alpha)^{-s} = \sum_{\fra\subset\cO_F} \chi(\fra)\bsN\!\fra^{-s}.
\end{align*}
Our assertion follows from the definition of the Hecke $L$-function and the fact that $c_\chi(\xi)=0$ for $\xi=1$.
\end{proof}
\begin{remark}\label{rem: class number one}
We assumed the condition on the narrow class number for simplicity.
By considering the Lerch zeta functions corresponding to additive characters in
${\Hecke}om_\bbZ(\fra,\bbC^\times)$ for general fractional ideals $\fra$ of $F$,
we may express the Hecke $L$-functions when the narrow class number of
$F$ is greater than \textit{one}.
\end{remark}
We will next define the Shintani zeta function
associated to a cone.
Note that we have a canonical isomorphism
\[
F\otimes\bbR\cong\bbR^I\coloneqq \prod_{\tau\in I}\bbR, \qquad \alpha\otimes 1 \mapsto (\alpha^\tau),
\]
where $I$ is the set of embeddings $\tau\colon F\hookrightarrow\bbR$
and we let $\alpha^\tau\coloneqq\tau(\alpha)$ for any embedding $\tau\in I$.
We denote by $\bbR^I_+\coloneqq\prod_{\tau\in I}\bbR_+$
the set of totally positive elements of $\bbR^I$, where $\bbR_+$ is the set of positive real numbers.
\begin{definition}
A rational closed polyhedral cone in $\bbR^I_+\cup\{0\}$, which we simply call a cone, is any set of the form
\[
\sigma_{\boldsymbol{\alpha}}\coloneqq\{ x_1 \alpha_1+\cdots+x_m\alpha_m \mid x_1,\ldots,x_m \in\bbR_{\geq0}\}
\]
for some ${\boldsymbol{\alpha}}=(\alpha_1,\ldots,\alpha_m)\in\cO_{F+}^m$.
In this case, we say that ${\boldsymbol{\alpha}}$ is a generator of $\sigma_{\boldsymbol{\alpha}}$.
By considering the case $m=0$, we see that $\sigma=\{0\}$ is a cone.
\end{definition}
We define the dimension $\dim\sigma$ of a cone $\sigma$ to be the dimension
of the $\bbR$-vector space generated by $\sigma$.
In what follows, we fix a numbering $I=\{\tau_1,\ldots,\tau_g\}$ of elements in $I$.
For any subset $R\subset\bbR_+^I$, we let
\[
\wh R\coloneqq\{(u_{\tau_1},\ldots,u_{\tau_g})\in\bbR_+^I \mid\exists\delta>0,\,0<\forall\delta'<\delta,(u_{\tau_1},\ldots,u_{\tau_{g-1}},
u_{\tau_g}-\delta')\in R\}.
\]
\begin{definition}
Let $\sigma$ be a cone, and let $\phi\colon\cO_F\rightarrow\bbC$ be a $\bbC$-valued function on $\cO_F$
which factors through $\cO_F/\frf$ for some nonzero ideal $\frf\subset\cO_F$.
We define the \textit{Shintani zeta function} $\zeta_\sigma(\phi,\bss)$ associated to a cone $\sigma$ and
function $\phi$ by the series
\begin{equation}\label{eq: Shintani zeta}
\zeta_\sigma(\phi,\bss)\coloneqq\sum_{\alpha\in\wh\sigma\cap\cO_F} \phi(\alpha) \alpha^{-\bss},
\end{equation}
where $\bss=(s_{\tau})\in\bbC^I$ and $\alpha^{-\bss}\coloneqq\prod_{\tau\in I}(\alpha^{\tau})^{-s_{\tau}}$.
The series \eqref{eq: Shintani zeta} converges if $\Re(s_{\tau})>1$ for any $\tau\in I$.
\end{definition}
By \cite{Shi76}*{Proposition 1}, the function $\zeta_\sigma(\phi,\bss)$ has a meromorphic continuation to any $\bss\in\bbC^I$.
If we let $\bss=(s,\ldots, s)$ for $s\in\bbC$, then we have
\begin{equation}\label{eq: diagonal}
\zeta_\sigma(\phi,(s,\ldots, s))=\sum_{\alpha\in\wh\sigma\cap\cO_F} \phi(\alpha) N(\alpha)^{-s}.
\end{equation}
Shintani constructed the generating function of values of $\zeta_\sigma(\xi, \bss)$
at nonpositive integers for additive characters $\xi\colon\cO_F\rightarrow\bbC^\times$ of finite order,
given as follows.
In what follows, we view $z\in F\otimes\bbC$ as an element
$z = (z_\tau)\in\bbC^I$ through the canonical isomorphism $F\otimes\bbC\cong\bbC^I$.
\begin{definition}\label{def: generating}
Let $\sigma=\sigma_{\boldsymbol{\alpha}}$ be a $g$-dimensional cone generated by
${\boldsymbol{\alpha}}=(\alpha_1,\ldots,\alpha_g)\in\cO_{F+}^g$,
and we let $P_{\boldsymbol{\alpha}}\coloneqq\{ x_1\alpha_1+\cdots+x_g\alpha_g\mid
\forall i\,\,0\leq x_i < 1\}$
be the parallelepiped spanned by $\alpha_1,\ldots,\alpha_g$.
We define $\sG_{\sigma}(z)$ to be the meromorphic function
on $F\otimes\bbC\cong\bbC^I$ given by
\[
\sG_\sigma(z)\coloneqq
\frac{\sum_{\alpha\in \wh P_{\boldsymbol{\alpha}}\cap\cO_F}e^{2\pi i\Tr(\alpha z)}}{{\bigl(1-e^{2\pi i\Tr(\alpha_1z)}}\bigr)
\cdots\bigl(1-e^{2\pi i\Tr(\alpha_gz)}\bigr)},
\]
where $\Tr(\alpha z)\coloneqq\sum_{\tau\in I}\alpha^\tau z_\tau$ for any $\alpha\in\cO_F$.
The definition of $\sG_\sigma(z)$
depends only on the cone and is independent of the choice of the generator ${\boldsymbol{\alpha}}$.
\end{definition}
\begin{remark}
If $F=\bbQ$ and $\sigma=\bbR_{\geq0}$, then we have
$
\sG_\sigma(z) = \frac{e^{2\pi iz}}{1-e^{2\pi i z}}.
$
\end{remark}
For $\bsk=(k_\tau)\in\bbN^I$, we denote $\partial^\bsk\coloneqq\prod_{\tau\in I}\partial_\tau ^{k_\tau}$,
where $\partial_\tau :=\frac{1}{2\pi i}\frac{\partial}{\partial z_\tau}$.
For $u\in F$,
we let $\xi_u$ be the finite additive character on $\cO_F$ defined by $\xi_u(\alpha)\coloneqq e^{2\pi i\Tr(\alpha u)}$.
We note that any additive character on $\cO_F$ with values in $\bbC^\times$ of finite order
is of this form for some $u\in F$.
The following theorem, based on the work of Shintani, is standard (see for example \cite{CN79}*{Th\'eor\`eme 5}, \cite{Col88}*{Lemme 3.2}).
\begin{theorem}\label{theorem: Shintani}
Let ${\boldsymbol{\alpha}}$ and $\sigma$ be as in Definition \ref{def: generating}.
For any $u\in F$ satisfying $\xi_u(\alpha_j)\neq1$ for $j=1,\ldots,g$,
we have
\[
\partial^\bsk \sG_\sigma(z)\big|_{z=u\otimes1}=\zeta_\sigma(\xi_u,-\bsk).
\]
\end{theorem}
Note that the condition $\xi_u(\alpha_j)\neq1$ for $j=1,\ldots,g$ ensures that $z=u\otimes 1$
does not lie on the poles of
the function $\sG_\sigma(z)$.
The Lerch zeta function $\cL(\xi\Delta, s)$ may be expressed as a finite sum of
functions $\zeta_\sigma(\xi,(s,\ldots, s))$
using the Shintani decomposition. We first review the definition of the Shintani decomposition.
We say that a cone $\sigma$ is \textit{simplicial}, if there exists a generator of $\sigma$
that is linearly independent over $\bbR$.
Any cone generated by a subset of such a generator is called a \textit{face} of $\sigma$.
A simplicial fan $\Phi$ is a set of simplicial cones such that
for any $\sigma\in\Phi$, any face of $\sigma$
is also in $\Phi$, and for any cones $\sigma,\sigma'\in\Phi$, the intersection $\sigma\cap\sigma'$ is
a common face of $\sigma$ and $\sigma'$.
A version of Shintani decomposition that we will use in this article is as follows.
\begin{definition}\label{def: Shintani}
A \textit{Shintani decomposition} is a simplicial fan $\Phi$
satisfying the following properties.
\begin{enumerate}
\item $\bbR_+^I\cup\{0\}=\coprod_{\sigma\in\Phi}\sigma^\circ$, where $\sigma^\circ$ is the relative interior of $\sigma$,
i.e., the interior of $\sigma$ in the $\bbR$-linear span of $\sigma$.
\item For any $\sigma\in\Phi$ and $\varepsilon\in\Delta$, we have $\varepsilon\sigma\in\Phi$.
\item The quotient $\Delta\backslash\Phi$ is a finite set.
\end{enumerate}
\end{definition}
We may obtain such decomposition
by slightly modifying the construction of Shintani \cite{Shi76}*{Theorem 1}
(see also \cite{Hid93}*{\S2.7 Theorem 1}, \cite{Yam10}*{Theorem 4.1}).
Another construction was given by Ishida \cite{Ish92}*{p.84}.
For any integer $q\geq0$,
we denote by $\Phi_{q+1}$ the subset of $\Phi$ consisting of cones of dimension $q+1$.
Note that by \cite{Yam10}*{Proposition 5.6}, $\Phi_g$ satisfies
\begin{equation}\label{eq: upper closure}
\bbR_+^I=\coprod_{\sigma\in\Phi_g}\wh\sigma.
\end{equation}
This gives the following result.
\begin{proposition}
Let $\xi\colon\cO_F\rightarrow\bbC^\times$ be a character of finite order, and $\Delta_\xi\subset\Delta$
its isotropic subgroup.
If $\Phi$ is a Shintani decomposition, then we have
\begin{equation}\label{eq: Shintani}
\cL(\xi\Delta, s) = \sum_{\sigma\in\Delta_\xi\backslash\Phi_g}\zeta_\sigma(\xi,(s,\ldots,s)).
\end{equation}
\end{proposition}
\begin{proof}
By \eqref{eq: upper closure}, if $C$ is a representative of $\Delta_\xi\backslash\Phi_g$,
then $\coprod_{\sigma\in C}\wh\sigma$ is a representative of the set $\Delta_\xi\backslash\bbR_+^I$.
Our result follows from the definition of the Lerch zeta function and \eqref{eq: diagonal}.
\end{proof}
The expression \eqref{eq: Shintani} is non-canonical, since it depends on the choice of the Shintani decomposition.
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0.125.4
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\section{Equivariant Coherent Cohomology}\label{section: equivariant}
In this section, we will first give the definition of equivariant sheaves and equivariant cohomology
of a scheme with an action of a group.
As in \S\ref{section: introduction}, we let
\begin{equation}\label{eq: algebraic torus}
\bbT\coloneqq {\Hecke}om_\bbZ(\cO_F,\bbG_m)
\end{equation}
be the algebraic torus over $\bbZ$ defined by Katz \cite[\S 1]{Katz81},
satisfying $\bbT(R)={\Hecke}om_\bbZ(\cO_F,R^\times)$ for any $\bbZ$-algebra $R$.
We will then construct the \textit{equivariant \v Cech complex},
which is an explicit complex which may be used to describe equivariant cohomology of $U\coloneqq\bbT\setminus\{1\}$
with action of $\Delta$.
\begin{remark}
In order to consider the values of Hecke $L$-functions when the narrow class number of $F$ is greater than
\textit{one} (cf. Remark \ref{rem: class number one}), then it would be necessary to consider the algebraic tori
\[
\bbT_\fra\coloneqq {\Hecke}om_\bbZ(\fra,\bbG_m)
\]
for general fractional ideals $\fra$ of $F$.
\end{remark}
We first review the basic facts concerning sheaves on schemes that are equivariant with respect to an action of a group.
Let $G$ be a group with identity $e$. A $G$-scheme is a scheme $X$ equipped with
a right action of $G$. We denote by
$[u]\colon X \rightarrow X$ the action of $u\in G$, so that $[uv] = [v]\circ[u]$ for any $u,v\in G$ holds.
In what follows, we let $X$ be a $G$-scheme.
\begin{definition}\label{def: equivariant structure}
A \textit{$G$-equivariant structure} on
an $\sO_X$-module $\sF$ is a family of isomorphisms
\[
\iota_u\colon [u]^* \sF \xrightarrow\cong \sF
\]
for $u\in G$, such that $\iota_e = \id_\sF$ and the diagram
\[\xymatrix{
[uv]^* \sF \ar[r]^-{\iota_{uv}}\ar@{=}[d] & \sF \\
[u]^*[v]^* \sF \ar[r]^-{[u]^*\iota_v} & [u]^*\sF \ar[u]_{\iota_u}
}\]
is commutative. We call $\sF$ equipped with a $G$-equivariant structure a \textit{$G$-equivariant sheaf}.
\end{definition}
Note that the structure sheaf $\sO_X$ itself is naturally a $G$-equivariant sheaf.
For any $G$-equivariant sheaf $\sF$ on $X$,
we define the equivariant global section by $\Gamma(X/G,\sF)\coloneqq
{\Hecke}om_{\bbZ[G]}(\bbZ,\Gamma(X, \sF)) = \Gamma(X,\sF)^G$.
Then the equivariant cohomology
$H^m(X/G, -)$ is defined to be the
$m$-th right derived functor of $\Gamma(X/G,-)$.
Suppose we have a group homomorphism $\pi\colon G\rightarrow H$.
For a $G$-scheme $X$ and an $H$-scheme $Y$, we say that a morphism $f\colon X\rightarrow Y$
of schemes is \textit{equivariant} with respect to $\pi$, if we have $f\circ[u]=[\pi(u)]\circ f$ for any $u\in G$.
If $\sF$ is a $H$-equivariant sheaf on $Y$ and $f$ is equivariant, then $f^*\sF$ is naturally an $G$-equivariant sheaf on $X$
with the equivariant structure given by $f^*\iota_{\pi(u)}\colon [u]^* (f^*\sF) = f^* ([\pi(u)]^* \sF) \rightarrow f^*\sF$
for any $u\in G$, and $f$ induces the pull-back homomorphism
\begin{equation}\label{eq: pullback}
f^*\colon H^m(Y/H,\sF) \rightarrow H^m(X/G, f^* \sF)
\end{equation}
on equivariant cohomology.
We now consider our case of the algebraic torus $\bbT$.
For any $\alpha \in \cO_F$, the morphism $\bbT(R) \rightarrow R^\times$ defined
by mapping $\xi \in \bbT(R)$ to $\xi(\alpha) \in R^\times$
induces a morphism of group schemes $t^\alpha\colon\bbT \rightarrow \bbG_m$, which gives a rational function of $\bbT$.
Then we have
\[
\bbT=\Spec\bbZ[t^\alpha\mid\alpha\in\cO_F],
\]
where $t^\alpha, t^{\alpha'}$ satisfies the relation $t^\alpha t^{\alpha'}=t^{\alpha+\alpha'}$ for any $\alpha,\alpha'\in\cO_F$.
If we take a basis $\alpha_1,\ldots,\alpha_g$ of $\cO_F$ as a $\bbZ$-module, then we have
\[
\Spec\bbZ[t^\alpha\mid\alpha\in\cO_F]=\Spec\bbZ[ t^{\pm\alpha_1},\ldots,t^{\pm \alpha_g}]\cong\bbG_m^g.
\]
The action of $\Delta$ on $\cO_F$ by multiplication induces an action of $\Delta$ on $\bbT$.
Explicitly, the isomorphism $[\varepsilon]\colon\bbT\rightarrow\bbT$
for $\varepsilon\in\Delta$ is given by $t^\alpha\mapsto t^{\varepsilon\alpha}$
for any $\alpha\in \cO_F$.
\begin{definition}\label{def: twist}
For any $\bsk=(k_{\tau})\in\bbZ^I$, we define a $\Delta$-equivariant sheaf $\sO_\bbT(\bsk)$ on $\bbT$
as follows.
As an $\sO_\bbT$-module we let
$\sO_\bbT(\bsk)\coloneqq\sO_\bbT$. The $\Delta$-equivariant structure
\[
\iota_\varepsilon\colon[\varepsilon]^*\sO_\bbT\cong\sO_\bbT
\]
is given by multiplication by
$\varepsilon^{-\bsk}\coloneqq\prod_{\tau\in I}(\varepsilon^{\tau})^{-k_{\tau}}$ for any
$\varepsilon\in\Delta$.
Note that for $\bsk, \bsk'\in\bbZ^I$,
we have $\sO_\bbT(\bsk)\otimes\sO_\bbT(\bsk')=\sO_\bbT(\bsk+\bsk')$.
For the case $\bsk=(k,\ldots,k)$, we have $\varepsilon^{-\bsk}=N(\varepsilon)^{-k}=1$ for any $\varepsilon\in\Delta$, hence
$\sO_\bbT(\bsk)=\sO_\bbT$.
\end{definition}
The open subscheme $U\coloneqq\bbT\setminus\{1\}$ also carries a natural $\Delta$-scheme structure.
We will now construct the \textit{equivariant \v Cech complex},
which may be used to express the cohomology of
$U$ with coefficients in a $\Delta$-equivariant quasi-coherent $\sO_U$-module $\sF$.
For any $\alpha\in\cO_{F}$, we let $U_\alpha\coloneqq\bbT\setminus \{t^{\alpha}=1\}$.
Then any $\varepsilon\in\Delta$ induces an isomorphism $[\varepsilon]\colon U_{\varepsilon\alpha}\rightarrow U_{\alpha}$.
We say that $\alpha\in\cO_{F+}$ is \textit{primitive} if $\alpha/N\not\in\cO_{F+}$ for any integer $N>1$.
In what follows, we let $A\subset\cO_{F+}$ be the set of primitive elements of $\cO_{F+}$.
Then
\begin{enumerate}
\item $\varepsilon A = A$ for any $\varepsilon\in\Delta$.
\item The set $\frU\coloneqq\{U_\alpha\}_{\alpha\in A}$ gives an affine open covering of $U$.
\end{enumerate}
We note that for any simplicial cone $\sigma$ of dimension $m$, there exists a generator ${\boldsymbol{\alpha}}\in A^m$,
unique up to permutation of the components.
Let $q$ be an integer $\geq 0$.
For any ${\boldsymbol{\alpha}}=(\alpha_0,\ldots,\alpha_{q})\in A^{q+1}$,
we let $U_{\boldsymbol{\alpha}}\coloneqq U_{\alpha_0}\cap\cdots\cap U_{\alpha_{q}}$,
and we denote by $j_{{\boldsymbol{\alpha}}}\colon U_{\boldsymbol{\alpha}}\hookrightarrow U$
the inclusion. We let
\[
\sC^q(\frU,\sF)\coloneqq \prod_{{\boldsymbol{\alpha}}\in A^{q+1}}^\alt j_{{\boldsymbol{\alpha}}*}j_{\boldsymbol{\alpha}}^*\sF
\]
be the subsheaf of
$ \prod_{{\boldsymbol{\alpha}}\in A^{q+1}} j_{{\boldsymbol{\alpha}}*}j_{\boldsymbol{\alpha}}^*\sF$ consisting of sections
$\bss=(s_{\boldsymbol{\alpha}})$ such that $s_{\rho({\boldsymbol{\alpha}})} = \sgn(\rho)s_{\boldsymbol{\alpha}}$ for any $\rho\in\frS_{q+1}$
and $s_{\boldsymbol{\alpha}}=0$ if $\alpha_i=\alpha_j$ for some $i\neq j$.
We define the differential
$
d^q\colon\sC^q(\frU,\sF)\rightarrow\sC^{q+1}(\frU,\sF)
$
to be the usual alternating sum
\begin{equation}\label{eq: Cech differential}
(d^qf)_{\alpha_0\cdots\alpha_{q+1}}\coloneqq\sum_{j=0}^{q+1}(-1)^j
f_{\alpha_0\cdots\check\alpha_{j}\cdots\alpha_{q+1}}\big|_{U_{(\alpha_0,\ldots,\alpha_{q+1})}\cap V}
\end{equation}
for any section $(f_{\boldsymbol{\alpha}})$ of $\sC^q(\frU,\sF)$ of each open set $V\subset U$.
If we let $\sF\hookrightarrow \sC^0(\frU,\sF)$ be the natural inclusion, then we have the exact sequence
\[\xymatrix{
0\ar[r]& \sF\ar[r]& \sC^0(\frU,\sF)\ar[r]^{d^0}&\sC^1(\frU,\sF)\ar[r]^{\quad d^1}&\cdots \ar[r]^{d^{q-1}\quad}&\sC^q(\frU,\sF)\ar[r]^{\quad d^q}&\cdots.
}\]
We next consider the action of $\Delta$.
For any ${\boldsymbol{\alpha}}\in A^{q+1}$ and $\varepsilon\in\Delta$, we have a commutative diagram
\[
\xymatrix{
U_{\varepsilon{\boldsymbol{\alpha}}}\ar@{^{(}->}[r]^{j_{\varepsilon{\boldsymbol{\alpha}}}}\ar[d]_{[\varepsilon]}^\cong&U\ar[d]^\cong_{[\varepsilon]}\\
U_{\boldsymbol{\alpha}}\ar@{^{(}->}[r]^{j_{\boldsymbol{\alpha}}} & U,
}
\]
where $\varepsilon{\boldsymbol{\alpha}}\coloneqq(\varepsilon\alpha_0,\ldots,\varepsilon\alpha_{q})$.
Then we have an isomorphism
\[
[\varepsilon]^* j_{{\boldsymbol{\alpha}}*}j_{\boldsymbol{\alpha}}^*\sF
\cong j_{\varepsilon{\boldsymbol{\alpha}}*}j_{\varepsilon{\boldsymbol{\alpha}}}^*[\varepsilon]^*\sF
\xrightarrow\cong j_{\varepsilon{\boldsymbol{\alpha}}*}j_{\varepsilon{\boldsymbol{\alpha}}}^*\sF,
\]
where the last isomorphism is induced by
the $\Delta$-equivariant structure $\iota_\varepsilon\colon[\varepsilon]^*\sF\cong\sF$.
This induces an isomorphism
$
\iota_\varepsilon\colon[\varepsilon]^*\sC^q(\frU,\sF)\xrightarrow\cong\sC^q(\frU,\sF),
$
which is compatible with the differential \eqref{eq: Cech differential}. Hence
$\sC^\bullet(A,\sF)$ is a complex of $\Delta$-equivariant sheaves on $U$.
\begin{proposition}\label{proposition: acyclic}
The sheaf $\sC^q(\frU,\sF)$ is acyclic with respect to the functor $\Gamma(U/\Delta, -)$.
\end{proposition}
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0.125.5
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The open subscheme $U\coloneqq\bbT\setminus\{1\}$ also carries a natural $\Delta$-scheme structure.
We will now construct the \textit{equivariant \v Cech complex},
which may be used to express the cohomology of
$U$ with coefficients in a $\Delta$-equivariant quasi-coherent $\sO_U$-module $\sF$.
For any $\alpha\in\cO_{F}$, we let $U_\alpha\coloneqq\bbT\setminus \{t^{\alpha}=1\}$.
Then any $\varepsilon\in\Delta$ induces an isomorphism $[\varepsilon]\colon U_{\varepsilon\alpha}\rightarrow U_{\alpha}$.
We say that $\alpha\in\cO_{F+}$ is \textit{primitive} if $\alpha/N\not\in\cO_{F+}$ for any integer $N>1$.
In what follows, we let $A\subset\cO_{F+}$ be the set of primitive elements of $\cO_{F+}$.
Then
\begin{enumerate}
\item $\varepsilon A = A$ for any $\varepsilon\in\Delta$.
\item The set $\frU\coloneqq\{U_\alpha\}_{\alpha\in A}$ gives an affine open covering of $U$.
\end{enumerate}
We note that for any simplicial cone $\sigma$ of dimension $m$, there exists a generator ${\boldsymbol{\alpha}}\in A^m$,
unique up to permutation of the components.
Let $q$ be an integer $\geq 0$.
For any ${\boldsymbol{\alpha}}=(\alpha_0,\ldots,\alpha_{q})\in A^{q+1}$,
we let $U_{\boldsymbol{\alpha}}\coloneqq U_{\alpha_0}\cap\cdots\cap U_{\alpha_{q}}$,
and we denote by $j_{{\boldsymbol{\alpha}}}\colon U_{\boldsymbol{\alpha}}\hookrightarrow U$
the inclusion. We let
\[
\sC^q(\frU,\sF)\coloneqq \prod_{{\boldsymbol{\alpha}}\in A^{q+1}}^\alt j_{{\boldsymbol{\alpha}}*}j_{\boldsymbol{\alpha}}^*\sF
\]
be the subsheaf of
$ \prod_{{\boldsymbol{\alpha}}\in A^{q+1}} j_{{\boldsymbol{\alpha}}*}j_{\boldsymbol{\alpha}}^*\sF$ consisting of sections
$\bss=(s_{\boldsymbol{\alpha}})$ such that $s_{\rho({\boldsymbol{\alpha}})} = \sgn(\rho)s_{\boldsymbol{\alpha}}$ for any $\rho\in\frS_{q+1}$
and $s_{\boldsymbol{\alpha}}=0$ if $\alpha_i=\alpha_j$ for some $i\neq j$.
We define the differential
$
d^q\colon\sC^q(\frU,\sF)\rightarrow\sC^{q+1}(\frU,\sF)
$
to be the usual alternating sum
\begin{equation}\label{eq: Cech differential}
(d^qf)_{\alpha_0\cdots\alpha_{q+1}}\coloneqq\sum_{j=0}^{q+1}(-1)^j
f_{\alpha_0\cdots\check\alpha_{j}\cdots\alpha_{q+1}}\big|_{U_{(\alpha_0,\ldots,\alpha_{q+1})}\cap V}
\end{equation}
for any section $(f_{\boldsymbol{\alpha}})$ of $\sC^q(\frU,\sF)$ of each open set $V\subset U$.
If we let $\sF\hookrightarrow \sC^0(\frU,\sF)$ be the natural inclusion, then we have the exact sequence
\[\xymatrix{
0\ar[r]& \sF\ar[r]& \sC^0(\frU,\sF)\ar[r]^{d^0}&\sC^1(\frU,\sF)\ar[r]^{\quad d^1}&\cdots \ar[r]^{d^{q-1}\quad}&\sC^q(\frU,\sF)\ar[r]^{\quad d^q}&\cdots.
}\]
We next consider the action of $\Delta$.
For any ${\boldsymbol{\alpha}}\in A^{q+1}$ and $\varepsilon\in\Delta$, we have a commutative diagram
\[
\xymatrix{
U_{\varepsilon{\boldsymbol{\alpha}}}\ar@{^{(}->}[r]^{j_{\varepsilon{\boldsymbol{\alpha}}}}\ar[d]_{[\varepsilon]}^\cong&U\ar[d]^\cong_{[\varepsilon]}\\
U_{\boldsymbol{\alpha}}\ar@{^{(}->}[r]^{j_{\boldsymbol{\alpha}}} & U,
}
\]
where $\varepsilon{\boldsymbol{\alpha}}\coloneqq(\varepsilon\alpha_0,\ldots,\varepsilon\alpha_{q})$.
Then we have an isomorphism
\[
[\varepsilon]^* j_{{\boldsymbol{\alpha}}*}j_{\boldsymbol{\alpha}}^*\sF
\cong j_{\varepsilon{\boldsymbol{\alpha}}*}j_{\varepsilon{\boldsymbol{\alpha}}}^*[\varepsilon]^*\sF
\xrightarrow\cong j_{\varepsilon{\boldsymbol{\alpha}}*}j_{\varepsilon{\boldsymbol{\alpha}}}^*\sF,
\]
where the last isomorphism is induced by
the $\Delta$-equivariant structure $\iota_\varepsilon\colon[\varepsilon]^*\sF\cong\sF$.
This induces an isomorphism
$
\iota_\varepsilon\colon[\varepsilon]^*\sC^q(\frU,\sF)\xrightarrow\cong\sC^q(\frU,\sF),
$
which is compatible with the differential \eqref{eq: Cech differential}. Hence
$\sC^\bullet(A,\sF)$ is a complex of $\Delta$-equivariant sheaves on $U$.
\begin{proposition}\label{proposition: acyclic}
The sheaf $\sC^q(\frU,\sF)$ is acyclic with respect to the functor $\Gamma(U/\Delta, -)$.
\end{proposition}
\begin{proof}
By definition, the functor $\Gamma(U/\Delta, -)$ is the composite of the functors
$\Gamma(U,-)$ and ${\Hecke}om_{\bbZ[\Delta]}(\bbZ,-)$.
Standard facts concerning the composition of functors shows that we have a spectral sequence
\[
E_2^{a,b}=H^a\bigl(\Delta, H^b(U, \sC^q(\frU,\sF))\bigr)\Rightarrow H^{a+b}(U/\Delta, \sC^q(\frU,\sF)).
\]
We first prove that $H^b(U, \sC^q(\frU,\sF))=0$ if $b\neq0$.
If we fix some total order on the set $A$, then we have
\[
\sC^q(\frU,\sF)\cong\prod_{\alpha_0<\cdots<\alpha_q} j_{{\boldsymbol{\alpha}}*}j^*_{\boldsymbol{\alpha}}\sF,
\]
and each component $j_{{\boldsymbol{\alpha}}*}j^*_{\boldsymbol{\alpha}}\sF$ is acyclic for the functor $\Gamma(U,-)$
since $U_{\boldsymbol{\alpha}}$ is affine.
Therefore $\sC^q(\frU,\sF)$ is acyclic by Lemma \ref{lem: 3.5} below.
It is now sufficient to prove that $H^a\bigl(\Delta, H^0(U, \sC^q(\frU,\sF))\bigr)=0$ for any integer $a\neq0$, where
\[
H^0(U, \sC^q(\frU,\sF)) = \prod_{{\boldsymbol{\alpha}}\in A^{q+1}}^\alt \Gamma(U,j_{{\boldsymbol{\alpha}}*}j_{\boldsymbol{\alpha}}^*\sF)
\cong\prod_{\alpha_0<\cdots<\alpha_q} \Gamma(U_{\boldsymbol{\alpha}},\sF).
\]
Assume that the total order on $A$ is preserved by the action of $\Delta$
(for example, we may take the order on $\bbR$ through an embedding $\tau\colon A\hookrightarrow\bbR$
for some $\tau\in I$).
Let $B$ be the subset of $A^{q+1}$ consisting of elements ${\boldsymbol{\alpha}} = (\alpha_0,\ldots,\alpha_q)$
such that $\alpha_0<\cdots<\alpha_q$.
Then action of $\Delta$ on $B$ is free.
We denote by $B_0$ a subset of $B$ representing the set
$\Delta\backslash B$,
so that any ${\boldsymbol{\alpha}}\in B$ may be written uniquely as ${\boldsymbol{\alpha}}=\varepsilon{\boldsymbol{\alpha}}_0$ for some
$\varepsilon\in\Delta$ and ${\boldsymbol{\alpha}}_0\in B_0$. We let
\[
M\coloneqq\prod_{{\boldsymbol{\alpha}}\in B_0} \Gamma(U_{{\boldsymbol{\alpha}}}, \sF),
\]
and we let
$
{\Hecke}om_\bbZ(\bbZ[\Delta], M)
$
be the coinduced module of $M$,
with the action of $\Delta$ given for any $\varphi\in{\Hecke}om_\bbZ(\bbZ[\Delta], M)$
by $\varepsilon\varphi(u)=\varphi(u\varepsilon)$ for any $u\in\bbZ[\Delta]$ and $\varepsilon\in\Delta$.
Then we have a $\bbZ[\Delta]$-linear isomorphism
\begin{equation}\label{eq: isomorphism vulcan}
H^0(U, \sC^q(\frU,\sF)) \xrightarrow\cong {\Hecke}om_\bbZ(\bbZ[\Delta], M)
\end{equation}
given by mapping any $(s_{\boldsymbol{\alpha}})\in H^0(U, \sC^q(A,\sF))$ to the $\bbZ$-linear homomorphism
\[
\varphi_{(s_{\boldsymbol{\alpha}})}(\delta)\coloneqq\Bigl(\iota_\delta\bigl([\delta]^*s_{\delta^{-1}{\boldsymbol{\alpha}}_0}\bigr)\Bigr)\in M
\]
for any $\delta\in\Delta$.
The compatibility of \eqref{eq: isomorphism vulcan} with the action of $\Delta$ is seen as follows.
By definition, the action of $\varepsilon\in\Delta$ on $(s_{\boldsymbol{\alpha}})\in H^0(U,\sC^q(A,\sF))$ is given by
$\varepsilon\bigl((s_{\boldsymbol{\alpha}})\bigr)
=\bigl(\iota_\varepsilon([\varepsilon]^* s_{\varepsilon^{-1}{\boldsymbol{\alpha}}})\bigr)$.
Hence noting that
\[
\iota_\delta\circ[\delta]^*\iota_\varepsilon= \iota_{\delta\varepsilon}\colon\Gamma(U_{\boldsymbol{\alpha}},[\delta\varepsilon]^*\sF)
\rightarrow\Gamma(U_{\boldsymbol{\alpha}},\sF)
\]
and
$
[\delta]^* \circ \iota_\varepsilon= [\delta]^*\iota_\varepsilon \circ [\delta]^*\colon\Gamma(U_{\boldsymbol{\alpha}},[\varepsilon]^*\sF)
\rightarrow\Gamma(U_{\boldsymbol{\alpha}},[\delta]^*\sF)
$
for any $\delta\in\Delta$, we have
\begin{align*}
\varphi_{\varepsilon(s_{\boldsymbol{\alpha}})}(\delta)&=
\Bigl(\iota_\delta\bigl([\delta]^*(\iota_\varepsilon([\varepsilon]^*s_{\varepsilon^{-1}\delta^{-1}{\boldsymbol{\alpha}}_0}))\bigr)\Bigr)
=\Bigl((\iota_\delta\circ[\delta]^*\iota_\varepsilon)\bigl([\delta\varepsilon]^*s_{\varepsilon^{-1}\delta^{-1}{\boldsymbol{\alpha}}_0}\bigr)\Bigr)\\
&=\bigl(\iota_{\delta\varepsilon}([\delta\varepsilon]^*s_{\varepsilon^{-1}\delta^{-1}{\boldsymbol{\alpha}}_0})\bigr)
=\varphi_{(s_{\boldsymbol{\alpha}})}(\delta\varepsilon)
\end{align*}
as desired. The fact that \eqref{eq: isomorphism vulcan} is an isomorphism
follows from the fact that $B_0$ is a representative of $\Delta\backslash B$.
By \eqref{eq: isomorphism vulcan} and Shapiro's lemma, we have $H^a(\Delta, H^0(U,\sC^q(A,\sF)))
\cong H^a(\{1\},M)=0$ for $a\neq0$ as desired.
\end{proof}
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The following Lemma \ref{lem: 3.5} was used in the proof of Proposition \ref{proposition: acyclic}.
\begin{lemma}\label{lem: 3.5}
Let $I$ be a scheme and let $(\sF_\lambda)_{\lambda\in\Lambda}$ be a family of quasi-coherent sheaves on $I$.
Then for any integer $m\geq0$, we have
\[
H^m\biggl(I, \prod_{\lambda\in\Lambda}\sF_\lambda\biggr) \cong\prod_{\lambda\in\Lambda}H^m(I, \sF_\lambda).
\]
\end{lemma}
\begin{proof}
Take an injective resolution $0\rightarrow\sF_\lambda\rightarrow I^\bullet_\lambda$ for each $\lambda\in\Lambda$.
We will prove that $0\rightarrow\prod_{\lambda\in\Lambda}\sF_\lambda\rightarrow \prod_{\lambda\in\Lambda}I^\bullet_\lambda$
is an injective resolution.
Since the product of injective objects is injective, it is sufficient to prove that $0 \rightarrow
\prod_{\lambda\in\Lambda}\sF_\lambda\rightarrow \prod_{\lambda\in\Lambda}I^\bullet_\lambda$ is exact.
For any affine open set $V$ of $I$, by affine vanishing, the global section
$0 \rightarrow \sF_\lambda(V) \rightarrow I^\bullet_\lambda(V)$ is exact, hence the product
\begin{equation}\label{eq: orange}
0 \rightarrow\prod_{\lambda\in\Lambda}\sF_\lambda(V) \rightarrow
\prod_{\lambda\in\Lambda} I^\bullet_\lambda(V)
\end{equation}
is also exact.
For any $x\in I$, if we take the direct limit of \eqref{eq: orange} with respect to open affine neighborhoods of $x$,
then we obtain the exact sequence
\[
0 \rightarrow
\Biggl(\prod_{\lambda\in\Lambda}\sF_\lambda\Biggr)_x\rightarrow
\Biggl( \prod_{\lambda\in\Lambda}I^\bullet_\lambda\Biggr)_x.
\]
This shows that
$0 \rightarrow
\prod_{\lambda\in\Lambda}\sF_\lambda\rightarrow \prod_{\lambda\in\Lambda}I^\bullet_\lambda$ is exact
as desired.
\end{proof}
Proposition \ref{proposition: acyclic} gives the following Corollary.
\begin{corollary}\label{cor: description}
We let $C^\bullet(\frU/\Delta,\sF)\coloneqq\Gamma(U/\Delta,\sC^\bullet(\frU,\sF))$.
Then for any integer $m\geq0$, the equivariant cohomology $H^m(U/\Delta,\sF)$
is given as
\[
H^m(U/\Delta,\sF) = H^m(C^\bullet(\frU/\Delta,\sF)).
\]
\end{corollary}
By definition, for any integer $q\in\bbZ$, we have
\[
C^q(\frU/\Delta,\sF)= \biggl(\prod^\alt_{{\boldsymbol{\alpha}}\in A^{q+1}}\Gamma(U_{\boldsymbol{\alpha}},\sF)\biggr)^\Delta.
\]
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0.125.7
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\section{Shintani Generating Class}\label{section: Shintani Class}
We let $\bbT$ be the algebraic torus of \eqref{eq: algebraic torus}, and let $U=\bbT\setminus\{1\}$.
In this section, we will use the description of equivariant cohomology of Corollary \ref{cor: description}
to define the Shintani generating class as a class in $H^{g-1}(U/\Delta,\sO_\bbT)$.
We will then consider the action of the differential operators $\partial_\tau$ on this class.
We first interpret the generating functions $\sG_\sigma(z)$ of Definition \ref{def: generating}
as rational functions on $\bbT$.
Let $\frD^{-1}\coloneqq\{u\in F\mid\Tr_{F/\bbQ}(u\cO_F)\subset\bbZ\}$ be the inverse different of $F$.
Then there exists an isomorphism
\begin{equation}\label{eq: uniformization}
(F\otimes\bbC)/\frD^{-1} \xrightarrow\cong\bbT(\bbC) = {\Hecke}om_\bbZ(\cO_F,\bbC^\times), \qquad z \mapsto \xi_z
\end{equation}
given by mapping any $z\in F\otimes\bbC$ to the character $\xi_z(\alpha)\coloneqq e^{2\pi i \Tr(\alpha z)}$ in
${\Hecke}om_\bbZ(\cO_F,\bbC^\times)$.
The function $t^\alpha$ on $\bbT(\bbC)$ corresponds through the isomorphism \eqref{eq: uniformization} to the function
$
e^{2\pi i \Tr(\alpha z)}
$
for any $\alpha\in\cO_F$.
Thus the following holds.
\begin{lemma}\label{lem: correspondence}
For ${\boldsymbol{\alpha}}=(\alpha_1,\ldots,\alpha_g)\in A^g$ and $\sigma\coloneqq\sigma_{\boldsymbol{\alpha}}$,
consider the rational function
\[
\cG_\sigma(t)\coloneqq\frac{\sum_{\alpha\in \wh P_{\boldsymbol{\alpha}}\cap\cO_F}t^\alpha}{{\bigl(1-t^{\alpha_1}\bigr)\cdots
\bigl(1-t^{\alpha_g}\bigr)}}
\]
on $\bbT$, where $P_{\boldsymbol{\alpha}}$ is again the parallelepiped spanned by $\alpha_1,\ldots,\alpha_g$.
Then $\cG_\sigma(t)$ corresponds to the function $\sG_{\sigma}(z)$ of
Definition \ref{def: generating} through the uniformization \eqref{eq: uniformization}.
Note that by definition, if we let $B\coloneqq\bbZ[t^{\alpha}\mid\alpha\in\cO_{F+}]$, then we have
\begin{equation}\label{eq: B}
\cG_\sigma(t)\in B_{\boldsymbol{\alpha}}\coloneqq
B\Bigl[\frac{1}{1-t^{\alpha_1}},\ldots,\frac{1}{1-t^{\alpha_g}}\Bigr].
\end{equation}
\end{lemma}
Again, we fix an ordering $I=\{\tau_1,\ldots,\tau_g\}$. For any ${\boldsymbol{\alpha}}=(\alpha_1,\ldots,\alpha_g)\in\cO_{F+}^g$,
let $\bigl(\alpha_j^{\tau_i}\bigr)$ be the matrix in $M_g(\bbR)$ whose $(i,j)$-component is $\alpha_j^{\tau_i}$.
We let $\sgn({\boldsymbol{\alpha}})\in\{0,\pm1\}$ be the signature of $\det\bigl(\alpha_j^{\tau_i}\bigr)$.
We define the \textit{Shintani generating class} $\cG$ as follows.
\begin{proposition}\label{prop: Shintani generating class}
For any ${\boldsymbol{\alpha}}=(\alpha_1,\ldots,\alpha_g)\in A^{g}$, we let
\[
\cG_{\boldsymbol{\alpha}}\coloneqq\sgn({\boldsymbol{\alpha}})\cG_{\sigma_{\boldsymbol{\alpha}}}(t)
\in\Gamma(U_{\boldsymbol{\alpha}},\sO_\bbT).
\]
Then we have
$
(\cG_{\boldsymbol{\alpha}}) \in C^{g-1}(A,\sO_\bbT).
$
Moreover, $(\cG_{\boldsymbol{\alpha}})$ satisfies the cocycle condition $d^{g-1}(\cG_{\boldsymbol{\alpha}})=0$, hence defines a class
\[
\cG\coloneqq [\cG_{\boldsymbol{\alpha}}]\in H^{g-1}(U/\Delta,\sO_\bbT).
\]
We call this class the Shintani generating class.
\end{proposition}
\begin{proof}
By construction, $(\cG_{\boldsymbol{\alpha}})$ defines
an element in $\Gamma(U, \sC^{g-1}(\frU,\sO_\bbT))=\prod^\alt_{{\boldsymbol{\alpha}}\in A^g} \Gamma(U_{\boldsymbol{\alpha}},\sO_\bbT)$.
Since $[\varepsilon]^*\cG_{\boldsymbol{\alpha}}=\cG_{\varepsilon{\boldsymbol{\alpha}}}$ for any $\varepsilon\in\Delta$,
we have
\[
(\cG_{\boldsymbol{\alpha}})\in \Gamma\bigl(U, \sC^{g-1}(\frU,\sO_\bbT)\bigr)^{\Delta}= C^{g-1}(\frU/\Delta,\sO_\bbT).
\]
To prove the cocycle condition $d^{g-1}(\cG_{\boldsymbol{\alpha}})=0$, it is sufficient to check that
\begin{equation}\label{eq: cocycle condition}
\sum_{j=0}^g(-1)^j\cG_{(\alpha_0,\ldots,\check\alpha_j,\ldots\alpha_g)}=0
\end{equation}
for any $\alpha_0,\ldots,\alpha_g\in A$.
By definition, the rational function $\cG_{\sigma_{\boldsymbol{\alpha}}}(t)$ maps to the formal power series
\[
\cG_{\sigma_{\boldsymbol{\alpha}}}(t) =\sum_{\alpha\in\wh\sigma_{\boldsymbol{\alpha}}\cap\cO_F}t^\alpha
\]
by taking the formal completion
$B_{\boldsymbol{\alpha}}\hookrightarrow\bbZ\llbracket t^{\alpha_1},\ldots,t^{\alpha_g}\rrbracket$, where $B_{\boldsymbol{\alpha}}$ is the ring defined in \eqref{eq: B}.
Since the map taking the formal completion is injective,
it is sufficient to check \eqref{eq: cocycle condition} for the associated formal power series.
By \cite{Yam10}*{Proposition 6.2}, we have
\[
\sum_{j=0}^g(-1)^j\sgn(\alpha_0,\ldots,\check\alpha_j,\ldots,\alpha_g)
\boldsymbol{1}_{\wh\sigma_{(\alpha_0,\ldots,\check\alpha_j,\ldots\alpha_g)}}\equiv0
\]
as a function on $\bbR_+^I$, where $\boldsymbol{1}_R$ is the characteristic function of $R\subset\bbR^I_+$ satisfying
$\boldsymbol{1}_R(x)=1$ if $x\in R$ and $\boldsymbol{1}_R(x)=0$ if $x\not\in R$.
Our assertion now follows by examining the formal power series expansion of
$\cG_{(\alpha_0,\ldots,\check\alpha_j,\ldots,\alpha_g)}$.
\end{proof}
We will next define differential operators $\partial_\tau$ for $\tau\in I$ on equivariant cohomology.
Since $t^\alpha = e^{2\pi i \Tr(\alpha z)}$ through \eqref{eq: uniformization} for any $\alpha\in\cO_F$, we have
\begin{equation}\label{eq: relation}
\frac{dt^\alpha}{t^\alpha} = \sum_{\tau\in I} 2\pi i \alpha^\tau dz_\tau.
\end{equation}
Let $\alpha_1,\ldots,\alpha_g$ be a basis of $\cO_F$.
For any $\tau\in I$, we let $\partial_\tau $ be the differential operator
\[
\partial_\tau \coloneqq\sum_{j=1}^g \alpha_j^\tau t^{\alpha_j}\frac{\partial}{\partial t^{\alpha_j}}.
\]
By \eqref{eq: relation}, we see that $\partial_\tau $ corresponds to the differential operator
$\frac{1}{2\pi i}\frac{\partial}{\partial z_\tau}$
through the uniformization \eqref{eq: uniformization}, and hence is independent of the choice of the basis $\alpha_1,\ldots,\alpha_g$.
By Theorem \ref{theorem: Shintani} and Lemma \ref{lem: correspondence}, we have the following result.
\begin{proposition}\label{prop: Shintani}
Let ${\boldsymbol{\alpha}}=(\alpha_1,\ldots,\alpha_g)\in A^g$ and $\sigma=\sigma_{\boldsymbol{\alpha}}$.
For any $\bsk=(k_\tau)\in\bbN^I$ and $\partial^\bsk\coloneqq\prod_{\tau\in I}\partial_\tau^{k_\tau}$,
we have
\[
\partial^\bsk\cG_\sigma(\xi) = \zeta_\sigma(\xi,-\bsk)
\]
for any torsion point $\xi\in U_{{\boldsymbol{\alpha}}}$.
\end{proposition}
The differential operator $\partial_\tau$ gives a morphism of abelian sheaves
\[
\partial_\tau\colon\sO_{\bbT_{F^\tau}}(\bsk)\rightarrow\sO_{\bbT_{F^\tau}}(\bsk-1_\tau)
\]
compatible with the action of $\Delta$ for any $\bsk\in\bbZ^I$,
where $\bbT_{F^\tau}\coloneqq\bbT\otimes F^\tau$
is the base change of $\bbT$ to $F^\tau$.
This induces a homomorphism
\[
\partial_\tau\colon H^m(U_{F^\tau}/\Delta,\sO_{\bbT_{F^\tau}}(\bsk))\rightarrow
H^m(U_{F^\tau}/\Delta,\sO_{\bbT_{F^\tau}}(\bsk-1_\tau))
\]
on equivariant cohomology.
\begin{lemma}\label{lem: differential}
Let $\wt F$ be the composite of $F^\tau$ for all $\tau\in I$.
The operators $\partial_\tau$ for $\tau\in I$, considered over $\wt F$, are commutative with each other.
Moreover, the composite
\[
\partial\coloneqq\prod_{\tau\in I}\partial_\tau\colon
\sO_{\bbT_{\wt F}} \rightarrow \sO_{\bbT_{\wt F}}(1,\ldots,1)=\sO_{\bbT_{\wt F}}
\]
is defined over $\bbQ$,
that is, it is a base change to $\wt F$ of a morphism of abelian sheaves
$
\partial\colon \sO_\bbT\rightarrow\sO_\bbT.
$
In particular, $\partial$ induces a homomorphism
\begin{equation}\label{eq: differential}
\partial\colon H^m(U/\Delta,\sO_\bbT) \rightarrow H^m(U/\Delta,\sO_\bbT).
\end{equation}
\end{lemma}
\begin{proof}
The commutativity is clear from the definition.
Since the group $\Gal(\wt F/\bbQ)$ permutes the operators $\partial_\tau$,
the operator $\partial$ is invariant under this action. This gives our assertion.
\end{proof}
Our main result, which we prove in \S \ref{section: specialization}, concerns the specialization of the classes
\begin{equation}\label{eq: main}
\partial^k\cG \in H^{g-1}\bigl(U/\Delta,\sO_\bbT\bigr)
\end{equation}
for $k\in\bbN$ at nontrivial torsion points of $\bbT$.
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\section{Specialization to Torsion Points}\label{section: specialization}
For any nontrivial torsion point $\xi$ of $\bbT$, let $\Delta_\xi\subset\Delta$ be the isotropic subgroup
of $\xi$. Then we may view $\xi\coloneqq\Spec\bbQ(\xi)$ as a $\Delta_\xi$-scheme with a trivial action of $\Delta_\xi$.
Then the natural inclusion $\xi\rightarrow U$ for $U\coloneqq\bbT\setminus\{1\}$
is compatible with the inclusion $\Delta_\xi\subset\Delta$,
hence the pullback \eqref{eq: pullback} induces the specialization map
\[
\xi^*\colon H^m(U/\Delta,\sO_\bbT)\rightarrow H^m(\xi/\Delta_\xi,\sO_\xi).
\]
The purpose of this section is to prove our main theorem, given as follows.
\begin{theorem}\label{theorem: main}
Let $\xi$ be a nontrivial torsion point of $\bbT$, and let $k$ be an integer $\geq0$.
If we let $\cG$ be the Shintani generating class defined in Proposition \ref{prop: Shintani generating class},
and if we let
$\partial^k\cG(\xi)\in H^{g-1}(\xi/\Delta_\xi,\sO_\xi)$ be image
by the specialization map $\xi^*$ of the class $\partial^k\cG$
defined in \eqref{eq: main}, then we have
\[
\partial^k\cG(\xi) = \cL(\xi\Delta,-k)
\]
through the isomorphism $H^{g-1}(\xi/\Delta_\xi,\sO_\xi)\cong\bbQ(\xi)$ given in Proposition \ref{prop: ecc} below.
\end{theorem}
We will prove Theorem \ref{theorem: main} at the end of this section.
The specialization map can be expressed explicitly in terms of
cocycles as follows.
We let $V_\alpha\coloneqq U_\alpha\cap \xi$ for any $\alpha\in A$.
Then $\frV\coloneqq\{V_\alpha\}_{\alpha\in A}$ is an affine open covering of $\xi$.
For any integer $q\geq 0$ and ${\boldsymbol{\alpha}}=(\alpha_0,\ldots,\alpha_{q})\in A^{q+1}$,
we let $V_{\boldsymbol{\alpha}}\coloneqq V_{\alpha_0}\cap\cdots\cap V_{\alpha_{q}}$ and
\begin{equation}\label{eq: alternating}
C^q\bigl(\frV/\Delta_\xi,\sO_{\xi}\bigr)\coloneqq
\biggl(\prod_{{\boldsymbol{\alpha}}\in A^{q+1}}^\alt\Gamma(V_{\boldsymbol{\alpha}},\sO_{\xi})\biggr)^{\Delta_\xi}.
\end{equation}
Here, note that $\Gamma(V_{\boldsymbol{\alpha}},\sO_\xi)=\bbQ(\xi)$ if $V_{\boldsymbol{\alpha}}\neq\emptyset$ and
$\Gamma(V_{\boldsymbol{\alpha}},\sO_\xi)=\{0\}$ otherwise.
The same argument as that of Corollary \ref{cor: description} shows that
we have
\begin{equation}\label{eq: similarly}
H^m(\xi/\Delta_\xi,\sO_\xi)\cong H^m\bigl(C^\bullet\bigl(\frV/\Delta_\xi,\sO_{\xi}\bigr)\bigr).
\end{equation}
We let $A_\xi$ be the subset of elements $\alpha\in A$ satisfying $\xi\in U_\alpha$.
This is equivalent to the condition that $\xi(\alpha)\neq 1$.
We will next prove in Lemma \ref{lemma: 5.2}
that the cochain complex $C^\bullet\bigl(\frV/\Delta_\xi,\sO_{\xi}\bigr)$ of \eqref{eq: alternating}
is isomorphic to the dual of the chain complex
$C_\bullet(A_\xi)$ defined as follows.
For any integer $q\geq0$, we let
\[
C_q(A_\xi)\coloneqq\bigoplus^\alt_{{\boldsymbol{\alpha}}\in A_\xi^{q+1}}\bbZ{\boldsymbol{\alpha}}
\]
be the quotient of $\bigoplus_{{\boldsymbol{\alpha}}\in A_\xi^{q+1}}\bbZ{\boldsymbol{\alpha}}$ by
the submodule generated by
\[
\{\rho({\boldsymbol{\alpha}})-\sgn(\rho){\boldsymbol{\alpha}}\mid {\boldsymbol{\alpha}}\in A_\xi^{q+1}, \rho\in\frS_{q+1}\}
\cup \{ {\boldsymbol{\alpha}}=(\alpha_0,\ldots,\alpha_q) \mid \text{$\alpha_i=\alpha_j$ for some $i\neq j$}\}.
\]
We denote by $\langle{\boldsymbol{\alpha}}\rangle$ the class represented by ${\boldsymbol{\alpha}}$ in $C_q(A_\xi)$.
We see that $C_q(A_\xi)$ has a natural action of $\Delta_\xi$ and is a free $\bbZ[\Delta_\xi]$-module.
In fact, a basis of $C_q(A_\xi)$ may be constructed in a similar way to the construction of $B_0$
in the proof of Proposition \ref{proposition: acyclic}.
Then $C_\bullet(A_\xi)$ is a complex of $\bbZ[\Delta_\xi]$-modules with respect to the standard differential operator
$d_q\colon C_q(A_\xi)\rightarrow C_{q-1}(A_\xi)$ given by
\[
d_q(\langle\alpha_0,\ldots,\alpha_{q}\rangle)\coloneqq\sum_{j=0}^{q}
(-1)^{j}\langle\alpha_0,\ldots,\check\alpha_{j},\ldots,\alpha_{q}\rangle
\]
for any ${\boldsymbol{\alpha}}=(\alpha_0,\ldots,\alpha_{q})\in A_\xi^{q+1}$.
If we let $d_0\colon C_0(A_\xi) \rightarrow \bbZ$ be the homomorphism defined
by $d_0(\langle \alpha\rangle)\coloneqq 1$ for any $\alpha\in A_\xi$, then
$C_\bullet(A_\xi)$ is a free resolution of $\bbZ$ with trivial $\Delta_\xi$-action.
We have the following.
\begin{lemma}\label{lemma: 5.2}
There exists a natural isomorphism of complexes
\[
C^\bullet(\frV/\Delta_\xi,\sO_{\xi})\xrightarrow\cong {\Hecke}om_{\Delta_\xi}(C_\bullet(A_\xi),\bbQ(\xi)).
\]
\end{lemma}
\begin{proof}
The natural isomorphism
\[
\prod_{{\boldsymbol{\alpha}}\in A^{q+1}}\Gamma(V_{\boldsymbol{\alpha}},\sO_\xi)=\prod_{{\boldsymbol{\alpha}}\in A^{q+1}_\xi}\bbQ(\xi)
\cong{\Hecke}om_\bbZ\left(\bigoplus_{{\boldsymbol{\alpha}}\in A_\xi^{q+1}}\bbZ{\boldsymbol{\alpha}},\bbQ(\xi)\right)
\]
induces an isomorphism between the submodules
\[
C^q(\frV/\Delta_\xi,\sO_\xi)=\left(\prod^\alt_{{\boldsymbol{\alpha}}\in A^{q+1}}\Gamma(V_{\boldsymbol{\alpha}},\sO_\xi) \right)^{\Delta_\xi}
\subset \prod_{{\boldsymbol{\alpha}}\in A^{q+1}}\Gamma(V_{\boldsymbol{\alpha}},\sO_\xi)
\]
and
\[
{\Hecke}om_{\Delta_\xi}(C_q(A_\xi),\bbQ(\xi))\subset{\Hecke}om_\bbZ
\left(\bigoplus_{{\boldsymbol{\alpha}}\in A_\xi^{q+1}}\bbZ{\boldsymbol{\alpha}},\bbQ(\xi)\right).
\]
Moreover, this isomorphism is compatible with the differential.
\end{proof}
We will next use a Shintani decomposition (see Definition \ref{def: Shintani}) to
construct a complex which is quasi-isomorphic to the complex $C_\bullet\bigl(A_\xi)$.
\begin{lemma}\label{lem: inclusion}
Let $\xi$ be as above.
There exists a Shintani decomposition $\Phi$ such that any $\sigma\in\Phi$
is of the form
$\sigma_{\boldsymbol{\alpha}}=\sigma$
for some ${\boldsymbol{\alpha}}\in A_\xi^{q+1}$.
\end{lemma}
\begin{proof}
Let $\Phi$ be a Shintani decomposition. We will deform $\Phi$ to construct a Shintani decomposition
satisfying our assertion.
Let $\Lambda$ be a finite subset of $A$ such that $\{\sigma_\alpha \mid \alpha\in \Lambda\}$
represents the quotient set $\Delta_\xi\backslash\Phi_1$.
If $\xi(\alpha)\neq 1$ for any $\alpha\in\Lambda$, then $\Phi$ satisfies our assertion since
$\alpha\in A_\xi$ if and only if $\xi(\alpha)\neq 1$.
Suppose that there exists $\alpha\in \Lambda$ such that $\xi(\alpha)=1$.
Since $\xi$ is a nontrivial character on $\cO_F$, there exists $\beta\in\cO_{F+}$ such that $\xi(\beta)\neq 1$.
Then for any integer $N$, we have $\xi(N\alpha+\beta)\neq 1$.
Let $\Phi'$ be the set of cones obtained by deforming $\sigma=\sigma_\alpha$ to $\sigma'\coloneqq\sigma_{N\alpha+\beta}$
and $\varepsilon\sigma$ to $\varepsilon\sigma'$ for any $\varepsilon\in\Delta_\xi$.
By taking $N$ sufficiently large, the amount of deformation can be made arbitrarily small so that
$\Phi'$ remains a fan.
By repeating this process, we obtain a Shintani decomposition satisfying the desired condition.
\end{proof}
In what follows, we fix a Shintani decomposition $\Phi$ satisfying the condition of Lemma \ref{lem: inclusion}.
Let $N\colon\bbR_+^I\rightarrow\bbR_+$ be the norm map defined by
$N((a_\tau))\coloneqq\prod_{\tau\in I}a_\tau$,
and we let
\[
\bbR_1^I\coloneqq\{(a_\tau)\in\bbR_+^I\mid N((a_\tau))=1\}
\]
be the subset of $\bbR^I_+$ of norm one.
For any $\sigma\in\Phi_{q+1}$, the intersection $\sigma\cap\bbR_1^I$
is a subset of $\bbR_1^I$ which is homeomorphic to a simplex of dimension $q$,
and the set $\{ \sigma\cap \bbR_1^I \mid \sigma\in\Phi_+\}$
for $\Phi_+\coloneqq\bigcup_{q\geq0}\Phi_{q+1}$
gives a simplicial decomposition of the topological
space $\bbR_1^I$.
In what follows, for any $\sigma\in\Phi_{q+1}$, we denote by $\langle\sigma\rangle$ the class $\sgn({\boldsymbol{\alpha}})\langle{\boldsymbol{\alpha}}\rangle$ in
$C_q(A_\xi)$, where ${\boldsymbol{\alpha}}\in A_\xi^{q+1}$ is a generator of $\sigma$. Recall that such a
generator ${\boldsymbol{\alpha}}$ is uniquely determined up to permutation from $\sigma$.
We then have the following.
\begin{lemma}\label{lemma: 5.4}
For any integer $q\geq0$, we let $C_q(\Phi)$ be the $\bbZ[\Delta_\xi]$-submodule of $C_q(A_\xi)$
generated by $\langle\sigma\rangle$ for all $\sigma\in\Phi_{q+1}$.
Then $C_\bullet(\Phi)$ is a subcomplex of $C_\bullet(A_\xi)$
which also gives a free resolution of $\bbZ$ as a $\bbZ[\Delta_\xi]$-module.
In particular, the natural inclusion induces a quasi-isomorphism of complexes
\[
C_\bullet(\Phi)\xrightarrow{\qis} C_\bullet(A_\xi)
\]
compatible with the action of $\Delta_\xi$.
\end{lemma}
\begin{proof}
Note that $C_q(\Phi)$ for any integer $q\geq0$
is a free $\bbZ[\Delta_\xi]$-module
generated by representatives of the quotient $\Delta_\xi\backslash \Phi_{q+1}$.
By construction, $C_\bullet(\Phi)$ can be identified with the chain complex associated to
the simplicial decomposition $\{\sigma\cap\bbR^I_1\mid \sigma\in\Phi_+\}$
of the topological space $\bbR^I_1$, hence we see that the complex $C_\bullet(\Phi)$ is exact and gives a free resolution
of $\bbZ$ as a $\bbZ[\Delta_\xi]$-module. Our assertion follows from the fact that $C_\bullet(A_\xi)$ also gives a free
resolution of $\bbZ$ as a $\bbZ[\Delta_\xi]$-module.
\end{proof}
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We will next use a Shintani decomposition (see Definition \ref{def: Shintani}) to
construct a complex which is quasi-isomorphic to the complex $C_\bullet\bigl(A_\xi)$.
\begin{lemma}\label{lem: inclusion}
Let $\xi$ be as above.
There exists a Shintani decomposition $\Phi$ such that any $\sigma\in\Phi$
is of the form
$\sigma_{\boldsymbol{\alpha}}=\sigma$
for some ${\boldsymbol{\alpha}}\in A_\xi^{q+1}$.
\end{lemma}
\begin{proof}
Let $\Phi$ be a Shintani decomposition. We will deform $\Phi$ to construct a Shintani decomposition
satisfying our assertion.
Let $\Lambda$ be a finite subset of $A$ such that $\{\sigma_\alpha \mid \alpha\in \Lambda\}$
represents the quotient set $\Delta_\xi\backslash\Phi_1$.
If $\xi(\alpha)\neq 1$ for any $\alpha\in\Lambda$, then $\Phi$ satisfies our assertion since
$\alpha\in A_\xi$ if and only if $\xi(\alpha)\neq 1$.
Suppose that there exists $\alpha\in \Lambda$ such that $\xi(\alpha)=1$.
Since $\xi$ is a nontrivial character on $\cO_F$, there exists $\beta\in\cO_{F+}$ such that $\xi(\beta)\neq 1$.
Then for any integer $N$, we have $\xi(N\alpha+\beta)\neq 1$.
Let $\Phi'$ be the set of cones obtained by deforming $\sigma=\sigma_\alpha$ to $\sigma'\coloneqq\sigma_{N\alpha+\beta}$
and $\varepsilon\sigma$ to $\varepsilon\sigma'$ for any $\varepsilon\in\Delta_\xi$.
By taking $N$ sufficiently large, the amount of deformation can be made arbitrarily small so that
$\Phi'$ remains a fan.
By repeating this process, we obtain a Shintani decomposition satisfying the desired condition.
\end{proof}
In what follows, we fix a Shintani decomposition $\Phi$ satisfying the condition of Lemma \ref{lem: inclusion}.
Let $N\colon\bbR_+^I\rightarrow\bbR_+$ be the norm map defined by
$N((a_\tau))\coloneqq\prod_{\tau\in I}a_\tau$,
and we let
\[
\bbR_1^I\coloneqq\{(a_\tau)\in\bbR_+^I\mid N((a_\tau))=1\}
\]
be the subset of $\bbR^I_+$ of norm one.
For any $\sigma\in\Phi_{q+1}$, the intersection $\sigma\cap\bbR_1^I$
is a subset of $\bbR_1^I$ which is homeomorphic to a simplex of dimension $q$,
and the set $\{ \sigma\cap \bbR_1^I \mid \sigma\in\Phi_+\}$
for $\Phi_+\coloneqq\bigcup_{q\geq0}\Phi_{q+1}$
gives a simplicial decomposition of the topological
space $\bbR_1^I$.
In what follows, for any $\sigma\in\Phi_{q+1}$, we denote by $\langle\sigma\rangle$ the class $\sgn({\boldsymbol{\alpha}})\langle{\boldsymbol{\alpha}}\rangle$ in
$C_q(A_\xi)$, where ${\boldsymbol{\alpha}}\in A_\xi^{q+1}$ is a generator of $\sigma$. Recall that such a
generator ${\boldsymbol{\alpha}}$ is uniquely determined up to permutation from $\sigma$.
We then have the following.
\begin{lemma}\label{lemma: 5.4}
For any integer $q\geq0$, we let $C_q(\Phi)$ be the $\bbZ[\Delta_\xi]$-submodule of $C_q(A_\xi)$
generated by $\langle\sigma\rangle$ for all $\sigma\in\Phi_{q+1}$.
Then $C_\bullet(\Phi)$ is a subcomplex of $C_\bullet(A_\xi)$
which also gives a free resolution of $\bbZ$ as a $\bbZ[\Delta_\xi]$-module.
In particular, the natural inclusion induces a quasi-isomorphism of complexes
\[
C_\bullet(\Phi)\xrightarrow{\qis} C_\bullet(A_\xi)
\]
compatible with the action of $\Delta_\xi$.
\end{lemma}
\begin{proof}
Note that $C_q(\Phi)$ for any integer $q\geq0$
is a free $\bbZ[\Delta_\xi]$-module
generated by representatives of the quotient $\Delta_\xi\backslash \Phi_{q+1}$.
By construction, $C_\bullet(\Phi)$ can be identified with the chain complex associated to
the simplicial decomposition $\{\sigma\cap\bbR^I_1\mid \sigma\in\Phi_+\}$
of the topological space $\bbR^I_1$, hence we see that the complex $C_\bullet(\Phi)$ is exact and gives a free resolution
of $\bbZ$ as a $\bbZ[\Delta_\xi]$-module. Our assertion follows from the fact that $C_\bullet(A_\xi)$ also gives a free
resolution of $\bbZ$ as a $\bbZ[\Delta_\xi]$-module.
\end{proof}
We again fix a numbering of elements in $I$ so that $I=\{\tau_1,\ldots,\tau_g\}$.
We let
\[
L\colon\bbR_+^I\rightarrow\bbR^g
\]
be the homeomorphism defined by $(x_\tau)\mapsto(\log x_{\tau_i})$. If we let $W\coloneqq\{ (y_{\tau_i})\in\bbR^g
\mid\sum_{i=1}^gy_{\tau_i}=0\}$, then $W$ is an $\bbR$-linear subspace of $\bbR^g$ of dimension $g-1$, and $L$ gives a
homeomorphism $\bbR_1^I\cong W\cong \bbR^{g-1}$.
For $\Delta_\xi\subset F$, the Dirichlet unit theorem (see for example \cite{Sam70}*{Theorem 1 p.61})
shows that the discrete subset $L(\Delta_\xi)\subset W$ is a free $\bbZ$-module of rank $g-1$,
hence we have
\[
\cT_\xi\coloneqq\Delta_\xi\backslash\bbR_1^I\cong \bbR^{g-1}/\bbZ^{g-1}.
\]
We consider the coinvariant
\[
C_q(\Delta_\xi\backslash\Phi)\coloneqq C_q(\Phi)_{\Delta_\xi}
\]
of $C_q(\Phi)$ with respect to the action of $\Delta_\xi$,
that is, the quotient of $C_q(\Phi)$ by the subgroup generated by $\langle\sigma\rangle-\langle\varepsilon\sigma\rangle$
for $\sigma\in\Phi_{q+1}$ and $\varepsilon\in\Delta_\xi$.
For any $\sigma\in\Phi_{q+1}$, we denote by $\ol\sigma$ the image of $\sigma$ in the quotient
$\Delta_\xi\backslash\Phi_{q+1}$, and we denote by $\langle\ol\sigma\rangle$
the image of $\langle\sigma\rangle$
in $C_q(\Delta_\xi\backslash\Phi)$,
which depends only on the class $\ol\sigma\in
\Delta_\xi\backslash\Phi_{q+1}$.
Then the set $\{ \Delta_\xi \backslash(\Delta_\xi\sigma\cap\bbR^I_1)
\mid \ol\sigma\in \Delta_\xi\backslash\Phi_+\}$ of subsets of $\cT_\xi$
gives a simplicial decomposition of $\cT_\xi$ and
$C_\bullet(\Delta_\xi\backslash \Phi)$
may be identified with the associated chain complex.
Hence we have
\begin{align*}
H_m(C_\bullet(\Delta_\xi\backslash\Phi))&= H_m(\cT_\xi,\bbZ),&
H^m\Bigl({\Hecke}om_\bbZ\bigl(C_\bullet(\Delta_{\xi}\backslash\Phi), \bbZ\bigr)\Bigr)&= H^m(\cT_\xi,\bbZ).
\end{align*}
Since $\cT_\xi\cong\bbR^{g-1}/\bbZ^{g-1}$,
the homology groups $H_m(\cT_\xi,\bbZ)$ for integers $m$ are free abelian groups,
and the pairing
\begin{equation}\label{eq: pairing}
H_m(\cT_\xi,\bbZ) \times H^m(\cT_\xi,\bbZ) \rightarrow \bbZ,
\end{equation}
obtained by associating to a cycle
$u\in C_m(\Delta_\xi\backslash\Phi)$ and a cocycle $\varphi \in {\Hecke}om_\bbZ\bigl(C_m(\Delta_\xi\backslash\Phi), \bbZ\bigr)$
the element $\varphi(u)\in\bbZ$, is perfect
(see for example \cite{Mun84}*{Theorem 45.8}).
The generator of the cohomology group
\[
H_{g-1}(\cT_\xi,\bbZ) = H_{g-1}\bigl(C_\bullet(\Delta_\xi\backslash\Phi)\bigr) \cong \bbZ
\]
is given by the fundamental class
\begin{equation}\label{eq: fundamental class}
\sum_{\ol\sigma\in\Delta_\xi\backslash\Phi_{g}} \langle\ol\sigma\rangle
\in C_{g-1}(\Delta_\xi\backslash\Phi),
\end{equation}
and the canonical isomorphism
\begin{equation}\label{eq: isom}
H^{g-1}(\cT_\xi,\bbQ(\xi)) = H^{g-1}\Bigl({\Hecke}om_\bbZ\bigl(C_\bullet(\Delta_\xi\backslash\Phi), \bbQ(\xi)\bigr)\Bigr) \cong \bbQ(\xi)
\end{equation}
induced by the fundamental class \eqref{eq: fundamental class} via the pairing \eqref{eq: pairing}
is given explicitly in terms of cocycles
by mapping any $\varphi\in {\Hecke}om_\bbZ(C_{g-1}(\Delta_\xi\backslash\Phi), \bbQ(\xi))$ to the element
$\sum_{\ol\sigma\in\Delta_\xi\backslash\Phi_g}\varphi(\langle\ol\sigma\rangle)\in \bbQ(\xi)$.
\begin{proposition}\label{prop: ecc}
Let $\eta\in H^{g-1}(\xi/\Delta_\xi,\sO_\xi)$ be represented by a cocycle
\[
(\eta_{\boldsymbol{\alpha}})\in C^{g-1}(\frV/\Delta_\xi,\sO_\xi)
=\biggl(\prod_{{\boldsymbol{\alpha}}\in A_\xi^{g}}^\alt
\bbQ(\xi)\biggr)^{\Delta_\xi}.
\]
For any cone $\sigma\in\Phi_g$, let
$\eta_\sigma\coloneqq \sgn({\boldsymbol{\alpha}})\eta_{{\boldsymbol{\alpha}}}$ for any ${\boldsymbol{\alpha}}\in A^g_\xi$ such that $\sigma_{\boldsymbol{\alpha}}=\sigma$.
Then the homomorphism mapping the cocycle $(\eta_{\boldsymbol{\alpha}})$ to
$
\sum_{\ol\sigma\in\Delta_\xi\backslash\Phi_{g}} \eta_{\sigma}
$
induces a canonical isomorphism
\begin{equation}\label{eq: isom main}
H^{g-1}(\xi/\Delta_\xi,\sO_\xi)\cong\bbQ(\xi).
\end{equation}
\end{proposition}
\begin{proof}
Since $C_q(\Phi)$ and $C_q(A_\xi)$ are free $\bbZ[\Delta_\xi]$-modules,
the quasi-isomorphism $C_\bullet(\Phi)\xrightarrow\qis C_\bullet(A_\xi)$ of Lemma \ref{lemma: 5.4} induces the
quasi-isomorphism
\[
{\Hecke}om_{\Delta_\xi}\bigl(C_\bullet(A_\xi),\bbQ(\xi)\bigr)\xrightarrow\qis {\Hecke}om_{\Delta_\xi}\bigl(C_\bullet(\Phi),\bbQ(\xi)\bigr).
\]
Combining this fact with Lemma \ref{lemma: 5.2} and \eqref{eq: similarly}, we see that
\[
H^{g-1}(\xi/\Delta_\xi,\sO_\xi)\cong H^{g-1}\bigl({\Hecke}om_{\Delta_\xi}\bigl(C_\bullet(\Phi),\bbQ(\xi)\bigr)\bigr).
\]
Since we have
$
{\Hecke}om_{\Delta_\xi}\bigl(C_\bullet(\Phi),\bbQ(\xi)\bigr)
= {\Hecke}om_{\bbZ}\bigl(C_\bullet(\Delta_\xi\backslash\Phi),\bbQ(\xi)\bigr),
$
our assertion follows from \eqref{eq: isom}.
\end{proof}
We will now prove Theorem \ref{theorem: main}.
\begin{proof}[Proof of Theorem \ref{theorem: main}]
By construction
and Lemma \ref{lem: differential}, the class $\partial^k\cG(\xi)$ is a class defined over $\bbQ(\xi)$ represented by the cocycle
$
(\partial^k\cG_{\boldsymbol{\alpha}}(\xi)) \in C^{g-1}(\frV/\Delta_\xi,\sO_{\xi}).
$
By Proposition \ref{prop: ecc} and Proposition \ref{prop: Shintani}, the class $\partial^k\cG(\xi)$ maps
through\eqref{eq: isom main} to
\[
\sum_{\sigma\in\Delta_\xi\backslash\Phi_g} \partial^k\cG_\sigma(\xi)
=\sum_{\sigma\in\Delta_\xi\backslash\Phi_g} \zeta_{\sigma}(\xi,(-k,\ldots,-k)).
\]
Our assertion now follows from \eqref{eq: Shintani}.
\end{proof}
\begin{corollary}
Assume that the narrow class number of $F$ is \textit{one},
and let $\chi\colon\Cl^+_F(\frf)\rightarrow\bbC^\times$ be a finite primitive Hecke character of $F$ of
conductor $\frf\neq(1)$.
If we let $U[\frf]\coloneqq\bbT[\frf]\setminus\{1\}$, then we have
\[
L(\chi, -k)=\sum_{\xi\in U[\frf]/\Delta}c_\chi(\xi)\partial^k\cG(\xi)
\]
for any integer $k\geq0$.
\end{corollary}
\begin{proof}
The result follows from Theorem \ref{theorem: main} and Proposition \ref{prop: Hecke}.
\end{proof}
The significance of this result is that
the special values at negative integers of \textit{any} finite Hecke character of $F$ may be expressed as a linear combination of special values of the derivatives of a single canonical cohomology class, the Shintani class $\cG$
in $H^{g-1}(U/\Delta,\sO_\bbT)$.
\begin{bibdiv}
\begin{biblist}
\bibselect{PolylogarithmBibliography}
\end{biblist}
\end{bibdiv}
\end{document}
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\begin{document}
\title{The Digital Twin Landscape at the Crossroads of Predictive Maintenance, Machine Learning and Physics Based Modeling}
\abstract{The concept of a digital twin has exploded in popularity over the past decade, yet confusion around its plurality of definitions, its novelty as a new technology, and its practical applicability still exists, all despite numerous reviews, surveys, and press releases. The history of the term digital twin is explored, as well as its initial context in the fields of product life cycle management, asset maintenance, and equipment fleet management, operations, and planning. A definition for a minimally viable framework to utilize a digital twin is also provided based on seven essential elements. A brief tour through DT applications and industries where DT methods are employed is also outlined. The application of a digital twin framework is highlighted in the field of predictive maintenance, and its extensions utilizing machine learning and physics based modeling. Employing the combination of machine learning and physics based modeling to form hybrid digital twin frameworks, may synergistically alleviate the shortcomings of each method when used in isolation. Key challenges of implementing digital twin models in practice are additionally discussed. As digital twin technology experiences rapid growth and as it matures, its great promise to substantially enhance tools and solutions for intelligent upkeep of complex equipment, are expected to materialize.
}
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\section{Introduction}
The concept of the {\em digital twin} (DT) has been increasingly mentioned over the last decade in both academic and industrial circles. The frequency of a web search topic of {\em digital twin} (includes similar search terms such as, {\em digital twins}, {\em digital twin definition}, {\em what is a digital twin}, etc.) has seen an approximately exponential rise in roughly the past decade (see Figure \ref{fig:DT_Trend_Analysis}). Publication of scholarly articles shows similar trends across several databases including Web of Science\texttrademark, Scopus\texttrademark, and Google Scholar\texttrademark. Yet, the definition of what a digital twin consists of has evolved since its initial introduction, for better or for worse, with some attaching various adjectives to broaden it, as well as others insisting on inclusion of tangential topics to stake novelty claims to the idea in specific technical fields.
This manuscript seeks to provide clarity on defining {\em digital twin}, through exploring the history of the term, its initial context in the fields of product life cycle management (PLM), asset maintenance, and fleet of equipment management, operations, and planning. A definition for a minimally viable digital twin framework is also provided to alleviate ambiguity of the DT. Furthermore, a brief tour through its applications across industries will be provided to clarify the context in which digital twins are used today. Thereafter, the application of digital twins in the fields of predictive maintenance, machine learning, and physics based modeling with a keen eye towards their intersections will be investigated. Finally, the challenges of implementing digital twins will be examined. The reader should gain a clear understanding of what is a digital twin, how is it applied, and what obstacles to its use lie ahead.
\begin{figure}
\caption{Trends of the query term {\em digital twin}
\label{fig:DT_Trend_Analysis}
\end{figure}
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\subsection{Defining Digital Twin and Related Terms}
\subsubsection{Digital Twin Definitions}
At first glance, the term {\em digital twin} conjures the idea of a simple model, a scaled replica or mathematical expression that is a representation of the physically real object or system, and that, certainly, is not a new concept. Over a century ago, Buckingham~\cite{buckingham1914physically} made use of dimensional analysis to lay the groundwork for engineers and physicists to use scaled physical models in place of the larger physical models they represented. These smaller {\em physical twins} of the larger physical object, such as a ship, an aircraft, a bridge or a building, allowed for testing of designs without having to recreate a full scale model, simply by utilizing equal dimensionless parameters invariant to scale, for example, the Reynolds number when analyzing fluid flow past the object. Decades later, with the advent of modern computers, these physical models were supplemented with computational (digital) models that were digitally drawn with computer aided design (CAD) while using governing equations that could be solved by discretizing the model design space into small volumes, elements, and/or nodes~\cite{Marinkovic2019FEM}. So is it relevant to ask if these design models also fall into the definition of a digital twin?
There are multiple definitions for the concept of DT which vary across different industries, and that can broadly encompass sophisticated computational physics-based models of parts, machine learning algorithms applied to recorded sensor data, CAD models, a repository for part and asset manufacturing and maintenance history, and/or scaled virtual reality environments (see Negri et al.~\cite{negri2017review} for a comprehensive list of different DT definitions); however the common theme connecting these definitions is that a virtual representation of a real physical system, machine, or product, spanning its entire life cycle, is created in order to track changes, provide traceability of parts and/or software, and typically connect embedded sensors and Internet of Things (IoT) devices with databases to document life cycle of the subject item. The term digital twin can also be synonymous with {\em digital database}; however, although potentially existing in a database, a digital twin can refer to the represented model or simulations (or their aggregate thereof) of the physical object or system. The first comprehensive definition of {\em digital twin} is often credited to NASA's 2012 {\em Modeling, Simulation, Information Technology \& Processing Roadmap} ~\cite{shafto2012modeling}, defined therein as:
\begin{quote}
A digital twin is an integrated multiphysics, multiscale simulation of a vehicle or system that uses the best available physical models, sensor updates, fleet history, etc., to mirror the life of its corresponding flying twin. The digital twin is ultra-realistic and may consider one or more important and interdependent vehicle systems, including propulsion/energy storage, avionics, life support, vehicle structure, thermal management/TPS [thermal protection system], etc. Manufacturing anomalies that may affect the vehicle may also be explicitly considered. In addition to the backbone of high-fidelity physical models, the digital twin integrates sensor data from the vehicle’s on-board integrated vehicle health management (IVHM) system, maintenance history and all available historical/fleet data obtained using data mining and text mining. By combining all of this information, the digital twin continuously forecasts the health of the vehicle/ system, the remaining useful life and the probability of mission success. The systems on board the digital twin are also capable of mitigating damage or degradation by recommending changes in mission profile to increase both the life span and the probability of mission success.~\cite{shafto2012modeling}
\end{quote}
Thus, in its initial definition, the digital twin concept was applied to the service life (planning, maintenance, and operation) of a complex asset, such as an aerospace or astronautical vehicle with thousands, if not millions, of individual parts assembled into a dense physical web of interacting functional systems. A similar DT concept was also proposed by Tuegel et al. for an aircraft, specifically~\cite{tuegel2011reengineering}. Such complex assets are subject to multi-year environmental degradation, requiring sustained maintenance, part replacement, mission dependent equipment swaps, and operational planning including fleet management, service downtime scheduling, and part and personnel logistics. Also accompanying the complex asset is a wealth of data, generated from onboard sensors. This data is often generated in the context of feedback for various control logic applications, but also for condition monitoring, warning indicators, and system alarms. These maintenance requirements and the amount of diagnostic data available align directly with the goals of predictive maintenance (PMx, sometimes also abbreviated PdM): the promise that analyzing and interpreting asset data will allow anticipation of the need for corrective maintenance, its convenient scheduling, and preventing equipment failures~\cite{errandonea2020digital,carvalho2019systematic,miller_system-level_2020}.
Another early conceptual framework for digital twins is heavily based on the concept of product lifecycle management (PLM): a systematic approach to managing the series of changes a product goes through, from its design and development, to its ultimate retirement, generational redesign, or disposal~\cite{terzi2010product}. PLM can be visualized as two interwoven cycles of product development, on the one hand, and product service and maintenance, on the other; the former taking place before an asset is used in the real world design intent, and the latter referring to the asset's service support during applicable use (Figure \ref{fig:DT-PLM}). The digital twin can then be conceptualized as a seamless link connecting the interrelated cycles by providing a common database to store designs, models, simulations, algorithms, data, and information tracked over time and throughout each cycle. A claim is made from Grieves~\cite{grieves2016origins} that the following definition preceded the previously mentioned digital twin definition by about a decade under the term {\em Mirrored Spaces Model} which is closely tied with the idea of PLM:
\begin{quote}
The Digital Twin is a set of virtual information constructs that fully describes a potential or actual physical manufactured product from the micro atomic level to the macro geometrical level. The Digital Twin would be used for predicting future behavior and performance of the physical product. At the Prototype stage, the prediction would be of the behavior of the designed product with components that vary between its high and low tolerances in order to ascertain that the as-designed product met the proposed requirements. In the Instance stage, the prediction would be a specific instance of a specific physical product that incorporated actual components and component history. The predictive performance would be based from current point in the product's lifecycle at its current state and move forward. Multiple instances of the product could be aggregated to provide a range of possible future states. Digital Twin Instances could be interrogated for the current and past histories. Irrespective of where their physical counterpart resided in the world, individual instances could be interrogated for their current system state: fuel amount, throttle settings, geographical location, structure stress, or any other characteristic that was instrumented. Multiple instances of products would provide data that would be correlated for predicting future states. For example, correlating component sensor readings with subsequent failures of that component would result in an alert of possible component failure being generated when that sensor pattern was reported. The aggregate of actual failures could provide Bayesian probabilities for predictive uses. \cite{grieves2016origins}
\end{quote}
In this context, the DT concept is a common realization that most modern machines, systems, and products are generating data from network connected sensors and their design and development typically span multiple engineering domains, as they include mechanical and structural hardware, electronics, embedded software, network communication, and often more. A significant design change or even a part replacement with nonidentical tolerances made in any of the engineering domains will necessitate and propagate significant design changes or changes in performance in other domains. A digital twin provides the connection between prior and future design and operations, to alleviate many communication and logistical problems arising from the intersection of disparate engineering and technical fields, which may exist during a product’s or system’s life cycle. Furthermore, the digital twin’s ability to be updated in real-time or near real-time from embedded and peripheral sensors, allows for deeper analysis of the performance and maintenance of the physical twin asset.
Rosen et al.~\cite{rosen_about_2015} describe the digital twin concept as the future of a trend in modeling and simulations that has evolved from individual solutions of specific problems to model based systems engineering (MSBE) with multi-physics and multi scale modeling, and then finally to seamless digital twin modeling, linked to operational data throughout the entire life cycle of an asset. Kritzinger et al.~\cite{kritzinger2018digital} describe levels of automation that distinguish traditional models from digital twin models; digital models have manual data analysis from the physical object and optional manual decision making from the digital object; digital shadows have automated digital threads connecting physical object data to the digital object but still have manual information flow from the digital object to the physical object; only the digital twin has automated data and information flow in each direction via the digital thread to and from each of the physical and digital objects. A comprehensive review~\cite{Qamsane2020digital} reveals at least ten different definitions for digital twins. A survey of 150 engineers asking what they thought a DT was, revealed the most popular definition was "a virtual replica of the physical asset which can be used to monitor and evaluate its performance" and found that its most useful application was thought to be in the field of predictive maintenance~\cite{Hamer2018feasibility}.
The variety and expanding inclusion of concepts may appear to define DT as a concept equivalent to a traditional {\em simulation} or even {\em data analysis} of sensor data; however, the novelty, in comparison to the 20th century definitions of similar terms, lies in two areas: a) the development of large, connected streams or {\em threads} of sensor data that may be analyzed to improve the understanding of the current system or machine (also a defining feature of {\em Industry 4.0}~\cite{lasi2014industry});
and b) the integration of multiple models describing the form, function and behavior of the physical system and its components at various scales and using diverse modeling paradigms.
\begin{figure}
\caption{The digital twin product lifecycle management (DT-PLM) concept. The two cycles of product development and product service and maintenance are interwoven. The digital twin model provides a seamless link in connecting these interrelated cycles by providing a common database to store designs, models, simulations, test data, as-manufactured data, algorithms, usage data, and other available information tracked over time and throughout each cycle as the physical twin asset moves along the cycles.}
\label{fig:DT-PLM}
\end{figure}
\subsubsection{Digital Thread, Industry 4.0, Digital Engineering}
Although sometimes used interchangeably with {\em digital twin}, {\em digital thread} has a more specific meaning:
\begin{quote}
The digital thread refers to the communication framework that allows a connected data flow and integrated view of the asset’s data throughout its lifecycle across traditionally siloed functional perspectives. The digital thread concept raises the bar for delivering ``the right information to the right place at the right time'' \cite{leiva2016demystifying}
\end{quote}
Digital thread is often considered synonymous with the {\em connected supply chain} that ``collects data on all parts, kits, processes, machines and tools in real time and then stores that data, enabling digital twins and complete traceability.'' \cite{leiva2016demystifying} Digital thread is also closely related to the concept of {\em Industry 4.0}:
\begin{quote}
The increasing digitalization of all manufacturing and manufacturing-supporting tools is resulting in the registration of an increasing amount of actor- and sensor-data which can support functions of control and analysis. Digital processes evolve as a result of the likewise increased networking of technical components and, in conjunction with the increase of the digitalization of produced goods and services, they lead to completely digitalized environments. Those are in turn driving forces for new technologies such as simulation, digital protection, and virtual or augmented reality. \cite{lasi2014industry}
\end{quote}
Yet another closely related term is {\em digital engineering}, which can be defined as:
\begin{quote}
Digital engineering is defined as an integrated digital approach that uses authoritative sources of systems data and models as a continuum across disciplines to support lifecycle activities from concept through disposal. Digital engineering applies similar emphasis on the continuous evolution of artifacts and the organizational integration of people and process throughout a development organization and integration team [\dots] Digital engineering is sometimes referred to as ``digital thread,'' which is understood as a more encompassing term, though the novelty of both terms has created some dispute over their exact overlap. Both digital thread and digital engineering are an extension of product lifecycle management, a common practice in private industry that involves the creation and storage of a system's lifecycle artifacts, in digital form, and which can be modified as a system evolves throughout its lifecycle. Digital thread and digital engineering both involve a single source of truth, referred to as the authoritative source of truth (ASoT), which contain artifacts maintained in a single repository, and stakeholders work from the same models rather than copies of models. \cite{Shepard2020digitalengineering}
\end{quote}
The relation between the concepts of digital twin and digital thread are further illustrated in Figure \ref{fig:DT_Fleet}. A digital twin may be created for each vehicle/asset in a fleet of vehicles/assets and fleet data analyzed by the digital twins may be leveraged into information for monitoring fleet health, decreasing downtime, managing inventory, optimizing operations, and performing simulation scenarios that study performance of the fleet.
\begin{figure}
\caption{Illustration of a fleet of digital twins. Digital threads connect each individual physical twin with its corresponding digital twin allowing for bidirectional data and information flow.}
\label{fig:DT_Fleet}
\end{figure}
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\subsubsection{Digital Thread, Industry 4.0, Digital Engineering}
Although sometimes used interchangeably with {\em digital twin}, {\em digital thread} has a more specific meaning:
\begin{quote}
The digital thread refers to the communication framework that allows a connected data flow and integrated view of the asset’s data throughout its lifecycle across traditionally siloed functional perspectives. The digital thread concept raises the bar for delivering ``the right information to the right place at the right time'' \cite{leiva2016demystifying}
\end{quote}
Digital thread is often considered synonymous with the {\em connected supply chain} that ``collects data on all parts, kits, processes, machines and tools in real time and then stores that data, enabling digital twins and complete traceability.'' \cite{leiva2016demystifying} Digital thread is also closely related to the concept of {\em Industry 4.0}:
\begin{quote}
The increasing digitalization of all manufacturing and manufacturing-supporting tools is resulting in the registration of an increasing amount of actor- and sensor-data which can support functions of control and analysis. Digital processes evolve as a result of the likewise increased networking of technical components and, in conjunction with the increase of the digitalization of produced goods and services, they lead to completely digitalized environments. Those are in turn driving forces for new technologies such as simulation, digital protection, and virtual or augmented reality. \cite{lasi2014industry}
\end{quote}
Yet another closely related term is {\em digital engineering}, which can be defined as:
\begin{quote}
Digital engineering is defined as an integrated digital approach that uses authoritative sources of systems data and models as a continuum across disciplines to support lifecycle activities from concept through disposal. Digital engineering applies similar emphasis on the continuous evolution of artifacts and the organizational integration of people and process throughout a development organization and integration team [\dots] Digital engineering is sometimes referred to as ``digital thread,'' which is understood as a more encompassing term, though the novelty of both terms has created some dispute over their exact overlap. Both digital thread and digital engineering are an extension of product lifecycle management, a common practice in private industry that involves the creation and storage of a system's lifecycle artifacts, in digital form, and which can be modified as a system evolves throughout its lifecycle. Digital thread and digital engineering both involve a single source of truth, referred to as the authoritative source of truth (ASoT), which contain artifacts maintained in a single repository, and stakeholders work from the same models rather than copies of models. \cite{Shepard2020digitalengineering}
\end{quote}
The relation between the concepts of digital twin and digital thread are further illustrated in Figure \ref{fig:DT_Fleet}. A digital twin may be created for each vehicle/asset in a fleet of vehicles/assets and fleet data analyzed by the digital twins may be leveraged into information for monitoring fleet health, decreasing downtime, managing inventory, optimizing operations, and performing simulation scenarios that study performance of the fleet.
\begin{figure}
\caption{Illustration of a fleet of digital twins. Digital threads connect each individual physical twin with its corresponding digital twin allowing for bidirectional data and information flow.}
\label{fig:DT_Fleet}
\end{figure}
\subsubsection{Minimally Viable Digital Twin Framework}
Now armed with a list of defined concepts, the task of answering, {\em what constitutes a minimally viable digital twin framework}, can be addressed. At bare minimum, a DT modeling framework (Figure \ref{fig:DT_Min_Framework}) must comprise the seven elements described below. It is worth noting that since the modeling framework may be implemented in stages, through the life-cycle of the asset, not all of the elements of the framework will be present in every stage (e.g., the physical asset may first be digitally designed) and can be considered as existing in the interim prototype stage until all elements are incorporated.
\begin{itemize}
\item Physical Twin Asset - object, system, or process occurring in the real world; without a corresponding physical twin, the digital twin model is simply a traditional exploratory simulation or design model.
\item Digital Twin - a digital representation of the physical twin asset, such as information models describing the asset's form, function and behavior, the current state/configuration of the physical asset, as well as pointers to a digital database of historical data from instrumentation and previous states/configurations. Without the digital twin representation, the physical twin merely exists with no data and models for analysis.
\item Instrumentation – sensors, detectors, and measurement tools that collectively generate digital data and knowledge about the physical asset or the environment in which it operates; instrumentation may be independent from or embedded on the physical asset and may even include manual inspection notes converted digitally for the digital twin to interpret and process. Without some sort of instrumentation, nothing is measured and no data is generated within the modeling framework.
\item Analysis - data analytics, computational models, simulations, algortihms, and decisions (human or machine); Analysis transforms digital twin data from/of the physical asset into information that is manually, automatically, or autonomously actionable. Without analysis, the digital twin simply mirrors the current state of the physical counterpart, with no actionable information generated for the physical twin.
\item Digital Thread – a digital connection (through wired/wireless communication networks) that provides a data link between the physical asset and the digital twin. Without a digital thread, there is no digital communication between the twins, and thus the configuration comprises, at best, an analog model.
\item Live Data – data from instrumentation that is streamed through the digital thread that changes over time indicating the changing state or status of the physical asset throughout its life cycle; {\em live} being defined as the maximum amount of capture latency that allows for the feedback to be actionable. Without live data, the model is static, temporary, or non-dynamic; such a model can only provide instantaneous, temporary solutions or prototypical off-line feedback.
\item Actionable Information – information generated from the analysis and relayed as an actionable feedback on the physical asset. (i.e., providing feedback on the operation, control, maintenance, and/or scheduling of use of the physical twin asset). With no actionable information being utilized by the physical twin, the physical twin {\em flies blind} with no external insights into its own state or condition.
\end{itemize}
With this framework, one can quickly distinguish what makes a digital twin model distinct from traditional design models and simulations: design models and simulations are used as testing grounds to create at least a physical asset prototype using simulated physics and computer aided design, whereas a digital twin model utilizes data directly from the field in the precise operating conditions and environment the asset is used. In addition, the authors believe the source of confusion in the multiple definitions of digital twin is alleviated by a clear distinction between digital twin and a digital twin modeling framework; the former is considered to be a digital representation of the states, conditions, environmental exposures, and configurations of the physical twin asset, while the latter (as described in the above system of seven components) provides the contextual framework in which the former is utilized. In this context, the digital twin modeling framework, which utilizes digital threads, is consistent with the idea of a {\em Cyber-Physical System} (CPS)~\cite{uhlemann_digital_2017-1}. The modeling philosophies are slightly different, though. In the CPS literature, emphasis is placed in the analysis/control tasks, and the digital representations used to assimilate instrumentation data and perform analysis (i.e., the digital twin) are typically control-oriented models of the physical asset purposefully designed to capture only the relevant behavior and properties needed to support those specific analysis tasks. The digital twin modeling framework, on the other hand, places less emphasis on the analysis tasks and more emphasis on ensuring that the digital twin is a faithful replica of the physical asset, supporting a variety of analysis task even beyond those that are originally envisioned during the design.
\begin{figure}
\caption{Minimally viable framework for a digital twin model. “Live data” is defined as a changing quantity over time with as little data capture latency as needed to allow for the information to be actionable in operations, which in turn, provides feedback on the use, control, maintenance, and/or use scheduling of the physical twin asset.}
\label{fig:DT_Min_Framework}
\end{figure}
A potential critique could be that this framework can be used to describe the relatively simple control logic algorithms (e.g., a vehicle cruise control or auto-pilot control) of a physical asset, or even a simple viewable dashboard of gauges that a human can view and interpret for informative feedback to control the asset. However, these simple examples miss the point that digital twins are meant to model complex interactions (multiphysics) to control complex assets (multiscale multisystems and multiprocesses). Ultimately, that is a minimal framework, and complexity implemented in digital twins can be envisaged to rise in sophistication along three main scales:
\begin{itemize}
\item {\bf Extent of autonomy and decision speed}: A basic model would be a manual supervisory one providing information to personnel who interprets the digital twin information and makes control decisions, while data or operation controls are uploaded/downloaded manually with decisions made on a coarse time scale of hours and days. An intermediate model would automate not just digital data processing but also simple stable controls with regular human interventions to mitigate unstable decision making with decisions made on the time scales of minutes to hours. An advanced autonomous model would automate the majority of decision making in real time (perhaps with only a few seconds of processing lag), instilling enough confidence to make human intervention and decision making rare.
\item {\bf Extent of component granularity}: A basic model would treat the asset as a single item, or perhaps concurrently consider a few components in isolation, and only allow for simple alarming or warning that a maintenance check by a human is needed for further investigation and troubleshooting. An intermediate model would separately model all major subsystems (e.g., avionics, engine, power equipment, hydraulics, structural frame, etc.) or known frequent problem areas, in an attempt to isolate expected faults and more easily identify anomalous behavior of major components and subsystems. An ultimate digital twin would model all components individually that make up the physical asset, as well as their potentially complex interactions in a holistic manner, living up to its "twin" title while allowing for extensive root cause analysis and adjudication of failures and faults down to individual components.
\item {\bf Extent of incorporated physics}: A basic model would focus on a single physics domain mode of failure such as shearing of a bolt due to excessive stress. An intermediate model would incorporate at least two domains of physics to identify failures dependent on multi-domain solutions (e.g., thermal cycling leading to fatigue driven structural failure). A high fidelity model would incorporate all relevant physics domains that can cause failures (e.g., a battery explosion involving electrical, chemical, thermal, and structural considerations).
\end{itemize}
The extent of component granularity and incorporated physics included are proportional to the amount and types of instrumentation (sensor) data that is used to monitor the health and upkeep of the physical twin asset. If there are relatively few and sparsely distributed sensors and detectors in, on, or around the physical asset, then there is a higher likelihood that the digital twin models may be attempting to solve an ill-posed problem: one where unique solutions do not exist or at least where solutions are not convergent at all or that converge to local optima only. One cannot hope to achieve accurate physical simulation of all or a large percentage of the components of a complex asset without instrumentation density that captures behavior of those components at the required granularity and level of detail. Similarly, instruments should be able to capture the relevant physical quantities that the physics model attempts to represent. As an example, a digital twin thermal cycling fatigue analysis model will have no input (and no solution) if there are no embedded temperature measuring devices capturing the real environmental exposure of the components attempting to be modeled. Therefore, accurate and useful digital twin models will directly rely on sensitive, reliable, and physically relevant instrumentation. Even with entirely data-driven statistical and machine learning techniques that do not explicitly model the physics of the asset, one can expect more reliable models if training data from a set of relevant physical sensor types are included, to avoid learning spurious correlations.
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\subsection{Digital Twin Literature Review of Reviews}
Many digital twin reviews exist. A recent Web of Science search for the title or abstract containing (("digital twin" OR "digital twins" OR "digital thread" OR "digital threads") AND ("review" OR "reviews" OR "survey" OR "surveys" OR "landscape" OR "landscapes")) returned 215 results. A similarly filtered Google Scholar\texttrademark{} search limited to review articles returned 305 results. A comprehensive review of digital twin review articles is in order, but it is beyond the scope of this manuscript. We only provide a brief overview with highlights below.
Lu et al.~\cite{lu2020digital} is the most cited review of the digital twin concept. The authors provide a systematic examination of the definition of digital twin and related terms, the enabling technologies, communication standards and network architecture of the digital thread, and current research issues and challenges. Nevertheless, the review is singularly focused on industrial manufacturing applications. Also, instead of demonstrating specific examples of DT, general advantages, such as "decision-making optimization", "data sharing", and "mass personalization" are outlined without quantitative metrics in support. Kritzinger et al.~\cite{kritzinger2018digital} is another frequently cited review of digital twin use in industrial manufacturing. The authors provide a classification schema of DT relevant articles based on the level of automation and model types defining terms such as "digital model" (manual data flow between physical and digital objects in both directions), "digital shadow" (automated data flow from physical to digital objects, but manual data flow from digital to manual objects), and "digital twin" (automated data flow between physical and digital objects in both directions)~\cite{kritzinger2018digital}. Only about one quarter of cited articles involved case studies.
Errandonea et al.~\cite{errandonea2020digital} provide a comprehensive review of digital twins used in the context of predictive maintenance. The paper identified 68 articles and conference papers that used DT for maintenance applications. Khan et al.~\cite{khan2020requirements} provide a look at the path towards {\em autonomous maintenance}, which explores requirements from sensors, machine learning, security, and material science that are needed in order to achieve highly automated and low human intervention digital twin models for equipment maintenance.
Khan et al.~\cite{khan2018review} is another comprehensive, as well as visionary, review of deep learning applied in the context of system health management (SHM). SHM includes PMx, yet is a more general term, encompassing diagnostics and anomaly detection in addition to the prognostics often performed in PMx. The authors identified 38 articles that use deep learning methods to analyze and interpret data from equipment to perform anomaly detection, fault and failure diagnosis and classification, remaining useful life estimation, and component degradation.
Rasheed et al.~\cite{rasheed2020digital} outline several hybrid methods for digital twins incorporating ML and Physics-Based Modeling (PBM); however, there are no current reviews on the intersection of digital twin predictive maintenance models incorporating both machine learning methods and physics based modeling.
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\section{Digital Twin Applications and Industries}
We now review possible digital twin applications and the examples from different industries, in order to better illustrate how the concepts described so far map into real-world usage scenarios.
\subsection{DT Applications}
We begin with a review of prototypical applications that are relevant to multiple different industries.
\paragraph{Risk assessment and decreased time to production:}
For manufacturing and production, a plant layout is a time intensive design and high cost endeavor, which inherently involves risk from inefficient allocations of time and resources. The digital twin creates an opportunity to decrease the time to production by effectively creating a high fidelity representation of the production line in a virtual space, while also exploring what-if design scenarios that can result in optimization of layout to maximize production output and/or minimizing costs~\cite{negri2017review}. This virtual simulation of the manufacturing line may also be used determine what variables of the system are important to monitor in the operation phase and further, whilst in the operation phase, the virtual digital twin, used initially for design, may then be updated with sensor data for the algorithms to analyze and produce prognostics for operations and maintenance.
\paragraph{Predictive maintenance:}
Predictive maintenance has been stated as both the original purpose for the concept of digital twin \cite{shafto2012modeling} as well as being the most popular application of the digital twin model \cite{errandonea2020digital}. The predictive maintenance application will be discussed in further detail in Section 3.
\paragraph{Real-time remote monitoring and fleet management:}
After the predictive maintenance models arrive at their updated predictions, fleet management may then be optimized and executed. Fleet management involves maintenance planning, use scheduling, and logistics as well as performance evaluation metrics. Verdouw et al.~\cite{verdouw2017digital} summarizes several digital twin applications in agriculture, including a fleet management application that tracks individual equipment location and energy use, accurate row tracking with various towed agricultural equipment, and evaluation of crop yield for individual machines. Major et al. provides a demonstrated example of real-time remote monitoring and control of a ship's crane, as well as the ship itself, with the long-term goal of improved fleet logistics and safety, and supervision from onshore monitoring centers~\cite{major2021real}.
\paragraph{Increased team collaboration and efficiency:}
Efficient collaboration between members of a project team is vital if a project is to stay within time and budget constraints. Data and information from assets, as it is updated, altered, and generated must be shared with project managers, multidisciplinary engineering groups, builders and/or manufacturers, and customers/consumers. The need for an integrated platform is substantial, especially when team collaborators are of various technical and skilled backgrounds. The digital twin framework provides such a platform that potentially allows near real time monitoring and information interrogation from a consolidated reliable source. Perhaps this is best illustrated in the field of construction; Lee et al.~\cite{lee2021integrated} disclose a blockchain framework for providing traceability of updates to the database that warehouses both the {\em as-planned} construction and the {\em as-built} project generated from GPS as-measured geometry, and material property testing (e.g., soil and building material). The blockchain traces and authenticates user updates that become a single real time source of truth for all users to interrogate. One of the biggest sources of delays in construction projects occurs from lack of adequately planning for supply chain logistics: getting the right building materials to the right place at the right time. Having a reliable and authenticated source of project status could allow optimization of supply orders and deliveries.
\subsection{DT Industries}
We now review specific applications within different industries.
\paragraph{Manufacturing:}
Digital twins have been extensively applied in industrial manufacturing, mainly in the form of a predictive maintenance model of large complex manufacturing machinery and in the form of a large scale simulation of production run machinery, the latter having the goal of decreasing time to production and assessing risk as is mentioned above. Uhlemann et al.~\cite{uhlemann_digital_2017-1} give an example of effective and efficient production layout planning for small to medium-sized enterprises where different generated layouts are compared using a digital twin simulated environment. PMx applications for manufacturing are also a target area for DT model application, focusing on the impact of upkeep on output production as well as correlated maintenance on upstream and downstream equipment~\cite{susto2014machine}. Rosen et al.~\cite{rosen_about_2015} explore production planning and control by developing a DT simulation framework to optimize the effects of production parameters on output production and manufacturing equipment maintenance.
\paragraph{Aerospace:}
The digital twin framework is heavily applied in aerospace and aviation in predictive maintenance and fleet management applications~\cite{glaessgen_digital_2012,karve_digital_2020,musso2020interacting,sisson2022digital,wang2020life,xiong2021digital,zaccaria2018fleet,zhou2021real,kapteyn2020toward,guivarch_creation_2019}. The large amount of sensor recorded data and up-to-date maintenance records create an environment in which predictive maintenance models can thrive, but also necessitates large scale data management to provide authoritative and easy-to-interrogate databases. Simulation studies evaluating performance, faults, or failure of assets utilizing digital twin models may be performed for various operating conditions of the assets and also under various ambient environmental conditions. Logistical plans may be developed and explored based on simulated site plans or various fleet sizes~\cite{west2018demonstrated}. Machine learning algorithms are employed to flexibly optimize logistical problems associated with predictive maintenance while also predicting the faults and failures of assets based on trends and patterns in the recorded sensor data. These faults may be identified from anomaly detectors that make use of recorded {\em healthy} or {\em nominal} operational data that contrast with anomalous data, which can be flagged in real time.
\paragraph{Architecture, Civil Engineering, Structures and Building Management:}
Building information models (BIM) are closely related to digital twins and are very well known in the architecture, engineering and construction management world. BIMs are semantically rich information models of buildings that can be used to easily visualize the 3D representation of the building along with key properties of the different components and systems. They have also been used to assimilate changes made to the physical building, and in turn, act as a guide during building design, construction and maintenance operations~\cite{coupry2021bim}. Structural health modeling is also a salient example of digital twin use for maintenance, repair and operations (MRO) of structures ~\cite{bigoni2022predictive,droz2021multi,taddei2018simulation, rosafalco2020fully}. Here, digital twins incorporating both analytics of data-driven machine learning of states attributed to sensor data and physics based models, which provide simulations for training models, work together to provide warnings and predictions of potentially adverse states of the structure.
\paragraph{Healthcare, Medicine, and the Human Digital Twin:}
In clinical settings, one is often surrounded by various complex and expensive equipment which is also the subject of maintenance, repair, and fleet management. Healthcare costs continue to rise, yet the high costs of preventive maintenance still dominate healthcare environments, where new technologies can be adopted slowly~\cite{shamayleh2020iot}. There is potential for large cost savings by adapting digital twin based predictive maintenance models. The digital twin concept has also been applied directly to the human patient as well with high fidelity cardiac models serving to help diagnose heart conditions that may lead to patient specific and personalized treatments~\cite{gillette2021framework}. \cite{corral2020digital} Yet another healthcare application is that of a digital twin model of humans in the context of wearable health monitors and trackers that collect data and provide personalized advice, diagnoses, and treatments based on data-driven prediction models~\cite{liu2019novel}. Along similar methodology, Barbiero et al.~\cite{barbiero2021graph} use a patient digital twin approach utilizing a graph neural network to forecast patient conditions (e.g., high blood pressure) based on clinical data from multiple levels of anatomy and physiology, such as cells, tissues, organs, and organ systems.
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\section{Interrelation between Predictive Maintenance and Digital Twin Framework}
\subsection{Distinguishing Types of Maintenance}
Predictive maintenance and the digital twin modeling framework share many common elements and goals. Predictive maintenance utilizes sensor data or maintenance history data from a physical asset to predict the probability of failure over a future interval. The prediction may be used to plan and execute future maintenance or operation of the physical asset. This cycle can be visualized as a control loop, (Figure \ref{fig:pmx-control-loop}) wherein data is generated from the instruments associated with the physical twin asset, passed through the digital thread via the data management block, analyzed by the digital twin models/algorithms, turned into actionable information to operate the physical twin asset, which then leads to new data generation, beginning the cycle anew. Thus, predictive maintenance models, when deployed, meet all the criteria of the definition of a minimally viable digital twin framework (Figure \ref{fig:DT_Min_Framework}). Therefore, it is often mentioned that predictive maintenance models are one of the most popular applications of digital twin~\cite{errandonea2020digital}. However, in the broader field of maintenance, repair, and overhaul (MRO), several distinct maintenance types exist depending on the desired level of safety, cost savings and available instrument data, and not all maintenance operations are fit to utilize a digital twin approach.
\begin{figure}
\caption{The layers of the predictive maintenance stack fit together in a control loop where data is generated, transmitted, analyzed, and acted on. This figure is reproduced with permission from Edman et al.~\cite{edman2021predictive}
\label{fig:pmx-control-loop}
\end{figure}
\paragraph{Reactive Maintenance:}
Reactive maintenance may be summarized by the adage, {\em if it isn't broke, don't fix it}. Equipment is allowed to run to failure. Failed parts are replaced or repaired. Repairs may be temporary in order to quickly regain operational status, while putting off until later more time consuming permanent repairs. Such an approach may save money and personnel time in the short term, but runs the risk of unpredictable interruption and more costly catastrophic failure in the future~\cite{swanson2001linking}. No data from the asset, even if any is recorded, is analyzed, therefore this approach does not utilize the digital twin framework.
\paragraph{Preventative Maintenance:}
Some assets cannot risk catastrophic failure from the perspective of both a total loss/destruction of the asset and loss of human life (i.e., such as catastrophic failure of an aircraft). Instead, the goal of preventive maintenance, also known as scheduled maintenance, is the exact opposite of the reactive approach, in that safety is of utmost importance. In preventive maintenance, there is little tolerance for risk of failure in any critical components. Maintenance plans are set by manufacturers with scheduled inspections and overhauls to occur well before estimated time of failure utilizing large safety margins~\cite{shafiee2015maintenance}. However, such precautions come with the penalty of prematurely replacing still usable components. Again, since no asset data is analyzed this approach does not utilize the digital twin framework.
\paragraph{Condition-Based Maintenance:}
Condition-based maintenance (CBM) falls between the {\em never replace before failure} philosophy of reactive maintenance and the {\em always replace before there is a remote chance of failure} philosophy of preventive maintenance. CBM can be synonymous with the term {\em diagnosis based maintenance} and is sometimes used interchangeably with PMx or with the term {\em condition monitoring}, although there is not a consensus on the equivalency of these terms, which is a source of confusion~\cite{si2011remaining}. However, CBM may have a subtle distinction from PMx, in that CBM requires a keen human observer and tends to use human senses (e.g., odd noises, smells, vibrations, visible erratic behaviors, etc.) to sense that the condition of the asset is in an anomalous state~\cite{nikolaev2019hybrid}. Other CBM methods could be simple heuristics applied to sensor data. An example heuristic would be accelerometer measurements to not exceed the threshold of $k$ standard deviations of signal from some past defined interval, otherwise an impending failure is expected. CBM is used to assess the state of the asset in the present moment based on a recent past interval of time and thus is typically not considered a predictive method. CBM rules are often static and sensitive to noise and artifacts that violate assumptions but do not increase risk of failure, which may lead to excessive maintenance~\cite{errandonea2020digital}. CBM methods can fall under the category of the digital twin framework if they make use of digital data, but often fail to meet the criteria, due to a lack of digital thread, if only analog signals are analyzed and actionable feedback is manually implemented.
\paragraph{Predictive Maintenance:}
Predictive maintenance is a proactive technique that uses real-time asset data (collected through sensors), historical performance data, and advanced analytics to forecast when asset failure will occur. Using data collected by sensors during normal operation, predictive maintenance software uses advanced algorithms to compare real-time data against known measurements, and accurately predicts asset failure. Advanced PMx techniques incorporate machine learning, which is summarized in several extensive reviews~\cite{carvalho2019systematic,miller_system-level_2020}. The result of PMx is that maintenance work can be scheduled and performed before an asset is expected to fail with minimal downtime or impact on operations~\cite{nikolaev2019hybrid}. PMx is synonymous with the the terms prognostic health management (PHM) and systems health management (SHM)~\cite{khan2018review}. As outlined above, predictive maintenance algorithms and models are an application of the digital twin modeling framework.
\paragraph{Prescriptive Maintenance:}
Prescriptive maintenance can be thought of as the implementation step after predictive maintenance~\cite{errandonea2020digital}. After an asset or fleet of assets are predicted (often with a probability or confidence window) to fail within a time interval into the future, the next task is to optimize the maintenance schedule that minimizes costs, minimizes equipment downtime, and maximizes logistical efficiencies (getting the right replacement component in the right place at the right time). Prescriptive maintenance is essentially the optimization and execution of the maintenance plan after predictive maintenance has been performed. Note that the concept of {\em prescriptive maintenance} is often included under the term {\em predictive maintenance}.
\paragraph{Diagnostics, Anomaly Detection, and Prognostics: What Went Wrong?, What Is Going Wrong?, What Will Go Wrong?}
Maintenance, repair, and operations is often divided into the distinct tasks of diagnostics, anomaly detection, and prognostics. ~\cite{khan2018review}
Diagnostics is the task primarily concerned with classifying anomalous behavior to known fault conditions (i.e., to answer the question of: what went wrong?). Faults, or the erroneous operation of equipment, are identified using a diagnostic framework. Anomaly detection is chiefly concerned with detection of unintended or unexpected functions of the monitored equipment. Anomalies may or may not cause or lead to faults or failures, but simply are significant deviations from nominal operation as recorded in the past. Ideally, all potential anomalies and diagnoses would be accounted for during the design and testing phase of the equipment development, yet in practice, complex or aging assets can fail in unanticipated modes and scenarios and therefore, in-the-field analysis of detected anomalies and their linking to discovered faults is at the heart of digital-twin data-driven techniques for diagnostics. Root cause analysis (RCA) is the systematic approach for diagnosing the fault (cause) that leads to a failure. Traditional RCA approaches have included manual analysis by subject matter experts (SMEs) using diagnostic fault trees, fishbone diagrams, and the 5 whys procedure~\cite{sivaraman2021multi}. However, these manual analysis methods become unwieldy with large systems that have many levels of component interaction. As equipment becomes more complex and sophisticated, the number or combinations and permutations of potential causal factors for certain fault events rapidly increases. Therefore, statistical tests and analytic methods, such as regression and the Pearson correlation coefficient, have been applied to capture relationships between recorded sensor variables~\cite{madhavan2021evidence,zhao2017advanced}. Nevertheless, even these methods have difficulty in dealing with non-linear patterns, as well as multi-variable dependence effects and multiple timing and lag effects~\cite{bonissone2009systematic}.
Finally, the task of prognostics deals with predicting a future state or condition (e.g., failure) of the equipment or component thereof. The prediction comes with uncertainties (or are probabilistic) that typically expand with the span of the forecast horizon. Remaining useful life (RUL) estimations of equipment components, effectively a time-to-failure prediction, are the most common type of prognostic indicator and traditionally, have been calculated using historical operational component data (e.g., similar components utilized across a fleet of assets) analyzed with statistical techniques such as regression, stochastic filtering (e.g., Kalman filter), covariate hazard (e.g., Cox proportional hazard), and hidden Markov models~\cite{si2011remaining}.
\section{Incorporating Machine Learning with the Digital Twin Framework for Predictive Maintenance}
Machine learning (ML) is a broad field of various analytical methods that learn by leveraging historical data to make decisions about data encountered in the future. Accordingly, since predictive maintenance, an application of digital twin models, utilizes instrumentation data to make diagnostic or prognostic decisions, ML has been often applied to PMx analyses~\cite{errandonea2020digital,khan2018review,nikolaev2019hybrid}. ML algorithms can be coarsely divided into three different types of learning: supervised, unsupervised, and semi-supervised. These different types are separated based on the type of data one can employ; supervised learning works with data that has labels (i.e., the true predictive outcome is linked to each observation); unsupervised learning has no access to ground-truth labels an utilizes techniques to group, cluster, or extract patterns that can be distinct indicators of an underlying phenomenon; while semi-supervised learning is a hybrid approach in which only a small portion of the data has labels and it may be possible to infer the missing ground-truth annotations.
Likewise, in PMx applications, the approach chosen will depend on the instrumentation data one can use and whether the data is labelled, that is, faults and failures, as well as nominal operation patterns, have been attributed to the recorded data. Ideally, data is collected from multiple sensors over a fleet of assets, over the entire life of each component, from initial normal operation, through a time of degradation, and then finally until failure. Despite increasing use of connected sensors embedded on complex assets, access to these ideal databases can be rare due to the storage requirements needed to warehouse prolonged acquisition data cycles. Also, many complex physical assets are rarely run until failure, due to the risk of a catastrophic consequences, including total loss of the asset.
\paragraph{Predictive Maintenance Workflow and Training from Nominal {\em Healthy} Data}
Booyse et al.~\cite{booyse2020deep} argue that the first phase in health monitoring is to be able to detect anomalous or faulty behavior from nominal healthy behavior. Typically this involves a workflow of data acquisition (DAQ) from the instrumentation, preprocessing of data to filter noise and remove artifact, identifying a set of distinct condition indicators that differentiate normal and anomalous data or previously known fault modes, training and testing an ML model to enable predictions on test data, and deploying the model to make predictions when encountering future data. Often, the acquired data is time series data (i.e., a recorded numeric variable, such as temperature, pressure, acceleration, concentration, strain, etc., as a function of time) and condition indicators, or features, can be based in the time-domain (e.g., mean, standard deviation, root-mean square, skewness, kurtosis, or morphological parameters of the shape of the time series signal, etc.), frequency-domain (e.g., power bandwidth, mean frequency, peak frequencies and amplitudes, etc.) and time-frequency domain (e.g., spectral entropy, spectral kurtosis, etc.). Condition indicators could also be derived from static or dynamic physical models. Determination of anomalous behavior may involve one or more distinctive features which may be extracted by unsupervised learning methods, such as clustering, or if many features are involved, dimension reduction maybe performed through, e.g., principal component analysis or other null-space basis model type. Yang et al.~\cite{yang2022causal} describe a causal correlation algorithm that is applied to Bayesian networks and potential for diagnosing causality of behaviors (e.g., faults) in large complicated networks that have non-linearities and are multivariate. As faults are identified and accumulated during the operational history of the asset, these events may then be compared to the identified groups or clusters, thus performing a diagnostic function of linking anomalous behavior with identified faults.
\paragraph{Similarity Models: When Data Encompasses Run to Failure Complete Histories}
If the acquired data is a complete history of a group (fleet) of similar equipment, spanning from its initial operation until failure, it lends itself to prognostication of failures using similarity models~\cite{Mikus2007},~\cite{Dubrawski2011}.
If one can map the current state of an asset on the specific time marks of usage trajectories observed from other assets whose actual outcomes are known, one could use the distribution of these outcomes as a reference for e.g., remaining useful life, time to specific type of failure, or other statistics of interest. Many of similarity based modeling techniques draw inspiration from medicine, where the individual patient's heath status assessment, diagnosis as well as prognosis are routinely mapped on the background of many similar cases observed before, sometimes including detection of distinct phenotypes of trajectories of evolution towards failure~\cite{Chen2015}, or from public health, where the task of detecting outbreaks of infectious diseases appears similar to the task of detecting onset of new types of failures spreading across the fleet of an equipment~\cite{Dubrawski2007}. Applications of this concept vary in how the similarity is defined and how the predictions are formed, and include statistical machine learning as well as neural network based methods ~\cite{adhikari2018machine,saha2019different,bektas2019neural}.
\paragraph{Survival Models: When Data Encompasses Only the End Time of Failure From Similar Equipment}
Sometimes only the time point of failure is known, and survival models such as Kaplan-Meier, Cox proportional hazard (CPH)~\cite{hrnjica2021survival}, or more advanced approaches using deep learning~\cite{chen2020predictive} can be utilized. Survival curves (probability of survival over time) for time-to-event (i.e., events are faults or failures in the PMx context) are generated from the failure history data. Distinct survival curves may be generated for groups of equipment that share similar covariates (similar condition indicators or properties, e.g., manufacture, operating conditions). Based on the survival curves, RUL may be estimated~\cite{hrnjica2021survival}.
\paragraph{Degradation models: When Data Encompasses Run to Known Threshold that Exceeds Safety Criteria}
Frequently, equipment is not run until failure, but instead until just before exceeding a safety threshold. A class of methods that reflect this concept is known as degradation models. They can be implemented as a linear stochastic process or an exponential stochastic process if the equipment experiences cumulative degradation effects~\cite{thiel2021cumulative}. Degradation models typically work with a single condition indicator, although data fusion techniques can be used to combine multiple indicators of degradation. However, what role maintenance and repair may have in not only censoring failure events, but also resetting or offsetting latent degradation states is largely an unexplored area of research~\cite{miller_system-level_2020}.
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\section{Incorporating Physics-Based Modeling and Simulations with Digital Twin Frameworks for Predictive Maintenance}
So far, when discussing PMx tasks and their specific applications, the digital twins at the core of the digital twin framework have been largely statistical models of the behavior of the physical asset, tailored specifically to address the PMx task in question. However, any careful reader may have observed already that there is ample opportunity to extend the reach of the process by leveraging more complex (e.g., multi-physics, multi-scale) models of the assets being twinned. Physics-based models (PBMs) are extensively utilized in the design-phase and are excellent candidates for such extensions. However, a drop-in replacement may be difficult as digital twins and PBMs differ in several fundamental ways:
\begin{enumerate}
\item PBMs are often used as a design tool in the early phases of the product or system development and are rarely returned to after the design, testing, and manufacturing phases are completed, perhaps only after product failure (failure analysis). Digital twins are updated and utilized through all phases from concept, design, and testing to implementation, customer support, and end of product life phases. Ensuring that PBMs can be easily updated throughout the life-cycle requires some adjustments.
\item Traditional PBMs are often built only for specifically assumed operating conditions for the product or the conditions are learned from limited bench-top lab testing or environmental data. The digital twin modeling framework relies on generated data from real world applications of products; simulations and analyses performed with the use of digital twins are enabled with data from the fully assembled, working product or system, operating in their real-world environment. Thus, continuously calibrating/update PBMs to enable this integration can pose a challenge.
\item Traditional PBMs are typically highly detailed representations of small parts or subsystems of an overall assemblage of parts or a system-of-systems. Digital twin analyses take advantage of data being collected from the working assembled system to predict faults and failures that occur due to the complex interactions of systems of parts and their environment. PBMs typically require orders of magnitude more computations to produce their predictions than the data-driven models typically used in PMx.
\end{enumerate}
It would be unfair to characterize the relationship of traditional PBMs and digital twin models as adversarial. Due to the benefits they provide, many of the design phase PBMs already form the initial record of the digital twin, or of the individual parts that comprise a digital twin of a system or asset. We now expand on the specific limitations of using PBMs in the digital twin modeling framework.
\paragraph{Limitations of Multi-Scale, Multi-Physics Models for the Digital Twin Framework}
A common refrain from digital twin skeptics may be that a realistic digital twin can never fully simulate a physical system or asset; however all models, including physics-based models (e.g., stress/strain analysis, computational fluid dynamics, electromagnetic scattering, and other complex processes) are based on assumptions and simplifications in an attempt to explain complex physical interactions. Yet, despite the simplifications, computational models have driven much of engineering progress over the past half century. PBMs often achieve accurate solutions by discretizing, splitting the volumes or areas into small regular elements and nodes that populate a model space governed by differential equations. Typically, these are second order (or higher), non-linear, partial differential equations, which have few, if any, general solutions or techniques. If the model space is of irregular geometry as well, a discretized model based on finite or boundary elements may be the only option for a reasonably accurate solution. Unfortunately, the accuracy of such models come at the cost of extensive computation time. Cerrone et al.~\cite{cerrone2014effects} estimated a single simulation run of crack propagation in a structural plate with several notches and holes had:
\begin{quote}
approximately 5.5 million degrees of freedom. Simulations were conducted on a 3.40 GHz, 4th generation Intel Core i7 processor. Abaqus/Explicit’s shared memory parallelization on four threads with a targeted time increment of $1\cdot10^{-6}$ seconds resulted in approximately a 4-day wall-clock run time~\cite{cerrone2014effects}.
\end{quote}
Of course, such a runtime is far from the promise of digital twins providing near real-time decision making from the constant stream of sensor data. To exacerbate, modern automobiles have tens of thousands of parts and large commercial aircraft may have millions of parts~\cite{airbus_a380_facts}. In addition, engineers frequently desire more than one type of {\em what if} simulation and would prefer to run a great number of varied simulations to explore variables and effects. These complicating factors would seem to relegate the concept of digital twins based entirely on physics models out of reach practically and economically~\cite{west2017digital,tuegel2012airframe}, or at least push its utility decades into the future. However, there are several novel forms of analysis that seek to hybridize physics based models with purely data-driven techniques, such as machine learning, which may result in more manageable computational costs of PBMs and automated learning of their parameters.
\paragraph{Combining Data-Driven Machine Learning Methods and Physics-Based Models to Address Each Other's Shortcomings}
While a digital twin definition often includes the concept of physics based models, there are well known limitations of such models as outlined above. The term {\em model} may be generalized to include trends and patterns directly learned from data.
For example, there is the related terminology of implicit digital twins (IDT) from Xiong et al.~\cite{xiong2021digital}:
\begin{quote}
However, the traditional DT method requires a definite physical model. The structure of the aero-engine system is complex, and the use of a physical-based model to implement a DT requires the establishment of its own model for each component unit, which complicates predictive maintenance and increases costs, let alone achieve accurate maintenance. To circumvent this limitation, this paper uses data-driven and deep learning technology to develop DT from sensor data and historical operation data of equipment and realizes reliable simulation data mapping through intelligent sensing and data mining (called implicit digital twin; IDT). By properly mapping the simulation data of aero-engine cluster to a certain parameter, combined with the deep learning method, various scenarios’ remaining useful life (RUL) can be predicted by adjusting the parameters.~\cite{xiong2021digital}
\end{quote}
\noindent In other words, a physical model may not be needed; an IDT model can simply be created through data-driven analysis (i.e., machine learning). Nevertheless, ML approaches have a few relevant limitations as well:
\begin{enumerate}
\item Scientific and engineering problems are often underconstrained (i.e., large number of variables, small number of samples) making learning reliable ML models from the corresponding data difficult.
\item Catastrophic failures are naturally infrequent and so they may be seldom, if ever, encountered in the recorded data. This issue can sometimes be alleviated using Bayesian forms of ML that allow incorporation of prior probability distributions to account for events that are not represented in data.
\item It can be easy for crossvalidation methods to misevaluate spurious relationships learned by data-driven frameworks as they can look deceptively well on training as well as tests sets.
\item Some ML methods, such as deep neural networks, are ``black boxes'' that provide few interpretable insights into the resulting models, and as such they may fail to convince their users that the obtained solutions are sufficiently systematic to be applicable to future, similar problems.
\item Data used for training ML models is most often just a limited projection of the reality that they are expected to capture. It is then easy for even advanced ML models to fail to follow common sense that comes natural to human domain experts, if important nuances of the underlying knowledge is not reflected in the training data.
\item Large amounts of labeled training data that is often required to produce reliable data-driven models can be expensive to acquire and/or time consuming to create.
\end{enumerate}
Some recently developed frameworks aim to combine the interpretability of PBMs with the data-driven analytical power of digital twins and machine learning.
Combining PBMs and ML seeks to overcome the problems of the long runtime of highly detailed and complex physical models on one hand, and the lack of interpretability and required large volumes of labeled training data on the other~\cite{kapteyn2020toward}. Hybrid ML and physics based solutions are likely to excel in solving analytic problems, such as planning maintenance of complex assets (e.g., vehicles, aircraft, buildings, manufacturing plants) where some data (e.g., small samples of rare occurrences, noisy and spatially or temporally sparse sensor recordings, etc.) and some knowledge of physics (e.g., missing boundary conditions, occurrence of physical interactions, etc.) exists, but neither alone are likely to contain enough information to solve complex diagnostics or prognostics problems with sufficiently useful accuracy or precision for practical application.
\paragraph{Hybrid Digital Twin Frameworks Combine Machine-Learning and Physics-Based Modeling}
There are several approaches to combining PBMs and ML, which may go by several different names, such as {\em physics informed machine learning} (PIML), {\em theory guided data science} (TGDS), {\em scientific machine learning} (SciML), which include physical constraints and known parameter relationships (e.g., an ODE, PDE, physical and material properties and relationships). There are three main strategies identified so far: physics informed neural networks, reduced order modeling, and simulated data generation for supplementing small data sets.
The work of adjusting PBMs to work well with ML (and in particular for the PMx tasks) has been so far limited to very simple components and systems. The R\&D community is yet to seriously consider scaling hybrid approaches to handle complex systems with multiple components requiring true multi-physics and multi-scale models. Work is needed on both figuring out how to automatically integrate multiple PBM models together, as well as how to automatically select the right hybridization strategy to make the resulting solutions run sufficiently fast without compromising fidelity of the resulting models.
\paragraph{Physics Informed Machine Learning (PIML) and Physics Informed Neural Networks (PINNs)} In 2019, Raissi et al.~\cite{raissi2019physics} proposed a deep learning framework that combines mathematical models and data by taking advantage of prior techniques for using neural networks as differentiation engines and differential equation solvers (e.g., Neural ODE~\cite{chen2018neural}). The main idea is that the physical relations and equations modeling them are leveraged to formulate a loss function for the ML algorithm to minimize violation of the principles of physics.
Specifically, a loss function can combine the usual data-driven component based on observed residuals, with physics-driven terms reflective of errors in the solutions of the governing ODE or PDE, and terms reflective of violations of any boundary or initial conditions.
Raissi et al.~\cite{raissi2019physics} provide several examples of solved dynamic as well as boundary value problems, including Schrodinger's, Navier Stokes, and Berger's equations. A similar approach by Jia et al.~\cite{jia2019physics} under the name of {\em physics guided neural networks} (PGNN) which was used to determine temperature distribution along the depth of a lake using both physical relations and sensor data.
\paragraph{Reduced Order Modeling}
Another method is to reduce the order, size, number of degrees of freedom (DoF), or dimensionality of the PBMs. This approach is often called {\em reduced order modeling} or {\em projection based modeling} or {\em lift and learn}~\cite{swischuk2019projection,kapteyn2020toward}.
Here, training data is generated by the PBMs, but only a few snapshots, e.g., a few individually solved time instances are used to reduce the computational power needed instead of solving over a complete time domain.
Then, a lower-dimensional basis is computed and the higher-order PDE model is projected onto the lower-dimensional space. Hartmann et al.~\cite{hartmann202012,hartmann2018model} give an excellent review on reduced order modeling and its role as a digital twin enabling technology, drastically reducing model complexity and computation time, all while maintaining high fidelity of solutions.
The reduced order model solutions can be arrived at rapidly for various boundary and/or initial conditions and are frequently used as a simulation database to which ML algorithms may be trained on to classify damage states or learn a regression to a continuous degradation model; the trained models can then be fed distributed sensor data from real-world assets in the field to perform equipment health monitoring tasks~\cite{bigoni2022predictive,kapteyn2020toward,hartmann2018model,droz2021multi,leser2020digital,taddei2018simulation,rosafalco2020fully}.
\section{Challenges of Digital Twin Implementation}
\paragraph{Sensor Robustness, Missing data, Poor Quality data, and Offline Sensors}
DT frameworks require live data, which is often generated by a dense array of sensors. Inevitably, one or more sensors will disconnect, contain periodic noise or artifact, or ironically, require maintenance. First, the DT models need to be able to detect and handle sensor signal dropout. If not accounted for in the model algorithms, faults may go unnoticed or misdiagnosed. One such way of dealing with data interruptions is to use {\em circuit breakers} attached to the sensors that trip when the signals go out of range. \cite{preuveneers2018robust} Another approach is to ensure adequate signal processing to remove and filter out unwanted noise and artifacts.
\paragraph{Workplace Adoption of the DT Framework}
Successful DT frameworks require a team effort of caring about data quality. Errors introduced through improper data entry or inadvertent part swaps will propagate throughout the DT framework. Improving user interfaces on data entry menus, as well as seeking out and requesting feedback from team members will demonstrate concerns for the daily user. Finally, sharing the goals, the rewards, and the output of the models with team members also helps reinforce positive feedback in the workplace.
\paragraph{Security Protocols}
It is estimated that the majority of network-connected digital twins will utilize at least five different kinds of integration endpoints, while each of the endpoints represents a potential area of security vulnerability. \cite{mullet2021review} Therefore, it is highly recommended that best practices be implemented wherever possible, including, but not limited to: end to end data encryption, restricted use of portable media such as portable hard drives and other portable media on the network, regular data backups (to offline locations if possible), automated system software patch and upgrade installs, password protected programmable logic contollers, managed user authentication and controlled access to digital twin assets~\cite{mullet2021review}.
\section{Summary}
This manuscript attempts to provide clarity on defining {\em digital twin}, through exploring the history of the term, its initial context in the fields of product life cycle management, asset maintenance, and equipment fleet management, operations, and planning. A definition for a minimally viable digital twin framework is also provided based on seven essential elements. A brief tour through DT applications and industries where DT methods are employed is also provided. Thereafter, the paper highlights the application of a digital twin framework in the field of predictive maintenance, and its extensions utilizing machine learning and physics based modeling. The authors submit that employing the combination of machine learning and physics based modeling to form hybrid digital twin frameworks, may synergistically alleviate the shortcomings of each method when used in isolation. Finally, the paper summarizes the key challenges of implementing digital twin models in practice. A few evident limitations notwithstanding, digital twin technology experiences rapid growth and as it matures, we expect its great promise to materialize and substantially enhance tools and solutions for intelligent upkeep of complex equipment.
\paragraph{Acknowledgement:}
This work was partially supported by the U.S.\ Army Contracting Command under Contracts W911NF20D0002 and W911NF22F0014 delivery order No. 4 and by a Space Technology Research Institutes grant from NASA’s Space Technology Research Grants Program.
\end{document}
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\begin{document}
\begin{abstract}
Let $\A \in \Reals^{n \times n}$ be a nonnegative irreducible square matrix and let $r(\A)$ be its spectral radius and Perron-Frobenius eigenvalue. Levinger asserted and several have proven that $r(t):=\spr((1{-}t) \A + t \A\tr)$ increases over $t \in [0,1/2]$ and decreases over $t \in [1/2,1]$. It has further been stated that $r(t)$ is concave over $t \in (0,1)$. Here we show that the latter claim is false in general through a number of counterexamples, but prove it is true for $\A \in \Reals^{2\times 2}$, weighted shift matrices (but not cyclic weighted shift matrices), tridiagonal Toeplitz matrices, and the 3-parameter Toeplitz matrices from Fiedler, but not Toeplitz matrices in general. A general characterization of the range of $t$, or the class of matrices, for which the spectral radius is concave in Levinger's homotopy remains an open problem.
\end{abstract}
\title{
Nonconcavity of the Spectral Radius \in Levinger's Theorem}
\pagestyle{myheadings}
\markboth{L. Altenberg \& J. E. Cohen}{Nonconcavity of the Spectral Radius in Levinger's Theorem}
\centerline{\emph{Dedicated to the memory of Bernard Werner Levinger (1928--2020)}}
\ \\
\noindent Keywords:
circuit matrix, convexity, direct sum, homotopy, nonuniform convergence, skew symmetric
\\
MSC2010: 15A18, 15A42, 15B05, 15B48, 15B57
\section{Introduction}
The variation of the spectrum of a linear operator as a function of variation in the operator has been extensively studied, but even in basic situations like a linear homotopy $(1{-}t) \X + t \Y$ between two matrices $\X, \Y$, the variational properties of the spectrum have not been fully characterized. We focus here on Levinger's theorem about the spectral radius over the convex combinations of a nonnegative matrix and its transpose, $(1{-}t) \A + t \A\tr$.
We refer to $\B(t) = (1{-}t) \A + t \A\tr$, $t \in [0,1]$, as \emph{Levinger's homotopy},\footnote{Also called Levinger's transformation \cite{Psarrakos:and:Tsatsomeros:2003:Perron}.} and the spectral radius of Levinger's homotopy as \emph{Levinger's function} $r(t) \eqdef \spr(\B(t)) = \spr((1{-}t) \A + t \A\tr)$.
On November 6, 1969, the \emph{Notices of the American Mathematical Society} received a three-line abstract from Bernard W. Levinger for his talk at the upcoming AMS meeting, entitled ``An inequality for nonnegative matrices.''\cite{Levinger:1970:Inequality} We reproduce it in full:
``\underline{Theorem.} Let $A \ge 0$ be a matrix with nonnegative components. Then $f(t) = p(tA + (1{-}t)A^T)$ is a monotone nondecreasing function of $t$, for $0 \le t \le 1/2$, where $p(C)$ denotes the spectral radius of the matrix $C$. This extends a theorem of Ostrowski. The case of constant $f(t)$ is discussed.''
Levinger presented his talk at the Annual Meeting of the American Mathematical Society at San Antonio in January 1970. Miroslav Fiedler and Ivo Marek were also at the meeting \cite{Marek:1974:Inequality}. Fiedler developed an alternative proof of Levinger's theorem and communicated it to Marek \cite{Marek:1978:Perron}. Fiedler did not publish his proof until 1995 \cite{Fiedler:1995:Numerical}. Levinger appears never to have published his proof.
Marek \cite{Marek:1978:Perron,Marek:1984:Perron} published the first proofs of Levinger's theorem, building on Fiedler's ideas to generalize it to operators on Banach spaces. Bapat \cite{Bapat:1987:Two} proved a generalization of Levinger's theorem for finite matrices. He showed that a necessary and sufficient condition for non-constant Levinger's function is that $\A$ have different left and right normalized (unit) eigenvectors (\emph{Perron vectors}) corresponding to the Perron-Frobenius eigenvalue (\emph{Perron root}).
Fiedler \cite{Fiedler:1995:Numerical} proved also that Levinger's function $r(t)$ is concave in some open neighborhood of $t=1/2$, and strictly concave when $\A$ has different left and right normalized Perron vectors. The extent of this open neighborhood was not elucidated.
Bapat and Raghavan \cite[p.~121]{Bapat:and:Raghavan:1997} addressed the concavity of Levinger's function in discussing ``an inequality due to Levinger, which essentially says that for any $\A \ge 0$, the Perron root, considered as a function along the line segment joining $\A$ and $\A\tr$, is concave.'' The inference about concavity would appear to derive from the theorem of \cite[Theorem 3]{Bapat:1987:Two} that $\spr(t \, \A + (1{-}t)\B\tr) \geq t\, r(\A) + (1{-}t)\, r(\B)$ for all $t \in [0, 1]$, when $\A$ and $\B$ have a common left Perron vector and a common right Perron vector. The same concavity conclusion with the same argument appears in \cite[Corollary 1.17]{Stanczak:Wiczanowski:and:Boche:2009:Fundamentals}.
However, concavity over the interval $t \in [0, 1]$ would require that
for all $t, h_1, h_2 \in [0, 1]$, $r(t \, \F(h_1) + (1{-}t) \F(h_2) ) \geq t\, r(\F(h_1)) + (1{-}t) r(\F(h_2))$,
where $\F(h) \eqdef h \, \A + (1{-} h)\B\tr$.
While Theorem 3.3.1 of \cite{Bapat:and:Raghavan:1997} proves this for $h_1 = 1$ and $h_2 = 0$, it cannot be extended generally to $h_1, h_2 \in (0,1)$ because $\F(h_1)$ and $\F(h_2)\tr$ will not necessarily have common left eigenvectors and common right eigenvectors.
Here, we show that the concavity claim is true for $2 \times 2$ and other special families of matrices. We also show that for each of these matrix families, counterexamples to concavity arise among matrix classes that are ``close'' to them, in having extra or altered parameters. Table \ref{Table:Comparison} summarizes our results.
\begin{table}[ht] \label{Table:Comparison}
\caption{Classes of nonnegative matrices with concave Levinger's function (left), and matrix classes ``close'' to them with nonconcave Levinger's function (right).}
{\small
\begin{tabular}{|lr|lr|}
\hline
{\bf Concave} & &{\bf Nonconcave} &\\
\hline
$2 \times 2$ &\!\!\!\!Theorem \ref{Theorem:2x2}& $3 \times 3$, $4 \times 4$ &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!Eqs.\ \eqref{eq:Ex1}, \eqref{eq:4x4} \\
Tridiagonal Toeplitz &\!\!\!\!Theorem \ref{Theorem:Tridiag} & 4-parameter Toeplitz &Eq.\ \eqref {eq:ToeplitzConvex}\\
Fiedler's 3-parameter Toeplitz &\!\!\!\!Theorem \ref{Theorem:FiedlerLevinger} & 4-parameter Toeplitz &Eq.\ \eqref {eq:ToeplitzConvex}\\
$n \times n$ weighted shift matrix &\!\!\!\!Theorem \ref{Theorem:Shift} & $n \times n$ cyclic weighted shift matrix &Eq.\ \eqref {eq:CyclicShift16} \\
\hline
\end{tabular}
}
\end{table}
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\section{Matrices that Violate Concavity}
\subsection{A Simple Example}
Let
\an{
\A &= \Pmatr{
0&1&0\\
0&0&0\\
0&0&2/5
}\notag
\stext{to give}
\B(t) &= (1{-}t)\A+t\A\tr
= \Pmatr{
0&1{-}t&0\\
t&0&0\\
0&0&2/5}. \label{eq:Ex1}
}
The eigenvalues of $\B(t)$ are $\{2/5, +\sqrt{t(1{-}t)}, -\sqrt{t(1{-}t)}\}$, plotted in Figure \ref{fig:Ex1}. On the interval $t \in [1/5, 4/5]$, $\spr(\B(t)) = \sqrt{t(1{-}t)}$ is strictly concave. On the intervals $t \in [0, 1/5]$ and $t \in [4/5,1]$, $\spr(\B(t))$ is constant. It is clear from the figure that $\spr(\B(t))$ is not concave in the neighborhood of $t=1/5$ (and $t=4/5$), since for all small $\epsilon > 0$,
\an{\label{eq:Ineq1}
\frac{1}{2}[ \spr(\B(1/5 - \epsilon) + \spr(\B(1/5 + \epsilon) ]> \spr(\B(1/5 )) = 2/5.
}
By the continuity of the eigenvalues in the matrix elements \cite[2.4.9]{Horn:and:Johnson:2013}, we can make $\B(t)$ irreducible and yet preserve inequality \eqref{eq:Ineq1} in a neighborhood of $t=1/5$ by adding a small enough positive perturbation to each element of $\A$.
\begin{figure}
\caption{Eigenvalues of the matrix $\B(t)$ from \eqref{eq:Ex1}
\label{fig:Ex1}
\end{figure}
The basic principle behind this counterexample is that the maximum of two concave functions need not be concave. Here $\B(t)$ is the direct sum of two block matrices. The eigenvalues of the direct sum are the union of the eigenvalues of the blocks, which are different functions of $t$. One block has a constant spectral radius and the other block has a strictly concave spectral radius. The spectral radius of $\B(t)$ is their maximum.
Another example of this principle is constructed by taking the direct sum of two $2 \times 2$ blocks, each of which is a Levinger homotopy of the matrix $\Pmatr{0&1\\0&0}$,
but for values of $t$ at opposite ends of the {unit} interval, one $2\times 2$ block, $\A_1$, with $t_1 = 511/512$ and the other $2\times 2$ block, $\A_2$, with $t_2 =1/8$. We take a weighted combination of the two blocks with weight $h$, $\A(h) = (1-h) \A_1 \oplus h \A_2$, to get:
\an{
\A(h) &=
\Pmatr{ 0 & (1{-}h) \frac{511}{512} & 0 & 0 \\
(1{-}h) \frac{1}{512} & 0 & 0 & 0 \\
0 & 0 & 0 & h \frac{1}{8} \\
0 & 0 & h\frac{7 }{8} & 0 } . \label{eq:4x4}
}
The eigenvalues of $\B(t,h) = (1{-}t) \A(h) + t \A(h)\tr$ are plotted in Figure \ref{fig:4x4}. We see that there is a narrow region of $h$ below $h=0.5$ where the maximum eigenvalue switches from block 2 to block 1 and back to block 2 with increasing $t \in [0,1]$, making $\spr(\B(t,h)) = \spr((1{-}t) \A(h) + t \A(h)\tr)$ at $h=0.4$ nonconcave with respect to the interval $t \in [0,1] $.
\begin{figure}
\caption{Eigenvalues of $\B(t,h)$ for a two-parameter homotopy: Levinger's homotopy $\B(t,h) = (1{-}
\label{fig:4x4}
\end{figure}
As in example \ref{eq:Ex1}, $\A(h)$ may be made irreducible by positive perturbation of the $0$ values without eliminating the nonconcavity.
The principle here may be codified as follows.
\begin{Proposition}
Let $\A = \A_1 \oplus \A_2 \in \Reals^{n \times n}$, where $\A_1$ and $\A_2$ are irreducible nonnegative square matrices.
{Then}
$\spr(t) \eqdef \spr((1-t)\A+ t \A\tr)$ is not concave in $t \in (0,1)$ if there exists $t^* \in (0,1)$ such that
\begin{enumerate}
\item $\spr((1-t^*)\A_1+ t^* \A_1\tr) = \spr((1-t^*)\A_2+ t^* \A_2\tr)$,\\ \ \\
{and} \ \\
\item $ \left. \df{}{t} \spr((1-t)\A_1+ t \A_1\tr)\right|_{t=t^*}
\neq \left. \df{}{t} \spr((1-t)\A_2+ t \A_2\tr)\right|_{t=t^*} $.
\end{enumerate}
\end{Proposition}
\begin{proof}
Let $r^* \eqdef r(t^*) = \spr((1-t^*)\A_1+ t^* \A_1\tr) = \spr((1-t^*)\A_2+ t^* \A_2\tr)$. Since the spectral radius of a nonnegative irreducible matrix is a simple eigenvalue by Perron-Frobenius theory, it is analytic in the matrix elements \cite[Fact 1.2]{Tsing:etal:1994:Analyticity}. Thus for each of $\A_1$ and $\A_2$, Levinger's function is analytic in $t$, and therefore has equal left and right derivatives around $t^*$. So we can set $s_1 = \dfinline{\spr((1-t)\A_1 + t \A_1\tr)}{t}|_{t=t^*}$ and $s_2 = \dfinline {\spr((1{-}t)\A_2+ t \A_2\tr)}{t}|_{t=t^*}$. Then
\ab{
\spr((1{-}t^* {-} \ep)\A_1+ (t^* {+} \ep) \A_1\tr) &= r^* + \ep s_1 + \Order(\ep^2), \\
\spr((1{-}t^* {-} \ep)\A_2+ (t^* {+} \ep) \A_2\tr) &= r^* + \ep s_2 + \Order(\ep^2).
}
For a small neighborhood around $t^*$,
\ab{
\spr(t^*{+}\ep) &= \spr( (1{-}t^* {-} \ep )\A+ (t^* {+} \ep) \A\tr) \\&
= \max\set{\spr((1{-}t^* {-} \ep)\A_1+ (t^* {+} \ep) \A_1\tr), \spr((1{-}t^* {-} \ep)\A_2+ (t^* {+} \ep) \A_2\tr)} \\&
= r^* + \Cases{
\ep \min(s_1, s_2) + \Order(\ep^2), &\qquad \ep < 0, \\
\ep \max(s_1, s_2) + \Order(\ep^2), &\qquad \ep > 0 .
}
}
A necessary condition for concavity is
$
\frac{1}{2}(\spr(t^*{+} \ep) + \spr(t^*{-}\ep)) \leq \spr(t^*).
$
However, for small enough $\ep > 0$, letting $\delta = \max(s_1, s_2) - \min(s_1, s_2) > 0$,
\ab{
\frac{\spr(t^*{+} \ep) + \spr(t^*{-}\ep)}{2}
&= r^* + \ep \frac{\max(s_1, s_2) - \min(s_1, s_2) }{2} + \Order(\ep^2) \\
&= r^* + \ep \delta / {2} + \Order(\ep^2)
> r^* .
}
The condition for concavity is thus violated.
\end{proof}
\subsection{Toeplitz Matrices}
The following nonnegative irreducible Toeplitz matrix has a nonconcave Levinger's function:
\an{\label{eq:ToeplitzConvex}
\A&=
\Pmatr{
5 & 0 & 6 & 0 \\
1 & 5 & 0 & 6 \\
0 & 1 & 5 & 0 \\
8 & 0 & 1 & 5
}
}
A plot of Levinger's function for \eqref{eq:ToeplitzConvex} is not unmistakably nonconcave, so instead we plot the second derivative of $\spr(\B(t))$ in Figure \ref{fig:ToeplitzConvex}, which is positive at the boundaries $t=0$ and $t=1$, and becomes negative in the interior.
\begin{figure}
\caption{The second derivative of Levinger's function for the Toeplitz matrix \eqref{eq:ToeplitzConvex}
\label{fig:ToeplitzConvex}
\end{figure}
\subsection{Weighted Circuit Matrices}
Another class of matrices where Levinger's function can be nonconcave is the weighted circuit matrix. A weighted circuit matrix is an $n \times n$ matrix in which there are $k\in[1,n]$ distinct integers $i_1, i_2, \ldots, i_k \in \{1, 2, \ldots, n\}$ such that all elements are zero except weights $c_j$, $j = 1, \ldots, k$, at matrix positions $(i_1, i_2), (i_2, i_3), \ldots, (i_{k-1}, i_k), (i_k, i_1)$, which form a circuit. We refer to a \emph{positive weighted circuit matrix} when the weights are all positive numbers.
When focusing on the spectral radius of a positive weighted circuit matrix, we may without loss of generality consider its non-zero principal submatrix, whose canonical permutation of the indices gives a \emph{positive cyclic weighted shift matrix}, $\A$, with elements
\an{
A_{ij} &= \Cases{
c_i > 0, &\qquad j = i \text{ mod } n + 1, \quad i \in \set{1, \ldots, n}, \\
0, & \qquad\text{otherwise.}
}\label{eq:CyclicShift}
}
Equation \eqref{eq:CyclicShift} defines a cyclic \emph{downshift} matrix, while an \emph{upshift} matrix results from replacing $j = i \text{ mod } n + 1$ with $i = j \text{ mod } n + 1$, which is equivalent for our purposes. Cyclic weighted shift matrices have the form
\ab{
\Pmatr{
0 & c_1 & 0 & 0 \\
0 & 0 & c_2 & 0 \\
0 & 0 & 0 & c_3 \\
c_4 & 0 & 0 & 0 \\
} .
}
If one of the weights $c_i$ is set to $0$, the matrix becomes a positive non-cyclic weighted shift matrix. In Section \ref{sec:WSM}, we show that Levinger's function of a positive non-cyclic weighted shift matrix is strictly concave. Cyclicity from a single additional positive element $c_i>0$ allows nonconcavity.
Here we provide an example of nonconcavity using a cyclic shift matrix with \emph{reversible weights}, which have been the subject of recent attention \cite{Chien:and:Nakazato:2020:Symmetry}. Figure \ref{fig:CyclicShift16} shows Levinger's function for a $16 \times 16$ cyclic weighted shift matrix with two-pivot reversible weights
\an{
c_j &= 16 + \sin\left(2 \pi \frac{j}{16}\right), &\quad j = 1, \ldots, 16 . \label{eq:CyclicShift16}
}
Levinger's function is convex for most of the interval $t \in [0,1]$, and is concave only in the small interval around $t=1/2$.
\begin{figure}
\caption{Nonconcave Levinger's function for a $16 \times 16$ two-pivot reversible cyclic weighted shift matrix with weights $c_j = 16 + \sin(2 \pi j /16)$, \eqref{eq:CyclicShift16}
\label{fig:CyclicShift16}
\end{figure}
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\section{Matrices with Concave Levinger's Function}
Here we show that several special classes of nonnegative matrices have concave Levinger's functions: $2\times 2$ matrices, non-cyclic weighted shift matrices, tridiagonal Toeplitz matrices, and Fiedler's 3-parameter Toeplitz matrices.
\subsection{\texorpdfstring{$2 \times 2$}{2 x 2}\ Matrices}
\begin{Theorem}\label{Theorem:2x2}
Let $\A \in \Reals^{2 \times 2}$ be nonnegative and irreducible. Then the spectral radius and Perron-Frobenius eigenvalue $r(t):=r((1{-}t) \A + t \A\tr)$ is concave over $t \in (0,1)$, strictly when $\A$ has different left and right Perron-Frobenius eigenvectors.
\end{Theorem}
\begin{proof}
Let $a, b,c,d \in (0,\infty), t\in (0,1)$,
and assume $b\ne c$ to assure that $\A\ne\A\tr$ and the left and right Perron-Frobenius eigenvectors are not colinear. Let
$$ \A:=\Pmatr{
a & b\\
c & d
},
\quad \B(t):=(1{-}t) \A+t \A\tr.$$
The Perron-Frobenius eigenvalue of $\B(t)$ is obtained by using the quadratic formula to solve the characteristic equation. After some simplification,
$$ r(t):= r(\B(t)) = \frac{a+d+\sqrt{(a-d)^2 + 4t(1{-}t)(b-c)^2 + 4bc }}{2}. $$
The first derivative with respect to $t$ is
$$r'(t)=\frac{{\left(1{-}2\,t\right)}\,{{\left(b-c\right)}}^2 }{\sqrt{(a-d)^2 + 4t(1{-}t)(b-c)^2 + 4bc}}. $$
The denominator above is positive for all $t\in(0,1)$ because of the assumption that $b\ne c$.
The second derivative is, again after some simplification,
\an{\label{eq:f''(t)}
r''(t)=-\frac{2\,(b-c)^2 \,\left((a-d)^2+(b+c)^2\right)}{{{\left((a-d)^2 + 4t(1{-}t)(b-c)^2 + 4bc\right)}}^{3/2} }<0.
}
The numerator in the fraction above is positive because $b\ne c$, and the minus sign in front of the fraction guarantees strict concavity for all $t\in(0,1)$.
\end{proof}
\subsection{Tridiagonal Toeplitz Matrices}
\begin{Theorem}[Tridiagonal Toeplitz Matrices]\label{Theorem:Tridiag}
Let $\A \in \Reals^{n \times n}$, $n \geq 2$, be a tridiagonal Toeplitz matrix with diagonal elements $b\geq 0$, subdiagonal elements $a \geq 0$, and superdiagonal elements $c \geq 0$, with $\max(a, \ c) > 0$. Then for $t \in (0, 1)$, $\spr( (1{-}t) \A + t \A\tr )$ is concave in $t$, increasing on $t \in (0,1/2)$, and decreasing on $t \in (1/2,1)$, all strictly when $a \neq c$.
\end{Theorem}
\begin{proof}
The eigenvalues of a tridiagonal Toeplitz matrix $\A$ with $a, c \neq 0$ are \cite[22-5.18]{Hogben:2014:Handbook} \cite[Theorem 2.4]{Bottcher:and:Grudsky:2005:Spectral}
\an{
\lambda_k(\A) &= b + 2 \sqrt{a c}\ \cos\left(\frac{k \pi}{n{+}1} \right). \label{eq:BG2005}
}
The matrix $(1{-}t) \A + t \A\tr$ has subdiagonal values $(1{-}t) a + t c$ and superdiagonal values $t a + (1{-}t) c$. Since at least one of $a,c$ is strictly positive, $(1{-}t) a + t c > 0$ and $t a + (1{-}t) c > 0$ for $t \in (0,1)$. Therefore \eqref{eq:BG2005} is applicable.
Writing $\lambda_k(t) \eqdef \lambda_k( (1{-}t) \A + t \A\tr)$, we obtain
\ab{
\lambda_k(t) &= b + 2 \sqrt{((1{-}t) a + t c) (t a + (1{-}t) c)}\ \cos\left(\frac{k \pi}{n{+}1} \right).
}
It is readily verified that the first derivatives are
\ab{
\df{}{t}\lambda_k(t) &= \cos\left(\frac{k \pi}{n{+}1} \right)
\frac{(a-c)^2 (1{-}2t)}{\sqrt{((1{-}t) a + t c) (t a + (1{-}t) c)}},
\stext{and the second derivatives are}
\ddf{}{t}\lambda_k(t) &= - \cos\left(\frac{k \pi}{n{+}1} \right)
\frac{(a^2-c^2)^2}{2 \big[ ( (1{-}t) a + t c) (t a + (1{-}t) c )\big]^{3/2} } .
}
Since $(1{-}t) a + t c > 0$ and $t a + (1{-}t) c > 0$ for $t \in (0,1)$, the denominators are positive. When $a=c$ both derivatives are identically zero. When $a \neq c$, the factors not dependent on $k$ are strictly positive for all $t \in (0,1)$ except for $t=1/2$ where the first derivative of all the eigenvalues vanishes.
Because the second derivatives have no sign changes on $t \in (0,1)$, and since $[ (1{-}t) a + t c][t a + (1{-}t) c ] > 0$, there are no inflection points. Therefore each eigenvalue is either convex in $t$ or concave in $t$, depending on the sign of $\cos( k \pi/(n+1) )$. The maximal eigenvalue is
\ab{
r(t) = \lambda_1(t) = b + 2 \sqrt{((1{-}t) a + t c) (t a + (1{-}t) c)}\ \cos(\pi/(n{+}1)) .
}
From its first derivative, since $ \cos(\pi/(n{+}1)) > 0$, $r(t)$ is increasing on $t \in (0, 1/2)$ and decreasing on $t \in (1/2,1)$, strictly when $a \neq c$. Since its second derivative is negative, $r(t)$ is concave in $t$ on $t \in (0,1)$, strictly when $a \neq c$.
\end{proof}
\subsection{Fiedler's Toeplitz Matrices}
Fiedler \cite[p. 180]{Fiedler:1995:Numerical} established this closed formula for the spectral radius of a special Toeplitz matrix.
\begin{Theorem}[Fiedler's 3-Parameter Toeplitz Matrices]\label{Theorem:Fiedler}
Consider a Toeplitz matrix $\A \in \Complex^{n \times n}$, $n \geq 3$,
with diagonal values $(v, 0, \ldots, 0, v, w, u, 0, \ldots, 0, u)$, with $v, w, u \in \Complex$:
\an{\label{eq:FiedlerFlipped}
\A&=\Pmatr{w & u & 0 & \cdots & 0 & u\\
v & w & u & 0 & \cdots & 0\\
0 & v & w & u & \cdots & 0\\
\vdots & \vdots & \ddots & \ddots & \ddots & \vdots \\
0 & 0 & \cdots & v & w & u \\
v & 0 & \cdots & 0 & v & w
}.
}
Let $\omega = e^{2 \pi i / n}$. The eigenvalues of $\A$ are
\ab{
\lambda_{j+1}(\A) &= w + \omega^j u^{(1{-}1/n)} v^{1/n} + \omega^{n-j} u^{1/n} v^{(1{-}1/n)}, \qquad j = 0, 1, \ldots, n{-}1.
}
\end{Theorem}
We apply Theorem \ref{Theorem:Fiedler} to the Levinger function.
\begin{Theorem}\label{Theorem:FiedlerLevinger}
Let $\A$ be defined as in \eqref{eq:FiedlerFlipped} with $u,v,w > 0$. Then $\spr(t) \eqdef \spr( (1{-}t) \A + t \A \tr)$ is concave in $t$ for $t \in (0,1)$, strictly if $u \neq v$.
\end{Theorem}
\begin{proof}
For $u, v, w > 0$, $\spr(\A) = \lambda_1(\A) = w + u^{(1{-}1/n)} v^{1/n} + u^{1/n} v^{(1{-}1/n)}$ from Theorem \ref{Theorem:Fiedler}.
Let $\B(t) = (1{-}t) \A + t \A\tr$. Then $\B(t)$ is again a Toeplitz matrix of the form \eqref {eq:FiedlerFlipped}, with diagonal values $(1{-}t)v{+}t u, 0, \ldots, 0, (1{-}t)v{+}t u, w, (1{-}t)u {+} t v,$ $0, \ldots, 0$, $(1{-}t)u{+}t v$ for matrix elements $A_{i, i{+}m}$, with $m \in \set{1{-}n, n{-}1}$, and $i \in $$\{\max(1, 1{-}m)$, $\ldots$, $\min(n, n{-}m)$$\}$. So again by Theorem \ref{Theorem:Fiedler},
\ab{
\spr(\B(t)) = w &+ [(1{-}t)u + t v]^{(1{-}1/n)} \ [(1{-}t)v+t u]^{1/n} \\ &
+ [(1{-}t)u + t v]^{1/n}\ [(1{-}t)v+t u]^{(1{-}1/n)} .
}
It is readily verified that
\ab{
&\ddf{}{t} \spr(\B(t)) \\
&= - \frac{n-1}{n^2 u^2 v^2} (u-v)^2 (u+v)^2 \\&
\quad \times
\left([(1{-}t)v + u]^{1/n} [(1{-}t)u + t v]^{(1{-}1/n)} + [(1{-}t)v + u]^{(1{-}1/n)} [(1{-}t)u + t v]^{1/n} \right) \\
& \leq 0,
}
with equality if and only if $u = v$.
\end{proof}
With the simple exchange of $A_{1n}$ and $A_{n1}$ in \eqref {eq:FiedlerFlipped}, $\A$ would become a circulant matrix, which has left and right Perron vectors colinear with the vector of all ones, $\ev$, and would therefore have a constant Levinger's function.
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\subsection{Weighted Shift Matrices} \label{sec:WSM}
An $n \times n$ weighted shift matrix, $\A$, has the form
\ab{
A_{ij} = \Cases{
c_i, &\qquad j=i+1, \quad i \in \set{1, \ldots, n-1},\\
0, & \qquad \text{otherwise},
}
}
where $c_i$ are the weights. It is obtained from a cyclic shift matrix be setting any one of the weights to $0$ and appropriately permuting the indices. Unless we explicitly use ``cyclic'', we mean \emph{non-cyclic shift matrix} when we write ``shift matrix''.
We will show that Levinger's function for positive weighted shift matrices is strictly concave. First we develop some lemmas.
\begin{Lemma}\label{Lemma:poly}
Let $\cv \in \Complex^{n+1}$ be a vector of complex numbers and $\alpha \in \Complex$, $\alpha \neq 0$. Then the roots of a polynomial $p(x) = \sum_{k=0}^n x^k \alpha^{n-k} c_k$ are $r_j = \alpha f_j(\cv)$, where $f_j: \Complex^{n+1} \goesto \Complex$, $j = 1, \ldots, n$.
\end{Lemma}
\begin{proof}
We factor and apply the Fundamental Theorem of Algebra to obtain
\ab{
p(x) &
= \sum_{k=0}^n x^k \alpha^{n-k} c_k
= \alpha^n \sum_{k=0}^n \Pfrac{x}{\alpha}^k c_k
= \alpha^n \prod_{j=1}^n \left( \frac{x}{\alpha} - f_j(\cv) \right).
}
Hence the roots of $p(x)$ are $\set{\alpha f_j(\cv)\ |\ j = 1, \ldots, n}$.
\end{proof}
\begin{Lemma}\label{Lemma:ab}
Let $\alpha, \beta \in \Complex \backslash 0$, $\A(\alpha, \beta) = [A_{ij}]$ be a hollow tridiagonal matrix, where $A_{ij} > 0$ for $j=i+1$ and $j=i-1$, $A_{ij}=0$ otherwise, and
\ab{
A_{ij} =
\Cases{
\alpha \, c_{ij}, & \qquad j=i+1, \quad i \in \set{1, \ldots, n-1},\\
\beta \, c_{ij}, & \qquad j=i-1, \quad i \in \set{2, \ldots, n},
}
}
so $\A(\alpha, \beta) $ has the form
\ab {
\A(\alpha,\beta) &=\Pmatr{
0 & \alpha \, c_{12} & 0 & \cdots & 0 & 0 & 0\\
\beta \, c_{21} & 0 & \alpha \, c_{23}& \cdots & 0 & 0 & 0\\
0 & \beta \, c_{32} & 0 & \ddots & 0 &0 & 0\\
\vdots & \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \ddots & 0 & \alpha \, c_ {n{-}2, n{-}1} & 0 \\
0 & 0 & 0 & \cdots & \beta \, c_{n{-}1, n{-}2} & 0 & \alpha \, c_ {n{-}1, n} \\
0 & 0 & 0 & \cdots & 0 & \beta \, c_ {n, n{-}1} & 0
}.
}
Let $\cv \in \Complex^{2(n-1)}$ represent the vector of $c_{ij}$ constants.
Then the eigenvalues of $\A$ are of the form $\sqrt{\alpha \beta}\: f_h(\cv) $, $h = 1, \ldots, n$, where $f_h\suchthat \Complex^{2(n-2)} \goesto \Complex$ are functions of the $c_{ij}$ constants that do not depend on $\alpha$ or $\beta$.
\end{Lemma}
\begin{proof}
The characteristic polynomial of $\A$ is
\ab{
p_\A(\lambda) &=
\det(\lambda\I - \A)\\
&=
\begin{vmatrix}
\lambda & -\alpha \, c_{12} & 0 & \cdots & 0 & 0 & 0\\
-\beta \, c_{21} & \lambda & -\alpha \, c_{23}& \cdots & 0 & 0 & 0\\
0 & -\beta \, c_{32} & \lambda & \ddots & 0 &0 & 0\\
\vdots & \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \ddots & \lambda & -\alpha \, c_ {n{-}2, n{-}1} & 0 \\
0 & 0 & 0 & \cdots & -\beta \, c_{n{-}1, n{-}2} & \lambda & -\alpha \, c_ {n{-}1, n} \\
0 & 0 & 0 & \cdots & 0 & -\beta \, c_ {n, n{-}1} & \lambda
\end{vmatrix}.
}
The characteristic polynomial has the recurrence relation
\an{
p_{\A_k}(\lambda) &
= \lambda \ p_{\A_{k-1}}(\lambda) - \alpha \, \beta \, c_{k,k{-}1} \ c_{k{-}1,k} \ p_{\A_{k-2}}(\lambda), &\quad k \in \set{3, \ldots, n}, \label{eq:rec}\\
\stext{with initial conditions}
p_{\A_2}(\lambda) &= \lambda^2 - \alpha \, \beta \ c_{12}\ c_{21}, \quad\text{and} \label{eq:A2} \\
p_{\A_1}(\lambda) &= \lambda, \label{eq:A1}
}
where $\A_k$ is the principal submatrix of $\A$ over indices $1, \ldots, k$.
We show by induction that for all $k \in \set{2, \ldots, n}$,
\an{
p_{\A_k}(\lambda) &
= \sum_{j=0}^k \lambda^j (\alpha \beta)^{ (k-j)/2} \, g_{jk}(\cv)
= \sum_{j=0}^k \lambda^j \sqrt{\alpha \beta}^{\, (k-j)} g_{jk}(\cv), \label{eq:IH}
}
where each $g_{jk}\suchthat \Complex^{2(n-1)} \goesto \Complex$, $k \in \set{2, \ldots, n}$, $j \in \set{0, \ldots, k}$, is a function of constants $\cv$.
From \eqref{eq:A2}, we see that \eqref{eq:IH} holds for $k=2$:
$
p(\A_2)(\lambda) = \lambda^2 - \alpha \, \beta \, c_{12}\, c_{21} .
$
For $k=3$, from the recurrence relation \eqref{eq:rec} and initial conditions \eqref{eq:A1}, \eqref{eq:A2}, we have
\ab{
p(\A_3)(\lambda) &
= \lambda\, p_{\A_{2}}(\lambda) - \alpha \beta \, c_{32} \, c_{23} \, p_{\A_{1}}(\lambda)
= \lambda (\lambda^2 - \alpha \beta \, c_{12}\, c_{21}) - \alpha \beta \, c_{32} \, c_{23} \, \lambda \\ &
= \lambda^3 - \lambda \sqrt{\alpha \beta}^{\, 2}( c_{12}\, c_{21} + c_{32} \, c_{23}),
}
which satisfies \eqref{eq:IH}. These are the basis steps for the induction.
For the inductive step, we need to show that if \eqref{eq:IH} holds for $k-1, k-2$ then it holds for $k$. Suppose that \eqref{eq:IH} holds for $2 \leq k-1, k-2 \leq n-1$. Then
\ab{
&p_{\A_k}(\lambda)
= \lambda\, p_{\A_{k-1}}(\lambda) - \alpha\beta \, c_{k,k{-}1} \, c_{k{-}1,k} \ p_{\A_{k-2}}(\lambda)\\ &
= \lambda \sum_{j=0}^{k{-}1} \lambda^j \sqrt{\alpha \beta}^{\, (k{-}1{-}j)} g_{j,k{-}1}(\cv)
- \alpha\beta \, c_{k,k{-}1} \, c_{k{-}1,k} \sum_{j=0}^{k-2} \lambda^j \sqrt{\alpha \beta}^{\, (k-2-j)} g_{j,k-2}(\cv) \\ &
= \sum_{j=1}^{k} \lambda^{j} \sqrt{\alpha \beta}^{\,( k{-}j)} g_{j-1,k{-}1}(\cv)
{-} \sum_{j=0}^{k{-}2} \lambda^j \sqrt{\alpha \beta}^{\, (k-j)} c_{k,k{-}1} \, c_{k{-}1,k} \ g_{j,k-2}(\cv) ,
}
which satisfies \eqref{eq:IH}. Thus by induction $p_{\A_n}(\lambda)$ satisfies \eqref{eq:IH}.
Then Lemma \ref{Lemma:poly} implies that the parameters $\set{\alpha, \beta}$ appear
as the linear factor $\sqrt{\alpha \beta}$
in each root of the characteristic polynomial of $\A(\alpha,\beta)$ --- its eigenvalues.
\end{proof}
\begin{Theorem}[Weighted Shift Matrices]\label{Theorem:Shift}
Levinger's function is strictly concave for nonnegative weighted shift matrices with at least one positive weight.
\end{Theorem}
\begin{proof}
Let the positive weighted shift matrix $\A$ be defined as
\ab{
A_{ij} = \Cases{
c_i \geq 0, & \qquad j=i+1, \quad i \in \set{1, \ldots, n-1},\\
0, & \qquad \text{otherwise},
}
}
where $c_i$ are the weights and $c_i >0$ for at least one $i = 1, \ldots, n-1$.
By Lemma \ref{Lemma:ab}, all the eigenvalues of Levinger's homotopy $\B(t) = (1{-}t) \A + t \A\tr$ are of the form $\lambda_i(\B(t)) = \sqrt{t(1{-}t)}\, f_i(\cv)$, where $\cv$ is the vector of weights, and $f_i\suchthat \Reals^{n-1} \goesto \Reals$, since $\B(t)$ is a direct sum of one or more (if some $c_i=0$) Jacobi matrices and these have real eigenvalues \cite[22.7.2]{Hogben:2014:Handbook}.
If at least one weight $c_i$ is positive, then $\B(t)$ has a principal submatrix $\Pmatr{0& (1{-}t) c_i\\t\, c_i&0}$ with a positive spectral radius for $t \in (0,1)$. Thus by \cite[Corollary 8.1.20(a)]{Horn:and:Johnson:2013}, $\spr(\B(t)) > 0$ for $t \in (0,1)$. Therefore for $t \in (0,1)$, $\spr(\B(t)) = \lambda_1(\B(t)) = \sqrt{t(1{-}t)}\: f_1(\cv) > 0$. Since $\sqrt{t(1{-}t)}$ is strictly concave in $t$ for $t \in (0,1)$, Levinger's function is strictly concave in $t$ for $t \in (0,1)$.
\end{proof}
\begin{Corollary}
Levinger's function is strictly concave for a nonnegative hollow tridiagonal matrix, $\A \in \Reals^{n \times n}$, in which $A_{ii}=0$ for $i \in \set{1, \ldots, n}$, and where for each $i \in \set{1, \ldots, n-1}$, $A_{i,i+1}\, A_{i+1,i} = 0$, and for at least one $i$, $A_{i,i+1} > 0$.
\end{Corollary}
\begin{proof}
$\A$ is derived from a weighted shift matrix by swapping some elements of the superdiagonal $A_{i,i+1}$ to the transposed position in the subdiagonal, $A_{i+1,i}$. The determinant of Levinger's homotopy $\det(\lambda \I - \B(t)) = \det(\lambda \I - (1-t) \A - t \A\tr)$ remains unchanged under such swapping because the term $\alpha \beta \, c_{k,k{-}1} \, c_{k{-}1,k}$ in \eqref{eq:rec}, which is $(1-t)t \, c_{k{-}1,k}^2$ in the weighted shift matrix, remains invariant under swapping as $t (1-t) \, c_{k,k{-}1}^2$.
\end{proof}
We complete the connection to positive weighted circuit matrices with this corollary.
\begin{Corollary}
By setting one or more, but not all, of the weights in a positive weighted circuit matrix to $0$, Levinger's function becomes strictly concave.
\end{Corollary}
\begin{proof}
A positive weighted circuit matrix where some but not all of the positive weights are changed to $0$ is, under appropriate permutation of the indices, a nonnegative weighted shift matrix to which Theorem \ref{Theorem:Shift} applies.
\end{proof}
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\begin{Theorem}[Weighted Shift Matrices]\label{Theorem:Shift}
Levinger's function is strictly concave for nonnegative weighted shift matrices with at least one positive weight.
\end{Theorem}
\begin{proof}
Let the positive weighted shift matrix $\A$ be defined as
\ab{
A_{ij} = \Cases{
c_i \geq 0, & \qquad j=i+1, \quad i \in \set{1, \ldots, n-1},\\
0, & \qquad \text{otherwise},
}
}
where $c_i$ are the weights and $c_i >0$ for at least one $i = 1, \ldots, n-1$.
By Lemma \ref{Lemma:ab}, all the eigenvalues of Levinger's homotopy $\B(t) = (1{-}t) \A + t \A\tr$ are of the form $\lambda_i(\B(t)) = \sqrt{t(1{-}t)}\, f_i(\cv)$, where $\cv$ is the vector of weights, and $f_i\suchthat \Reals^{n-1} \goesto \Reals$, since $\B(t)$ is a direct sum of one or more (if some $c_i=0$) Jacobi matrices and these have real eigenvalues \cite[22.7.2]{Hogben:2014:Handbook}.
If at least one weight $c_i$ is positive, then $\B(t)$ has a principal submatrix $\Pmatr{0& (1{-}t) c_i\\t\, c_i&0}$ with a positive spectral radius for $t \in (0,1)$. Thus by \cite[Corollary 8.1.20(a)]{Horn:and:Johnson:2013}, $\spr(\B(t)) > 0$ for $t \in (0,1)$. Therefore for $t \in (0,1)$, $\spr(\B(t)) = \lambda_1(\B(t)) = \sqrt{t(1{-}t)}\: f_1(\cv) > 0$. Since $\sqrt{t(1{-}t)}$ is strictly concave in $t$ for $t \in (0,1)$, Levinger's function is strictly concave in $t$ for $t \in (0,1)$.
\end{proof}
\begin{Corollary}
Levinger's function is strictly concave for a nonnegative hollow tridiagonal matrix, $\A \in \Reals^{n \times n}$, in which $A_{ii}=0$ for $i \in \set{1, \ldots, n}$, and where for each $i \in \set{1, \ldots, n-1}$, $A_{i,i+1}\, A_{i+1,i} = 0$, and for at least one $i$, $A_{i,i+1} > 0$.
\end{Corollary}
\begin{proof}
$\A$ is derived from a weighted shift matrix by swapping some elements of the superdiagonal $A_{i,i+1}$ to the transposed position in the subdiagonal, $A_{i+1,i}$. The determinant of Levinger's homotopy $\det(\lambda \I - \B(t)) = \det(\lambda \I - (1-t) \A - t \A\tr)$ remains unchanged under such swapping because the term $\alpha \beta \, c_{k,k{-}1} \, c_{k{-}1,k}$ in \eqref{eq:rec}, which is $(1-t)t \, c_{k{-}1,k}^2$ in the weighted shift matrix, remains invariant under swapping as $t (1-t) \, c_{k,k{-}1}^2$.
\end{proof}
We complete the connection to positive weighted circuit matrices with this corollary.
\begin{Corollary}
By setting one or more, but not all, of the weights in a positive weighted circuit matrix to $0$, Levinger's function becomes strictly concave.
\end{Corollary}
\begin{proof}
A positive weighted circuit matrix where some but not all of the positive weights are changed to $0$ is, under appropriate permutation of the indices, a nonnegative weighted shift matrix to which Theorem \ref{Theorem:Shift} applies.
\end{proof}
What kind of transition does Levinger's function make during the transition from a cyclic weighted shift matrix with nonconcave Levinger's function to a weighted shift matrix with its necessarily concave Levinger's function, as one of the weights is lowered to $0$?
Does the convexity observed in Figure \ref{fig:CyclicShift16} at the boundaries $t=0$ and $t=1$ flatten and become strictly concave for some positive value of that weight?
We examine this transition for the cyclic shift matrix in example \eqref{eq:CyclicShift16} (Figure \ref{fig:CyclicShift16}). The minimal weight is $c_{12} = 16 + \sin\left(2 \pi \frac{12}{16}\right) = 15$. Figure \ref{fig:C12} plots Levinger's function as $c_{12}$ is divided by factors of $2^{8}$.
Figure \ref{fig:Shift-Matrix_Limit} plots the second derivatives of Levinger's function. We observe non-uniform convergence to the $c_{12}=0$ curve. As $c_{12}$ decreases, the second derivative converges to the $c_{12}=0$ curve over wider and wider intervals of $t$, but outside of these intervals the second derivative \emph{diverges} from the $c_{12}=0$ curve, attaining larger values near and at the boundaries $t=0$ and $t=1$ with smaller $c_{12}$. Meanwhile for $c_{12}=0$, Levinger's function is proportional to $\sqrt{t(1-t)}$, the second derivative of which goes to $-\infty$ as $t$ goes to $0$ or $1$. When $c_{12}> 0$, $\B(0)$ and $\B(1)$ are irreducible, and when $c_{12}=0$, $\B(t)$ is irreducible for $t \in (0,1)$. But for $c_{12}=0$, $\B(0)$ and $\B(1)$ are reducible matrices. While the eigenvalues are always continuous functions of the elements of the matrix, the derivatives of the spectral radius need not be, and in this case, we see an unusual example of nonuniform convergence in the second derivative of the spectral radius.
\begin{figure}
\caption{Levinger's function for the cyclic weighted shift matrix from \eqref{eq:CyclicShift16}
\label{fig:C12}
\end{figure}
\begin{figure}\label{fig:Shift-Matrix_Limit}
\end{figure}
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\section{Matrices with Constant Levinger's Function}
Bapat \cite{Bapat:1987:Two} and Fiedler \cite{Fiedler:1995:Numerical} identified matrices with colinear left and right Perron vectors as having constant Levinger's function. Here we make explicit a property implied by this constraint that appears not to have been described. We use the centered representation of Levinger's homotopy. The \emph{symmetric} part of a square matrix $\A$ is
\an{
\bm{S}(\A) \eqdef (\A + \A\tr) / 2 . \label{eq:SymPart}
}
The \emph{skew symmetric} part of $\A$ is
\an{
\K(\A) \eqdef (\A - \A\tr)/2 . \label{eq:SkewPart}
}
Then $\A = \bm{S}(\A) + \K(\A)$. Levinger's homotopy in this centered representation is now, suppressing the $\A$ argument,
\ab{
\C(p) &\eqdef \bm{S} + p \, \K, \qquad p \in [-1, 1],
\stext{and Levinger's function is}
c(p) &\eqdef r((p+1)/2)
= \spr(\bm{S} + p \, \K).
}
The range of $p$ in this centered representation may be extended beyond $[-1,1]$, while maintaining $\C(p) \geq \0$, to the interval $p \in [-\alpha, \alpha]$ where
\ab{
\alpha = \min_{i,j} \frac{A_{ij} + A_{ji}}{| A_{ji} - A_{ij} | } \geq 1.
}
\begin{Theorem}\label{Theorem:SK}
Let $\A \in \Reals^{n \times n}$ be irreducible and nonnegative. Then $\spr((1{-}t) \A + t \A\tr)$ is constant in $t \in [0,1]$ if and only if the Perron vector of $\A + \A\tr$ is in the null space of $\A - \A\tr$.
\end{Theorem}
\begin{proof}
\cite{Bapat:1987:Two} and \cite{Fiedler:1995:Numerical} proved that $\spr((1{-}t) \A + t \A\tr)$ is constant in $t \in [0,1]$ if and only if the left and right Perron vectors of $\A$ are colinear. Suppose the left and right Perron vectors of $\A$ are colinear. Without loss of generality, they can be normalized to sum to $1$ in which case they are identical. Let the left and right Perron vectors of $\A$ be $\x$.
Then
\ab{
\frac{1}{2} (\A + \A\tr) \x &= \spr(\A) \ \x,
\stext{and}
(\A - \A\tr) \x &= \spr(\A)\ (\x - \x) = \0.
}
Hence $\x$ is the Perron vector of $\A + \A\tr$ and $\x > \0$ is in the null space of $\A - \A\tr$.
For the converse, let the Perron vector of $\A + \A\tr$ be $\x > \0$, and let $\x$ be in the null space of $\A - \A\tr$. Then
\ab{
(\A + \A\tr ) \x &= \spr(\A{+}\A\tr)\ \x
\text{ and }
(\A - \A\tr ) \x = \A \x - \A\tr \x = \0,
}
{which gives}
\ab{
\A \x &
= \frac{1}{2}[(\A + \A\tr ) +( \A - \A\tr)] \x
= \frac{1}{2}\spr(\A{+}\A\tr)\ \x + \0
= \frac{\spr(\A{+}\A\tr)}{2} \ \x
\stext{and}
\A\tr \x &
= \frac{1}{2}[(\A + \A\tr ) - ( \A - \A\tr)] \x
= \frac{1}{2}\spr(\A{+}\A\tr)\ \x - \0
= \frac{\spr(\A{+}\A\tr)}{2}\ \x
}
hence $\x$ is a Perron vector of $\A$ and of $\A\tr$.
\end{proof}
\begin{Corollary}
Let $\bm{S} = \bm{S}\tr \in \Reals^{n\times n}$ be a nonnegative irreducible symmetric matrix, and $\K = - \K\tr \in \Reals^{n\times n}$ be a nonsingular skew symmetric matrix such that $\A= \bm{S} + \K \geq \0$. Then $n$ is even and $\A$ has a non-constant Levinger's function.
\end{Corollary}
\begin{proof}
If $\K$ is a nonsingular skew symmetric matrix, $n$ must be even, since odd-order skew symmetric matrices are always singular \cite[2-9.27]{Hogben:2014:Handbook}. If $\C(p) := \bm{S} + p \, \K$ with $\K$ nonsingular, then because the null space of $\K$ is $\{\0\}$, $\C(p)$ must have a non-constant Levinger's function $c(p)$ by Theorem \ref{Theorem:SK}.
\end{proof}
The following corollary pursues the observation made by an anonymous reviewer that a matrix $\A$ with colinear left and right Perron vectors is orthogonally similar to a direct sum $\Pmatr{\spr(\A)} \oplus \F$ for some square matrix $\F$. This entails that the skew symmetric part of $\A$ is orthogonally similar to $\Pmatr{\spr(\A) - \spr(\A)} \oplus (\F-\F\tr) / 2 = \Pmatr{0} \oplus (\F-\F\tr) / 2$, and is thus singular.
\begin{Corollary}
Let $\bm{S} = \bm{S}\tr \in \Reals^{n\times n}$ be a nonnegative irreducible symmetric matrix, and $\K = - \K\tr \in \Reals^{n\times n}$ be a skew symmetric matrix, such that $\A = \bm{S} + \K \geq \0$. Let $\Q = (\Q\tr)^{-1}$ be an orthogonal matrix that diagonalizes $\bm{S}$ to
\ab{
\Lam &\eqdef \Q\tr \bm{S} \Q = \Pmatr
{\spr(\bm{S}) & 0 & \cdots & 0\\
0 & \lambda_2 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \lambda_n
} .
}
Then $\A$ has a constant Levinger's function if and only if
\an{\label{eq:KQP}
\K_1 &\eqdef \Q\tr\K \Q = \Pmatr
{0 & \0\tr \\
\0 &\K_2
} = \Pmatr{0} \oplus \K_2,
}
where $\K_2 = - \K_2\tr \in \Reals^ {n{-}1 \times n{-}1} $ and $\0\tr = (0 \ldots 0) \in \Reals^{n{-}1}$.
\end{Corollary}
\begin{proof}
Since $\bm{S}$ is real and symmetric, $\bm{S} = \Q \Lam \Q\tr$ is in Jordan canonical form. Let $\x > \0$ be the normalized Perron vector of $\bm{S}$. Then $\x = [\Q]_1$ is the first column of $\Q$, and the other columns of $\Q$ are orthogonal to $\x$, so $\x\tr \Q = (1\:0 \cdots 0)$. The necessary and sufficient condition from Theorem \ref{Theorem:SK} for $\A$ to have constant Levinger's function is that $\x\tr\K = \0\tr$, equivalent to
\ab{
\x\tr \K &
= \x\tr \Q\K_1\Q\tr
= (1\: 0 \cdots 0) \K_1\Q\tr
= \0\tr.
}
Since $\Q$ is orthogonal, it has null space $\{\0\}$, so $ (1\: 0 \cdots 0) \K_1\Q\tr = \0\tr$ if and only if $ (1\: 0 \cdots 0) \K_1 = \0\tr$, which is the top row of $\K_1$. $\K_1$ and $\K_2$ must be skew symmetric since $\K$ is skew symmetric, as can be seen immediately from transposition. The skew symmetry of $\K_1$ implies its first column must also be all zeros as its first row is, establishing the form given in \eqref{eq:KQP}.
\end{proof}
\section{Conclusions}
We have shown that it is not in general true that the spectral radius along a line from a nonnegative square matrix $\A$ to its transpose --- Levinger's function --- is concave. Our counterexamples to concavity have a simple principle in the case of direct sums of block matrices, namely, that the maximum of two concave functions need not be concave. However, for the other examples we present --- Toeplitz matrices, and positive circuit or cyclic weighted shift matrices --- whatever principles underly the nonconcavity remain to be discerned. Also remaining to be discerned are the properties of matrix families --- a few of which we have presented here --- that guarantee concave Levinger functions. A general characterization of the range of $t$ for which the spectral radius is concave in Levinger's homotopy remains an open problem.
\section*{Biographical Note}
Bernard W. Levinger (Berlin, Germany, September 3, 1928 -- Fort Collins, Colorado, USA, January 17, 2020) and his family fled Nazi Germany to England in 1936, to Mexico in 1940, and to the United States in 1941, which initially placed them in an immigration prison and deported them to Mexico, but which ultimately allowed their immigration, whereupon they settled in New York City. Levinger graduated from Bronx High School of Science and earned a doctorate in mathematics from New York University. He was Professor of Mathematics and Professor Emeritus at Colorado State University, Fort Collins. He leaves a large family, including his wife Lory of more than 65 years.\cite{Levinger:1928-2020}
\end{document}
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\begin{document}
\chapter*{The spread of finite and infinite groups \\ {\normalfont\large Scott Harper\footnotemark}}
\footnotetext[1]{School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK \newline \url{[email protected]}}
\section*{Abstract}\trivlist\item[]
It is well known that every finite simple group has a generating pair. Moreover, Guralnick and Kantor proved that every finite simple group has the stronger property, known as $\frac{3}{2}$-generation, that every nontrivial element is contained in a generating pair. Much more recently, this result has been generalised in three different directions, which form the basis of this survey article. First, we look at some stronger forms of $\frac{3}{2}$-generation that the finite simple groups satisfy, which are described in terms of spread and uniform domination. Next, we discuss the recent classification of the finite $\frac{3}{2}$-generated groups. Finally, we turn our attention to infinite groups, focusing on the recent discovery that the finitely presented simple groups of Thompson are also $\frac{3}{2}$-generated, as are many of their generalisations. Throughout the article we pose open questions in this area, and we highlight connections with other areas of group theory.
\varepsilonndtrivlist\addvspace{26pt}
\section{Introduction} \label{s:intro}
Every finite simple group can be generated by two elements. This well-known result was proved for most finite simple groups by Steinberg in 1962 \cite{ref:Steinberg62} and completed via the Classification of Finite Simple Groups (see \cite{ref:AschbacherGuralnick84}). Much more is now known about generating pairs for finite simple groups. For instance, for any nonabelian finite simple group $G$, almost all pairs of elements generate $G$ \cite{ref:KantorLubotzky90,ref:LiebeckShalev95}, $G$ has an invariable generating pair \cite{ref:GuralnickMalle12JLMS,ref:KantorLubotzkyShalev11}, and, with only finitely many exceptions, $G$ can be generated by a pair of elements where one has order $2$ and the other has order either $3$ or $5$ \cite{ref:LiebeckShalev96Ann,ref:LubeckMalle99}.
The particular generation property of finite simple groups that this survey focuses on was established by Guralnick and Kantor \cite{ref:GuralnickKantor00} and independently by Stein \cite{ref:Stein98}. They proved that if $G$ is a finite simple group, then every nontrivial element of $G$ is contained in a generating pair. Groups with this property are said to be \varepsilonmph{$\frac{3}{2}$-generated}. We will survey the recent work (mostly from the past five years) that addresses natural questions arising from this theorem.
Section~\ref{s:finite} focuses on finite groups and considers recent progress towards answering two natural questions. Do finite simple groups satisfy stronger versions of $\frac{3}{2}$-generation? Which other finite groups are $\frac{3}{2}$-generated? Regarding the first, in Sections~\ref{ss:finite_spread} and~\ref{ss:finite_udn}, we will meet two strong versions of $\frac{3}{2}$-generation, namely (uniform) spread and total/uniform domination. Regarding the second, Section~\ref{ss:finite_bgh} presents the recent classification of the finite $\frac{3}{2}$-generated groups established by Burness, Guralnick and Harper in 2021 \cite{ref:BurnessGuralnickHarper21}. All these ideas are brought together as we discuss the generating graph in Section~\ref{ss:finite_graph}. Section~\ref{ss:finite_app} rounds off the first half by highlighting applications of spread to word maps, the product replacement graph and the soluble radical of a group.
Section~\ref{s:infinite} focuses on infinite groups and, in particular, whether any results on the $\frac{3}{2}$-generation of finite groups extend to the realm of infinite groups. After discussing this in general terms in Sections~\ref{ss:infinite_intro} and~\ref{ss:infinite_soluble}, our focus shifts to the finitely presented infinite simple groups of Richard Thompson in Sections~\ref{ss:infinite_thompson_introduction} to~\ref{ss:infinite_thompson_t}. Here we survey the ongoing work of Bleak, Donoven, Golan, Harper, Hyde and Skipper, which reveals strong parallels between the $\frac{3}{2}$-generation of these infinite simple groups and the finite simple groups. Section~\ref{ss:infinite_thompson_introduction} serves as an introduction to Thompson's groups for any reader unfamiliar with them.
This survey is based on my one-hour lecture at \varepsilonmph{Groups St Andrews 2022} at the University of Newcastle, and I thank the organisers for the opportunity to present at such an enjoyable and interesting conference. I have restricted this survey to the subject of spread and have barely discussed other aspects of generation. Even regarding the spread of finite simple groups, much more could be said, especially regarding the methods involved in proving the results. Both of these omissions from this survey are discussed amply in Burness' survey article from \varepsilonmph{Groups St Andrews 2017} \cite{ref:Burness19}, which is one reason for deciding to focus in this article on the progress made in the past five years.
\textbf{Acknowledgements. } The author wrote this survey when he was first a Heilbronn Research Fellow and then a Leverhulme Early Career Fellow, and he thanks the Heilbronn Institute for Mathematical Research and the Leverhulme Trust. He thanks Tim Burness, Charles Cox, Bob Guralnick, Jeremy Rickard and a referee for their helpful comments, and he also thanks Guralnick for his input on Application~3, especially his suggested proof of Theorem~\ref{thm:x_radical}.
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\section{Finite Groups} \label{s:finite}
\subsection{Generating pairs} \label{ss:finite_intro}
It is easy to write down a pair of generators for each alternating group $A_n$: for instance, if $n$ is odd, then $A_n = \langle (1 \, 2 \, 3), (1 \, 2 \, \dots \, n) \rangle$. In 1962, Steinberg \cite{ref:Steinberg62} proved that every finite simple group of Lie type is $2$-generated, by exhibiting an explicit pair of generators. In light of the Classification of Finite Simple Groups, once the sporadic groups were all shown to be $2$-generated, it became known that every finite simple group is $2$-generated \cite{ref:AschbacherGuralnick84}. Since then, numerous stronger versions of this theorem have been proved (see Burness' survey \cite{ref:Burness19}).
Even as early as 1962, Steinberg raised the possibility of stronger versions of his $2$-generation result \cite{ref:Steinberg62}:
\trivlist\item[]\small
``It is possible that one of the generators can be chosen of order 2, as is the case for the projective unimodular group, or even that one of the generators can be chosen as an arbitrary element other than the identity, as is the case for the alternating groups. Either of these results, if true, would quite likely require methods much more detailed than those used here.''
\varepsilonndtrivlist\normalsize
That is, Steinberg is suggesting the possibility that for a finite simple group $G$ one might be able to replace just the existence of $x,y \in G$ such that $\langle x, y \rangle = G$, with the stronger statement that for all nontrivial elements $x \in G$ there exists $y \in G$ such that $\langle x,y \rangle = G$. He alludes to the fact that this much stronger condition is known to hold for the alternating groups, which was shown by Piccard in 1939 \cite{ref:Piccard39}. In the following example, we will prove this result on alternating groups, but with different methods than Piccard used.
\begin{example} \label{ex:alternating}
Let $G = A_n$ for $n \mathfrak{g}eq 5$. We will focus on the case $n \varepsilonquiv 0 \mod{4}$ and then address the remaining cases at the end.
Write $n=4m$ and let $s$ have cycle shape $[2m-1,2m+1]$, that is, let $s$ be a product of disjoint cycles of lengths $2m-1$ and $2m+1$. Visibly, $s$ is contained in a maximal subgroup $H \leqslant G$ of type $(\mathbb{S}m{2m-1} \times \mathbb{S}m{2m+1}) \cap G$. We claim that no further maximal subgroups of $G$ contain $s$. Imprimitive maximal subgroups are ruled out since $2m-1$ and $2m+1$ are coprime. In addition, a theorem of Marggraf \cite[Theorem~13.5]{ref:Wielandt64} ensures that no proper primitive subgroup of $A_n$ contains a $k$-cycle for $k < \frac{n}{2}$, so $s$ is contained in no primitive maximal subgroups as a power of $s$ is a $(2m-1)$-cycle.
Now let $x$ be an arbitrary nontrivial element of $G$. Choosing $g$ such that $x$ moves some point from the $(2m-1)$-cycle of $s^g$ to a point in the $(2m+1)$-cycle of $s^g$ gives $x \not\in H^g$. This means that no maximal subgroup of $G$ contains both $x$ and $s^g$, so $\langle x, s^g \rangle = G$. In particular, every nontrivial element of $G$ is contained in a generating pair.
We now address the other cases, but we assume that $n \mathfrak{g}eq 25$ for exposition. If $n \varepsilonquiv 2 \mod{4}$, then we choose $s$ with cycle shape $[2m-1,2m+3]$ (where $n=4m+2$) and proceed as above but now the unique maximal overgroup has type $(\mathbb{S}m{2m-1} \times \mathbb{S}m{2m+3}) \cap G$. A similar argument works for odd $n$. Here $s$ has cycle shape $[m-2,m,m+2]$ if $n=3m$, $[m+1,m+1,m-1]$ if $n=3m+1$ and $[m+2,m,m]$ if $n=3m+2$, and the only maximal overgroups of $s$ are the three obvious intransitive ones. For each $1 \neq x \in G$, it is easy to find $g \in G$ such that $x$ misses all three maximal overgroups of $s^g$ and hence deduce that $\langle x, s^g \rangle = G$.
\varepsilonnd{example}
In 2000, Guralnick and Kantor \cite{ref:GuralnickKantor00} gave a positive answer to the longstanding question of Steinberg by proving the following.
\begin{theorem} \label{thm:guralnick_kantor}
Let $G$ be a finite simple group. Then every nontrivial element of $G$ is contained in a generating pair.
\varepsilonnd{theorem}
We say that a group $G$ is \varepsilonmph{$\frac{3}{2}$-generated} if every nontrivial element of $G$ is contained in a generating pair. The author does not know the origin of this term, but it indicates that the class of $\frac{3}{2}$-generated groups includes the class of $1$-generated groups and is included in the class of $2$-generated groups. This is somewhat analogous to the class of $\frac{3}{2}$-transitive permutation groups introduced by Wielandt \cite[Section~10]{ref:Wielandt64}, which is included in the class of $1$-transitive groups and includes the class of $2$-transitive groups.
Let us finish this section by briefly turning from simple groups to simple Lie algebras. Here we have a theorem of Ionescu \cite{ref:Ionescu76}, analogous to Theorem~\ref{thm:guralnick_kantor}.
\begin{theorem}\label{thm:ionescu}
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$. Then for all $x \in \mathfrak{g} \setminus 0$ there exists $y \in \mathfrak{g}$ such that $x$ and $y$ generate $\mathfrak{g}$ as a Lie algebra.
\varepsilonnd{theorem}
In fact, Bois \cite{ref:Bois09} proved that every classical finite dimensional simple Lie algebra in characteristic other than 2 or 3 has this $\frac{3}{2}$-generation property, but Goldstein and Guralnick \cite{ref:GoldsteinGuralnick} have proved that $\mathfrak{sl}_n$ in characteristic 2 does not.
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\subsection{Spread} \label{ss:finite_spread}
Let us now introduce the concept that gives this article its name.
\begin{definition} \label{def:spread}
Let $G$ be a group.
\begin{enumerate}
\item The \varepsilonmph{spread} of $G$, written $s(G)$, is the supremum over integers $k$ such that for any $k$ nontrivial elements $x_1, \dots, x_k \in G$ there exists $y \in G$ such that $\langle x_1, y \rangle = \cdots = \langle x_k, y \rangle = G$.
\item The \varepsilonmph{uniform spread} of $G$, written $u(G)$, is the supremum over integers $k$ for which there exists $s \in G$ such that for any $k$ nontrivial elements $x_1, \dots, x_k \in G$ there exists $y \in s^G$ such that $\langle x_1, y \rangle = \cdots = \langle x_k, y \rangle = G$.
\varepsilonnd{enumerate}
\varepsilonnd{definition}
The term spread was introduced by Brenner and Wiegold in 1975 \cite{ref:BrennerWiegold75}, but the term uniform spread was not formally introduced until 2008 \cite{ref:BreuerGuralnickKantor08}.
Note that $s(G) > 0$ if and only if every nontrivial element of $G$ is contained in a generating pair. Therefore, spread gives a way of quantifying how strongly a group is $\frac{3}{2}$-generated. Uniform spread captures the idea that the complementary element $y$, while depending on the elements $x_1, \dots, x_k$, can be chosen somewhat uniformly for all choices of $x_1, \dots, x_k$: it can always be chosen from the same prescribed conjugacy class. In Section~\ref{ss:finite_udn}, we will see a way of measuring how much more uniformity in the choice of $y$ we can insist on. Observe that Example~\ref{ex:alternating} actually shows that $u(A_n) \mathfrak{g}eq 1$ for all $n \mathfrak{g}eq 5$.
By Theorem~\ref{thm:guralnick_kantor}, every finite simple group $G$ satisfies $s(G) > 0$. What more can be said about the (uniform) spread of finite simple groups? The main result is the following proved by Breuer, Guralnick and Kantor \cite{ref:BreuerGuralnickKantor08}.
\begin{theorem}\label{thm:breuer_guralnick_kantor}
\hspace{-2mm} Let $G$ be a nonabelian finite simple group. Then $s(G) \mathfrak{g}eq u(G) \mathfrak{g}eq 2$. Moreover, $s(G) = 2$ if and only if $u(G)=2$ if and only if
\[
G \in \{ A_5, A_6, \Omega^+_8(2) \} \cup \{ {\rm Sp}_{2m}(2) \mid m \mathfrak{g}eq 3 \}.
\]
\varepsilonnd{theorem}
The asymptotic behaviour of (uniform) spread is given by the following theorem of Guralnick and Shalev \cite[Theorem~1.1]{ref:GuralnickShalev03}. The version of this theorem stated in \cite{ref:GuralnickShalev03} is given just in terms of spread, but the result given here follows immediately from their proof (see \cite[Lemma~2.1--Corollary~2.3]{ref:GuralnickShalev03}).
\begin{theorem}\label{thm:guralnick_shalev}
Let $(G_i)$ be a sequence of nonabelian finite simple groups such that $|G_i| \to \infty$. Then $s(G_i) \to \infty$ if and only if $u(G_i) \to \infty$ if and only if $(G_i)$ has no infinite subsequence consisting of either
\begin{enumerate}
\item alternating groups of degree all divisible by a fixed prime
\item symplectic groups over a field of fixed even size or odd-dimensional orthogonal groups over a field of fixed odd size.
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
Given that $s(G_i) \to \infty$ if and only if $u(G_i) \to \infty$, we ask the following. (Note that $s(G) - u(G)$ can be arbitrarily large, see Theorem~\ref{thm:spread_psl2}(iv) for example.)
\begin{question} \label{que:spread_uniform}
Does there exist a constant $c$ such that for all nonabelian finite simple groups $G$ we have $s(G) \leqslant c \cdot u(G)$?
\varepsilonnd{question}
There are explicit upper bounds that justify the exceptions in parts~(i) and~(ii) of Theorem~\ref{thm:guralnick_shalev}. Indeed, $s(\mathrm{Sp}_{2m}(q)) \leqslant q$ for even $q$ and $s(\Omega_{2m+1}(q)) \leqslant \frac{1}{2}(q^2+q)$ for odd $q$ (see \cite[Proposition~2.5]{ref:GuralnickShalev03} for a geometric proof). For alternating groups of composite degree $n > 4$, if $p$ is the least prime divisor of $n$, then $s(A_n) \leqslant \binom{2p+1}{3}$ (see \cite[Proposition~2.4]{ref:GuralnickShalev03} for a combinatorial proof). For even-degree alternating groups, the situation is clear: $s(A_n)=4$, but much less is known in odd degrees (see \cite[Section~3.1]{ref:GuralnickShalev03} for partial results).
\begin{question} \label{que:spread_alt}
What is the (uniform) spread of $A_n$ when $n$ is odd?
\varepsilonnd{question}
The spread of even-degree alternating groups was determined by Brenner and Wiegold in the paper where they first introduced the notion of spread. They also studied the spread of two-dimensional linear groups, but their claimed value for $s(\mathrm{PSL}_2(q))$ was only proved to be a lower bound. Further work by Burness and Harper demonstrates that this is not an upper bound when $q \varepsilonquiv 3 \mod{4}$, where they prove the following (see \cite[Theorem~5 \& Remark~5]{ref:BurnessHarper20}).
\begin{theorem} \label{thm:spread_psl2}
Let $G = \mathrm{PSL}_2(q)$ with $q \mathfrak{g}eq 11$.
\begin{enumerate}
\item If $q$ is even, then $s(G) = u(G) = q-2$.
\item If $q \varepsilonquiv 1 \mod{4}$, then $s(G) = u(G) = q-1$.
\item If $q \varepsilonquiv 3 \mod{4}$, then $s(G) \mathfrak{g}eq q-3$ and $u(G) \mathfrak{g}eq q-4$.
\item If $q \varepsilonquiv 3 \mod{4}$ is prime, then $s(G) \mathfrak{g}eq \frac{1}{2}(3q-7)$ and $s(G)-u(G) = \frac{1}{2}(q+1)$.
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
\begin{question} \label{que:spread_psl2}
What is the (uniform) spread of $\mathrm{PSL}_2(q)$ when $q \varepsilonquiv 3 \mod{4}$?
\varepsilonnd{question}
In short, determining the spread of simple groups is difficult. We conclude by commenting that the precise value of the spread of only two sporadic groups is known, namely $s(\mathrm{M}_{11}) = 3$ \cite{ref:Woldar07} (see also \cite{ref:BradleyHolmes07}) and $s(\mathrm{M}_{23}) = 8064$ \cite{ref:BradleyHolmes07, ref:Fairbairn12JGT}.
In contrast, the exact spread and uniform spread of symmetric groups is known. In a series of papers in the late 1960s \cite{ref:Binder68,ref:Binder70,ref:Binder70MZ,ref:Binder73}, Binder determined the spread of $\mathbb{S}m{n}$ and also showed that $u(\mathbb{S}m{n}) \mathfrak{g}eq 1$ unless $n \in \{4,6\}$ (Binder used different terminology). However, the uniform spread of symmetric groups was only completely determined in a 2021 paper of Burness and Harper \cite{ref:BurnessHarper20}; indeed, showing that $u(\mathbb{S}m{n}) \mathfrak{g}eq 2$ for even $n > 6$ involves both a long combinatorial argument and a CFSG-dependent group theoretic argument (see \cite[Theorem~3 \& Remark~3]{ref:BurnessHarper20}). We say more on $\mathbb{S}m{6}$ in Example~\ref{ex:spread_s6}.
\begin{theorem} \label{thm:spread_sym}
Let $G = \mathbb{S}m{n}$ with $n \mathfrak{g}eq 5$. Then
\[
s(G) = \left\{\begin{array}{ll}
2 & \text{if $n$ is even} \\
3 & \text{if $n$ is odd} \\
\varepsilonnd{array} \right.\quad \text{and} \quad
u(G) = \left\{\begin{array}{ll}
0 & \text{if $n=6$} \\
2 & \text{otherwise.} \\
\varepsilonnd{array} \right.
\]
\varepsilonnd{theorem}
\textbf{Methods. A probabilistic approach. } As we turn to discuss the key method behind these results, we return to Example~\ref{ex:alternating} where we proved that $u(G) \mathfrak{g}eq 1$ when $G = A_n$ for even $n > 6$. We found an element $s \in G$ contained in a unique maximal subgroup $H$ of $G$. Since $G$ is simple, $H$ is corefree, so $\bigcap_{g \in G} H^g = 1$, which means that for each nontrivial $x \in G$ there exists $g \in G$ such that $x \not\in H^g$. This implies that $\langle x, s^g \rangle = G$, so $s^G$ witnesses $u(G) \mathfrak{g}eq 1$. This argument can be generalised in two ways: one yields Lemma~\ref{lem:spread}, giving a better lower bound on the uniform spread of $G$ and the other yields Lemma~\ref{lem:udn_bases}, pertaining to the uniform domination number of $G$, which we will meet in the next section.
Lemma~\ref{lem:spread} takes a probabilistic approach, so we need some notation. For a finite group $G$ and elements $x,s \in G$, we write
\begin{equation} \label{eq:q}
Q(x,s) = \frac{|\{ y \in s^G \mid \langle x,y \rangle \neq G \}|}{|s^G|},
\varepsilonnd{equation}
which is the probability a uniformly random conjugate of $s$ does not generate with $x$, and write $\mathcal{M}(G,s)$ for the set of maximal subgroups of $G$ that contain $s$.
\begin{lemma} \label{lem:spread}
Let $G$ be a finite group and let $s \in G$.
\begin{enumerate}
\item For $x \in G$,
\[
Q(x,s) \leqslant \sum_{H \in \mathcal{M}(G,s)}^{} \frac{|x^G \cap H|}{|x^G|}.
\]
\item For a positive integer $k$, if $Q(x,s) < \frac{1}{k}$ for all prime order elements $x \in G$, then $u(G) \mathfrak{g}eq k$ is witnessed by $s^G$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{proof}
For (i), let $x \in G$. Then $\langlex,s^g\rangle \neq G$ if and only if $x \in H^g$, or equivalently $x^{g^{-1}} \in H$, for some $H \in \mathcal{M}(G,s)$. Therefore, \[
Q(x,s) = \frac{|\{y \in s^G \mid \langle x, y \rangle \neq G \}|}{|s^G|} \leqslant \sum_{H \in \mathcal{M}(G,s)} \frac{|x^G \cap H|}{|x^G|}.
\]
For (ii), fix $k$. To prove that $u(G) \mathfrak{g}eq k$ is witnessed by $s^G$, it suffices to prove that for all elements $x_1,\dots,x_k \in G$ of prime order there exists $y \in s^G$ such that $\langlex_i,y\rangle=G$ for all $1 \leqslant i \leqslant k$. Therefore, let $x_1,\dots,x_k \in G$ have prime order. If $Q(x_i,s) < \frac{1}{k}$ for all $1 \leqslant i \leqslant k$, then
\[
\frac{|\{y \in s^G \mid \text{$\langle x_i, y \rangle = G$ for all $1 \leqslant i \leqslant k$} \}|}{|s^G|} \mathfrak{g}eq 1 - \sum_{i=1}^{k}Q(x_i,s) > 0,
\]
so there exists $y \in s^G$ such that $\langlex_i,y\rangle=G$ for all $1 \leqslant i \leqslant k$.
\varepsilonnd{proof}
Therefore, to obtain lower bounds on the uniform spread (and hence spread) of a finite group, it is enough to (a) identify an element whose maximal overgroups $H$ are tightly constrained, and then (b) for each such $H$ and for all prime order $x \in G$, bound the quantity $\frac{|x^G \cap H|}{|x^G|}$.
The ratio $\frac{|x^G \cap H|}{|x^G|}$ is the well-studied \varepsilonmph{fixed point ratio}. More precisely, $\frac{|x^G \cap H|}{|x^G|}$ is nothing other than the proportion of points in $G/H$ fixed by $x$ in the natural action of $G$ on $G/H$. These fixed point ratios, in the context of primitive actions of almost simple groups, have seen many applications via probabilistic methods, not just to spread, but also to base sizes (e.g. the Cameron--Kantor conjecture) and monodromy groups (e.g. the Guralnick--Thompson conjecture), see Burness' survey article \cite{ref:Burness18}.
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\begin{theorem} \label{thm:spread_sym}
Let $G = \mathbb{S}m{n}$ with $n \mathfrak{g}eq 5$. Then
\[
s(G) = \left\{\begin{array}{ll}
2 & \text{if $n$ is even} \\
3 & \text{if $n$ is odd} \\
\varepsilonnd{array} \right.\quad \text{and} \quad
u(G) = \left\{\begin{array}{ll}
0 & \text{if $n=6$} \\
2 & \text{otherwise.} \\
\varepsilonnd{array} \right.
\]
\varepsilonnd{theorem}
\textbf{Methods. A probabilistic approach. } As we turn to discuss the key method behind these results, we return to Example~\ref{ex:alternating} where we proved that $u(G) \mathfrak{g}eq 1$ when $G = A_n$ for even $n > 6$. We found an element $s \in G$ contained in a unique maximal subgroup $H$ of $G$. Since $G$ is simple, $H$ is corefree, so $\bigcap_{g \in G} H^g = 1$, which means that for each nontrivial $x \in G$ there exists $g \in G$ such that $x \not\in H^g$. This implies that $\langle x, s^g \rangle = G$, so $s^G$ witnesses $u(G) \mathfrak{g}eq 1$. This argument can be generalised in two ways: one yields Lemma~\ref{lem:spread}, giving a better lower bound on the uniform spread of $G$ and the other yields Lemma~\ref{lem:udn_bases}, pertaining to the uniform domination number of $G$, which we will meet in the next section.
Lemma~\ref{lem:spread} takes a probabilistic approach, so we need some notation. For a finite group $G$ and elements $x,s \in G$, we write
\begin{equation} \label{eq:q}
Q(x,s) = \frac{|\{ y \in s^G \mid \langle x,y \rangle \neq G \}|}{|s^G|},
\varepsilonnd{equation}
which is the probability a uniformly random conjugate of $s$ does not generate with $x$, and write $\mathcal{M}(G,s)$ for the set of maximal subgroups of $G$ that contain $s$.
\begin{lemma} \label{lem:spread}
Let $G$ be a finite group and let $s \in G$.
\begin{enumerate}
\item For $x \in G$,
\[
Q(x,s) \leqslant \sum_{H \in \mathcal{M}(G,s)}^{} \frac{|x^G \cap H|}{|x^G|}.
\]
\item For a positive integer $k$, if $Q(x,s) < \frac{1}{k}$ for all prime order elements $x \in G$, then $u(G) \mathfrak{g}eq k$ is witnessed by $s^G$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{proof}
For (i), let $x \in G$. Then $\langlex,s^g\rangle \neq G$ if and only if $x \in H^g$, or equivalently $x^{g^{-1}} \in H$, for some $H \in \mathcal{M}(G,s)$. Therefore, \[
Q(x,s) = \frac{|\{y \in s^G \mid \langle x, y \rangle \neq G \}|}{|s^G|} \leqslant \sum_{H \in \mathcal{M}(G,s)} \frac{|x^G \cap H|}{|x^G|}.
\]
For (ii), fix $k$. To prove that $u(G) \mathfrak{g}eq k$ is witnessed by $s^G$, it suffices to prove that for all elements $x_1,\dots,x_k \in G$ of prime order there exists $y \in s^G$ such that $\langlex_i,y\rangle=G$ for all $1 \leqslant i \leqslant k$. Therefore, let $x_1,\dots,x_k \in G$ have prime order. If $Q(x_i,s) < \frac{1}{k}$ for all $1 \leqslant i \leqslant k$, then
\[
\frac{|\{y \in s^G \mid \text{$\langle x_i, y \rangle = G$ for all $1 \leqslant i \leqslant k$} \}|}{|s^G|} \mathfrak{g}eq 1 - \sum_{i=1}^{k}Q(x_i,s) > 0,
\]
so there exists $y \in s^G$ such that $\langlex_i,y\rangle=G$ for all $1 \leqslant i \leqslant k$.
\varepsilonnd{proof}
Therefore, to obtain lower bounds on the uniform spread (and hence spread) of a finite group, it is enough to (a) identify an element whose maximal overgroups $H$ are tightly constrained, and then (b) for each such $H$ and for all prime order $x \in G$, bound the quantity $\frac{|x^G \cap H|}{|x^G|}$.
The ratio $\frac{|x^G \cap H|}{|x^G|}$ is the well-studied \varepsilonmph{fixed point ratio}. More precisely, $\frac{|x^G \cap H|}{|x^G|}$ is nothing other than the proportion of points in $G/H$ fixed by $x$ in the natural action of $G$ on $G/H$. These fixed point ratios, in the context of primitive actions of almost simple groups, have seen many applications via probabilistic methods, not just to spread, but also to base sizes (e.g. the Cameron--Kantor conjecture) and monodromy groups (e.g. the Guralnick--Thompson conjecture), see Burness' survey article \cite{ref:Burness18}.
To address task (a), one applies the well-known and extensive literature on the subgroup structure of almost simple groups. For (b), one appeals to the bounds on fixed point ratios of primitive actions of almost simple groups, the most general of which is \cite[Theorem~1]{ref:LiebeckSaxl91} of Liebeck and Saxl. This states that
\begin{equation}\label{eq:fpr}
\frac{|x^G \cap H|}{|x^G|} \leqslant \frac{4}{3q}
\varepsilonnd{equation}
for any almost simple group of Lie type $G$ over $\mathbb{F}_q$, maximal subgroup $H \leqslant G$ and nontrivial element $x \in G$, with known exceptions. This is essentially best possible, since $\frac{|x^G \cap H|}{|x^G|} \approx q^{-1}$ when $q$ is odd, $G = \mathrm{PGL}_n(q)$, $H$ is the stabiliser of a $1$-space of $\mathbb{F}_q^n$ and $x$ lifts to the diagonal matrix $[-1,1,1, \dots, 1] \in \mathrm{GL}_n(q)$. However, there are much stronger bounds that take into account the particular group $G$, subgroup $H$ or element $x$ (see \cite[Section~2]{ref:Burness18} for a survey).
Bounding uniform spread via Lemma~\ref{lem:spread} was the approach introduced by Guralnick and Kantor in their 2000 paper \cite{ref:GuralnickKantor00} where they prove that $u(G) \mathfrak{g}eq 1$ for all nonabelian finite simple groups $G$. Clearly this approach also easily yields further probabilistic information and we refer the reader to Burness' survey article \cite{ref:Burness19} for much more on this approach. We will give just one example, which we will return to later in the article (see \cite[Example~3.9]{ref:Burness19}).
\begin{example} \label{ex:spread_e8}
Let $G = E_8(q)$ and let $s$ generate a cyclic maximal torus of order $\Phi_{30}(q) = q^8+q^7-q^5-q^4-q^3+q+1$. Weigel proved that $\mathcal{M}(G,s) = \{ H \}$ where $H = N_G(\langles\rangle) = \langles\rangle:30$ (see \cite[Section~4(j)]{ref:Weigel92}). Applying Lemma~\ref{lem:spread} with the bound in \varepsilonqref{eq:fpr}, for all nontrivial $x \in G$ we have $u(G) \mathfrak{g}eq 1$ since
\[
\sum_{H \in \mathcal{M}(G,s)} \frac{|x^G \cap H|}{|x^G|} \leqslant \frac{4}{3q} \leqslant \frac{2}{3} < 1.
\]
However, we can do better: $|x^G \cap H| \leqslant |H| \leqslant q^{14}$ and $|x^G| > q^{58}$ for all nontrivial elements $x \in G$, so $u(G) \mathfrak{g}eq q^{44}$ since
\[
\sum_{H \in \mathcal{M}(G,s)} \frac{|x^G \cap H|}{|x^G|} < \frac{1}{q^{44}}.
\]
\varepsilonnd{example}
While the overwhelming majority of results on (uniform) spread are established via the probabilistic method encapsulated in Lemma~\ref{lem:spread}, there are cases where this approach fails, as the following example highlights.
\begin{example} \label{ex:spread_sp}
Let $m \mathfrak{g}eq 3$ and let $G = \mathrm{Sp}_{2m}(2)$. By Theorem~\ref{thm:breuer_guralnick_kantor}, we know that $u(G)=2$. However, if $x$ is a transvection, then $Q(x,s) > \frac{1}{2}$ for all $s \in G$. This is proved in \cite[Proposition~5.4]{ref:BreuerGuralnickKantor08}, and we give an indication of the proof. Every element of $G = \mathrm{Sp}_{2m}(2)$ is contained in a subgroup of type ${\rm O}^+_{2m}(2)$ or ${\rm O}^-_{2m}(2)$ (see \cite{ref:Dye79}, for example).
Assume that $s$ is contained in a subgroup $H \cong {\rm O}^-_{2m}(2)$. The groups $\mathrm{Sp}_{2m}(2)$ and ${\rm O}^\varphim_{2m}(2)$ contain $2^{2m}-1$ and $2^{2m-1} \mp 2^{m-1}$ transvections, respectively, so
\[
Q(x,s) \mathfrak{g}eq \frac{2^{2m-1}+2^{m-1}}{2^{2m}-1} = \frac{2^{m-1}}{2^m-1} > \frac{1}{2}.
\]
A more involved argument gives $Q(x,s) > \frac{1}{2}$ if $s$ is contained in a subgroup of type ${\rm O}^+_{2m}(2)$ but none of type ${\rm O}^-_{2m}(2)$, relying on $s$ being reducible here.
\varepsilonnd{example}
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\subsection{Uniform domination} \label{ss:finite_udn}
We began by observing that any finite simple group $G$ is $\frac{3}{2}$-generated, that is
\begin{equation} \label{eq:generation}
\text{for all $x \in G \setminus 1$ there exists $y \in G$ such that $\langlex,y\rangle = G$.}
\varepsilonnd{equation}
We then looked to strengthen \varepsilonqref{eq:generation} by increasing the scope of the first quantifier. Recall that the \varepsilonmph{spread} of $G$, denoted $s(G)$, is the greatest $k$ such that
\[
\text{for all $x_1, \dots, x_k \in G$ there exists $y \in G$ such that $\langlex_1,y\rangle = \cdots = \langle x_k,y\rangle = G$.}
\]
We also have a related notion: the \varepsilonmph{uniform spread} of $G$, denoted $u(G)$, is the greatest $k$ for which there exists an element $s \in G$ such that
\[
\text{for all $x_1, \dots, x_k \in G$ there exists $y \in s^G$ such that $\langlex_1,y\rangle = \cdots = \langle x_k,y\rangle = G$.}
\]
The notion of uniform spread inspires us to strengthen \varepsilonqref{eq:generation} by narrowing the range of the second quantifier. That is, we say that the \varepsilonmph{total domination number} of $G$, denoted $\mathfrak{g}amma_t(G)$, is the least size of a subset $S \subseteq G$ such that
\[
\text{for all $x \in G \setminus 1$ there exists $y \in S$ such that $\langlex,y\rangle = G$.}
\]
Again we have a related notion: the \varepsilonmph{uniform domination number} of $G$, denoted $\mathfrak{g}amma_u(G)$, is the least size of a subset $S \subseteq G$ of conjugate elements such that
\[
\text{for all $x \in G \setminus 1$ there exists $y \in S$ such that $\langlex,y\rangle = G$.}
\]
These latter two concepts were introduced by Burness and Harper in \cite{ref:BurnessHarper19} and studied further in \cite{ref:BurnessHarper20}. The terminology is motivated by the generating graph (see Section~\ref{ss:finite_graph}).
Let $G$ be a nonabelian finite simple group. Clearly $2 \leqslant \mathfrak{g}amma_t(G) \leqslant \mathfrak{g}amma_u(G)$, and since $u(G) \mathfrak{g}eq 1$, there exists a conjugacy class $s^G$ such that $\mathfrak{g}amma_u(G) \leqslant |s^G|$. However, the class exhibited in Guralnick and Kantor's proof of $u(G) \mathfrak{g}eq 1$ is typically very large (for groups of Lie type, $s$ is usually a regular semisimple element), so it is natural to seek tighter upper bounds on $\mathfrak{g}amma_u(G)$. The following result of Burness and Harper does this \cite[Theorems~2, 3 \& 4]{ref:BurnessHarper19} (see \cite[Theorem~4(i)]{ref:BurnessHarper20} for the refined upper bound in (iii)).
\begin{theorem} \label{thm:udn}
Let $G$ be a nonabelian finite simple group.
\begin{enumerate}
\item If $G = A_n$, then $\mathfrak{g}amma_u(G) \leqslant 77 \log_2{n}$.
\item If $G$ is classical of rank $r$, then $\mathfrak{g}amma_u(G) \leqslant 7r+70$.
\item If $G$ is exceptional, then $\mathfrak{g}amma_u(G) \leqslant 5$.
\item If $G$ is sporadic, then $\mathfrak{g}amma_u(G) \leqslant 4$.
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
In this generality, these bounds are optimal up to constants. For example, if $n \mathfrak{g}eq 6$ is even, then $\log_2{n} \leqslant \mathfrak{g}amma_t(A_n) \leqslant \mathfrak{g}amma_u(A_n) \leqslant 2 \log_2{n}$, and if $G$ is $\mathrm{Sp}_{2r}(q)$ with $q$ even or $\Omega_{2r+1}(q)$ with $q$ odd, then $r \leqslant \mathfrak{g}amma_t(G) \leqslant \mathfrak{g}amma_u(G) \leqslant 7r$ \cite[Theorems~3(i) \& 6.3(iii)]{ref:BurnessHarper20}. Regarding the bounds in (iii) and (iv), for sporadic groups, $\mathfrak{g}amma_u(G) = 4$ is witnessed by $G = {\rm M}_{11}$ \cite[Theorem~3]{ref:BurnessHarper19}, but the best lower bound for exceptional groups is $\mathfrak{g}amma_u(G) \mathfrak{g}eq 3$ given by $G = F_4(q)$ \cite[Lemma~6.17]{ref:BurnessHarper20}.
\begin{question} \label{que:udn_alt}
Does there exist a constant $c$ such that for all $n \mathfrak{g}eq 5$ we have $\log_p{n} \leqslant \mathfrak{g}amma_t(A_n) \leqslant \mathfrak{g}amma_u(A_n) \leqslant c \log_p{n}$ where $p$ is the least prime divisor of $n$?
\varepsilonnd{question}
By \cite[Theorem~4(ii)]{ref:BurnessHarper20}, we know that $\mathfrak{g}amma_t(A_n) \mathfrak{g}eq \log_p{n}$, so to provide an affirmative answer to Question~\ref{que:udn_alt}, it suffices to prove that $\mathfrak{g}amma_u(A_n) \leqslant c \log_p{n}$.
\begin{question} \label{que:udn_lie}
Does there exist a constant $c$ such that for all finite simple groups of Lie type $G$ other than $\mathrm{Sp}_{2r}(q)$ with $q$ even and $\Omega_{2r+1}(q)$ with $q$ odd, we have $\mathfrak{g}amma_u(G) \leqslant c$?
\varepsilonnd{question}
By Theorem~\ref{thm:udn}, to answer Question~\ref{que:udn_lie}, it suffices to consider classical groups of large rank, and it was shown in \cite[Theorem~6.3(ii)]{ref:BurnessHarper19} that $c=15$ suffices for some families of these groups. Affirmative answers to Questions~\ref{que:udn_alt} and~\ref{que:udn_lie} would answer Question~\ref{que:udn_tdn} too.
\begin{question} \label{que:udn_tdn}
Does there exist a constant $c$ such that for all nonabelian finite simple groups $G$ we have $\mathfrak{g}amma_u(G) \leqslant c \cdot \mathfrak{g}amma_t(G)$?
\varepsilonnd{question}
The smallest possible value of $\mathfrak{g}amma_u(G)$ is $2$ (since $G$ is not cyclic), and an almost complete classification of when this is achieved was given in \cite[Corollary~7]{ref:BurnessHarper20}.
\begin{theorem} \label{thm:udn_two}
Let $G$ be a nonabelian finite simple group. Then $\mathfrak{g}amma_u(G) = 2$ only if $G$ is one of the following
\begin{enumerate}
\item $A_n$ for prime $n \mathfrak{g}eq 13$
\item $\mathrm{PSL}_2(q)$ for odd $q \mathfrak{g}eq 11$
\item[] $\mathrm{PSL}^\varepsilon_n(q)$ for odd $n$, but not $n=3$ with $(q,\varepsilon) \in \{ (2,+), (4,+), (3,-), (5,-) \}$
\item[] ${\rm PSp}_{4m+2}(q)^\ast$ for odd $q$ and $m \mathfrak{g}eq 2$, and ${\rm P}\Omega^\varphim_{4m}(q)^\ast$ for $m \mathfrak{g}eq 2$
\item ${}^2B_2(q)$, ${}^2G_2(q)$, ${}^2F_4(q)$, ${}^3D_4(q)$, ${}^2E_6(q)$, $E_6(q)$, $E_7(q)$ , $E_8(q)$
\item ${\rm M}_{23}$, ${\rm J}_1$, ${\rm J}_4$, ${\rm Ru}$, ${\rm Ly}$, ${\rm O'N}$, ${\rm Fi}_{23}$, ${\rm Th}$, $\mathbb{B}$, $\mathbb{M}$ or ${\rm J}_3^\ast$, ${\rm He}^\ast$, ${\rm Co}_1^\ast$, ${\rm HN}^\ast$.
\varepsilonnd{enumerate}
Moreover, $\mathfrak{g}amma_u(G) = 2$ in all the cases without an asterisk.
\varepsilonnd{theorem}
We will say that a subset $S \subseteq G$ of conjugate elements of $G$ is a \varepsilonmph{uniform dominating set} of $G$ if for all nontrivial $x \in G$ there exists $y \in S$ such that $\langle x, y \rangle = G$, so $\mathfrak{g}amma_u(G)$ is the smallest size of a uniform dominating set of $G$. For groups $G$ such that $\mathfrak{g}amma_u(G) = 2$, we know that there exists a uniform dominating set of size two. How abundant are such subsets? To this end, let $P(G,s,2)$ be the probability that two random conjugates of $s$ form a uniform dominating set for $G$, and let $P(G) = \max\{ P(G,s,2) \mid s \in G \}$. Then we have the following probabilistic result \cite[Corollary~8 \& Theorem~9]{ref:BurnessHarper20}.
\begin{theorem} \label{thm:udn_prob}
Let $(G_i)$ be a sequence of nonabelian finite simple groups such that $|G_i| \to \infty$. Assume that $\mathfrak{g}amma_u(G_i) = 2$, and $G_i \not\in \{ {\rm PSp}_{4m+2}(q) \mid \text{odd $q$, $m \mathfrak{g}eq 2$}\} \cup \{ {\rm P}\Omega^\varphim_{4m}(q) \mid \text{$m \mathfrak{g}eq 2$} \} \cup \{ {\rm J}_3, {\rm He}, {\rm Co}_1, {\rm HN} \}$. Then
\[
P(G_i) \to \left\{ \begin{array}{ll} \frac{1}{2} & \text{if $G = \mathrm{PSL}_2(q)$} \\ 1 & \text{otherwise.} \varepsilonnd{array} \right.
\]
Moreover, $P(G_i) \leqslant \frac{1}{2}$ if and only if $G_i =\mathrm{PSL}_2(q)$ for $q \varepsilonquiv 3 \mod{4}$ or $G_i \in \{ A_{13}, {\rm PSU}_5(2), {\rm Fi}_{23} \}$.
\varepsilonnd{theorem}
\textbf{Methods. Bases of permutation groups. }
Let us now discuss the methods used in \cite{ref:BurnessHarper19,ref:BurnessHarper20} to bound $\mathfrak{g}amma_u(G)$. Here there is a very pleasing connection with an entirely different topic in permutation group theory: bases. For a group $G$ acting faithfully on a set $\Omega$, a subset $B \subseteq \Omega$ is a \varepsilonmph{base} if the pointwise stabiliser $G_{(B)}$ is trivial. Since $G$ acts faithfully, the entire domain $\Omega$ is a base, so we naturally ask for the smallest size of a base, which we call the \varepsilonmph{base size} $b(G,\Omega)$. To turn this combinatorial notion into an algebraic one, we observe that when $G$ acts on $G/H$, a subset $\{ Hg_1, \dots, Hg_c \}$ is a base if and only if $\cap_{i=1}^c H^{g_i} = 1$, so $b(G,G/H)$ is the smallest number of conjugates of $H$ whose intersection is trivial.
Bases have been studied for over a century, and the base size has been at the centre of several recently proved conjectures, such as Pyber's conjecture that there is a constant $c$ such that $\frac{\log|G|}{\log|\Omega|} \leqslant b(G, \Omega) \leqslant c \frac{\log|G|}{\log|\Omega|}$ for all primitive groups $G \leqslant \mathrm{Sym}(\Omega)$ (see \cite{ref:DuyanHalasiMaroti18}), and Cameron's conjecture that $b(G, \Omega) \leqslant 7$ for nonstandard primitive almost simple groups $G \leqslant \mathrm{Sym}(\Omega)$ (see \cite{ref:BurnessLiebeckShalev09}). There is an ambitious ongoing programme of work, initiated by Saxl, to provide a complete classification of the primitive groups $G \leqslant \mathrm{Sym}(\Omega)$ with $b(G,\Omega) = 2$. There are numerous partial results in this direction, and we give just one, as we will use it below. Burness and Thomas \cite{ref:BurnessThomas} proved that if $G$ is a simple group of Lie type and $T$ is a maximal torus, then $b(G,G/N_G(T)) = 2$ apart from a few known low rank exceptions.
The following result is the bridge that connects bases with uniform domination (see \cite[Corollaries~2.2 \&~2.3]{ref:BurnessHarper19}).
\begin{lemma} \label{lem:udn_bases}
Let $G$ be a finite group and let $s \in G$.
\begin{enumerate}
\item Assume that $\mathcal{M}(G,s) = \{ H \}$ and $H$ is corefree. Then the smallest uniform dominating set $S \subseteq s^G$ satisfies $|S|=b(G,G/H)$.
\item Assume that $H \in \mathcal{M}(G,s)$ is corefree. Then every uniform dominating set $S \subseteq s^G$ satisfies $|S| \mathfrak{g}eq b(G,G/H)$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{proof}
For (i), note that $x \in H$ if and only if $\langle x, s \rangle \neq G$. Hence, $\{ s^{g_1}, \dots, s^{g_c} \}$ is a uniform dominating set if and only if $\bigcap_{i=1}^{c} H^{g_i} = 1$, or said otherwise, if and only if $\{ g_1, \dots, g_c \}$ is a base for $G$ acting on $G/H$. The result follows.
For (ii), if $x \in H$, then $\langle x, s \rangle \neq G$. Therefore, if $\{ s^{g_1}, \dots, s^{g_c} \}$ is a uniform dominating set, then $\bigcap_{i=1}^{c} H^{g_i} = 1$, so $\{ g_1, \dots, g_c \}$ is a base for $G$ acting on $G/H$ and, consequently, $c \mathfrak{g}eq b(G, G/H)$.
\varepsilonnd{proof}
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\begin{theorem} \label{thm:udn_prob}
Let $(G_i)$ be a sequence of nonabelian finite simple groups such that $|G_i| \to \infty$. Assume that $\mathfrak{g}amma_u(G_i) = 2$, and $G_i \not\in \{ {\rm PSp}_{4m+2}(q) \mid \text{odd $q$, $m \mathfrak{g}eq 2$}\} \cup \{ {\rm P}\Omega^\varphim_{4m}(q) \mid \text{$m \mathfrak{g}eq 2$} \} \cup \{ {\rm J}_3, {\rm He}, {\rm Co}_1, {\rm HN} \}$. Then
\[
P(G_i) \to \left\{ \begin{array}{ll} \frac{1}{2} & \text{if $G = \mathrm{PSL}_2(q)$} \\ 1 & \text{otherwise.} \varepsilonnd{array} \right.
\]
Moreover, $P(G_i) \leqslant \frac{1}{2}$ if and only if $G_i =\mathrm{PSL}_2(q)$ for $q \varepsilonquiv 3 \mod{4}$ or $G_i \in \{ A_{13}, {\rm PSU}_5(2), {\rm Fi}_{23} \}$.
\varepsilonnd{theorem}
\textbf{Methods. Bases of permutation groups. }
Let us now discuss the methods used in \cite{ref:BurnessHarper19,ref:BurnessHarper20} to bound $\mathfrak{g}amma_u(G)$. Here there is a very pleasing connection with an entirely different topic in permutation group theory: bases. For a group $G$ acting faithfully on a set $\Omega$, a subset $B \subseteq \Omega$ is a \varepsilonmph{base} if the pointwise stabiliser $G_{(B)}$ is trivial. Since $G$ acts faithfully, the entire domain $\Omega$ is a base, so we naturally ask for the smallest size of a base, which we call the \varepsilonmph{base size} $b(G,\Omega)$. To turn this combinatorial notion into an algebraic one, we observe that when $G$ acts on $G/H$, a subset $\{ Hg_1, \dots, Hg_c \}$ is a base if and only if $\cap_{i=1}^c H^{g_i} = 1$, so $b(G,G/H)$ is the smallest number of conjugates of $H$ whose intersection is trivial.
Bases have been studied for over a century, and the base size has been at the centre of several recently proved conjectures, such as Pyber's conjecture that there is a constant $c$ such that $\frac{\log|G|}{\log|\Omega|} \leqslant b(G, \Omega) \leqslant c \frac{\log|G|}{\log|\Omega|}$ for all primitive groups $G \leqslant \mathrm{Sym}(\Omega)$ (see \cite{ref:DuyanHalasiMaroti18}), and Cameron's conjecture that $b(G, \Omega) \leqslant 7$ for nonstandard primitive almost simple groups $G \leqslant \mathrm{Sym}(\Omega)$ (see \cite{ref:BurnessLiebeckShalev09}). There is an ambitious ongoing programme of work, initiated by Saxl, to provide a complete classification of the primitive groups $G \leqslant \mathrm{Sym}(\Omega)$ with $b(G,\Omega) = 2$. There are numerous partial results in this direction, and we give just one, as we will use it below. Burness and Thomas \cite{ref:BurnessThomas} proved that if $G$ is a simple group of Lie type and $T$ is a maximal torus, then $b(G,G/N_G(T)) = 2$ apart from a few known low rank exceptions.
The following result is the bridge that connects bases with uniform domination (see \cite[Corollaries~2.2 \&~2.3]{ref:BurnessHarper19}).
\begin{lemma} \label{lem:udn_bases}
Let $G$ be a finite group and let $s \in G$.
\begin{enumerate}
\item Assume that $\mathcal{M}(G,s) = \{ H \}$ and $H$ is corefree. Then the smallest uniform dominating set $S \subseteq s^G$ satisfies $|S|=b(G,G/H)$.
\item Assume that $H \in \mathcal{M}(G,s)$ is corefree. Then every uniform dominating set $S \subseteq s^G$ satisfies $|S| \mathfrak{g}eq b(G,G/H)$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{proof}
For (i), note that $x \in H$ if and only if $\langle x, s \rangle \neq G$. Hence, $\{ s^{g_1}, \dots, s^{g_c} \}$ is a uniform dominating set if and only if $\bigcap_{i=1}^{c} H^{g_i} = 1$, or said otherwise, if and only if $\{ g_1, \dots, g_c \}$ is a base for $G$ acting on $G/H$. The result follows.
For (ii), if $x \in H$, then $\langle x, s \rangle \neq G$. Therefore, if $\{ s^{g_1}, \dots, s^{g_c} \}$ is a uniform dominating set, then $\bigcap_{i=1}^{c} H^{g_i} = 1$, so $\{ g_1, \dots, g_c \}$ is a base for $G$ acting on $G/H$ and, consequently, $c \mathfrak{g}eq b(G, G/H)$.
\varepsilonnd{proof}
Let us explain how Lemma~\ref{lem:udn_bases} applies. Part~(i) gives an upper bound: if we can find $s \in G$ such that $\mathcal{M}(G,s) = \{H\}$ and $b(G,G/H) \leqslant c$, then $\mathfrak{g}amma_u(G) \leqslant c$. Part~(ii) gives a lower bound: if we can show that for all $s \in G$ there exists $H \in \mathcal{M}(G,s)$ with $b(G,G/H) \mathfrak{g}eq c$, then $\mathfrak{g}amma_u(G) \mathfrak{g}eq c$. We give two examples to show how we do this in practice.
\begin{example} \label{ex:udn_e8}
Let $G = E_8(q)$ and let $s$ generate a cyclic maximal torus of order $\Phi_{30}(q) = q^8+q^7-q^5-q^4-q^3+q+1$. As noted in Example~\ref{ex:spread_e8}, $\mathcal{M}(G,s) = \{ H \}$ where $H$ is the normaliser of the torus $\langles\rangle$. Now, applying Burness and Thomas' result \cite[Theorem~1]{ref:BurnessThomas} mentioned above, we see that $b(G,G/H)=2$, so Lemma~\ref{lem:udn_bases} implies that $\mathfrak{g}amma_u(G) = 2$.
\varepsilonnd{example}
\begin{example} \label{ex:udn_an}
Let $n > 6$ be even and let $G = A_n$. We will give upper and lower bounds on $\mathfrak{g}amma_u(G)$ via Lemma~\ref{lem:udn_bases}.
Seeking an upper bound on $\mathfrak{g}amma_u(G)$, let $s = (1 \, 2 \, \dots \, l)(l+1 \, l+2 \, \dots \, n)$ where $l \in \{\frac{n}{2}-1, \frac{n}{2}-2\}$ is odd. As we showed in Example~\ref{ex:alternating}, $\mathcal{M}(G,s) = \{H\}$ where $H \cong (\mathbb{S}m{l} \times \mathbb{S}m{n-l}) \cap A_n$. The action of $A_n$ on $A_n/H$ is just the action of $A_n$ on the set of $l$-subsets of $\{1, 2, \dots, n\}$. The base size of this action was studied by Halasi, and by \cite[Theorem~4.2]{ref:Halasi12}, we have $b(G,G/H) \leqslant \left\lceil \log_{\lceil n/l \rceil} n \right\rceil \cdot (\lceil n/l \rceil - 1) \leqslant 2 \log_2{n}$. Applying Lemma~\ref{lem:udn_bases}(i) gives $\mathfrak{g}amma_u(G) \leqslant 2\log_2{n}$.
Turning to a lower bound, note that every element of $G$ is contained in a subgroup $K$ of type $(\mathbb{S}m{k} \times \mathbb{S}m{n-k}) \cap A_n$ for some $0 < k < n$. By \cite[Theorem~3.1]{ref:Halasi12}, we have $b(G, G/K) \mathfrak{g}eq \log_2{n}$. Applying Lemma~\ref{lem:udn_bases}(ii) gives $\mathfrak{g}amma_u(G) \mathfrak{g}eq \log_2{n}$.
\varepsilonnd{example}
We now address the general case where $s$ is not contained in a unique maximal subgroup of $G$. In the spirit of how uniform spread was studied, a probabilistic approach is adopted. Write $Q(G,s,c)$ for the probability that a random $c$-tuple of elements of $s^G$ does not give a uniform dominating set of $G$ and write $\mathcal{P}(G)$ for the set of prime order elements of $G$. The main lemma is \cite[Lemma~2.5]{ref:BurnessHarper19}.
\begin{lemma} \label{lem:udn_prob}
Let $G$ be a finite group, let $s \in G$ and let $c$ be a positive integer.
\begin{enumerate}
\item For all positive integers $c$, we have
\[
Q(G,s,c) \leqslant \sum_{x \in \mathcal{P}(G)} \left(\sum_{H \in \mathcal{M}(G,s)}\frac{|x^G \cap H|}{|x^G|}\right)^c.
\]
\item For a positive integer $c$, if $Q(G,s,c) < 1$, then $\mathfrak{g}amma_u(G) \leqslant c$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{proof}
Part~(ii) is immediate. For part~(i), $\{s^{g_1}, \dots, s^{g_c}\}$ is not a uniform dominating set of $G$ if and only if there exists a prime order element $x \in G$ such that $\langle x, s^{g_i} \rangle \neq G$ for all $1 \leqslant i \leqslant c$. Since $Q(x,s)$ is the probability that $x$ does not generate $G$ with a random conjugate of $s$ (see \varepsilonqref{eq:q}), this implies that $Q(G,s,c) \leqslant \sum_{x \in \mathcal{P}(G)}^{} Q(x,s)^c$. The result follows from Lemma~\ref{lem:spread}(i).
\varepsilonnd{proof}
Could Lemma~\ref{lem:udn_prob} yield a better bound than Lemma~\ref{lem:udn_bases} when $s$ satisfies $\mathcal{M}(G,s) = \{H\}$? In this case, $Q(G,s,c)$ is nothing other than the probability that a random $c$-tuple of elements of $G/H$ form a base and $\sum_{x \in \mathcal{P}(G)} \left(\frac{|x^G \cap H|}{|x^G|}\right)^c$ is the upper bound for $Q(G,s,c)$ used, first by Liebeck and Shalev \cite{ref:LiebeckShalev99} and then by numerous others since, to obtain upper bounds on the base size $b(G,G/H)$. Therefore, Lemma~\ref{lem:udn_prob} has nothing new to offer in this special case.
We conclude with an example of Lemma~\ref{lem:udn_prob} in action. This establishes a (typical) special case of Theorem~\ref{thm:udn}(ii).
\begin{example} \label{ex:udn_prob}
Let $n \mathfrak{g}eq 10$ be even and let $G = \mathrm{PSL}_n(q)$. We proceed similarly to Example~\ref{ex:udn_an}, by fixing odd $l \in \{\frac{n}{2}-1, \frac{n}{2}-2\}$ and then letting $s$ lift to a block diagonal matrix \scalebox{0.75}{$\left( \begin{array}{cc} A & 0 \\ 0 & B \varepsilonnd{array} \right)$} where $A \in \mathrm{SL}_l(q)$ and $B \in \mathrm{SL}_{n-l}(q)$ are irreducible. The order of $s$ is divisible by a \varepsilonmph{primitive prime divisor} of $q^{n-l}-1$ (a prime divisor coprime to $q^k-1$ for each $1 \leqslant k < n-l$). Using the framework of Aschbacher's theorem on the subgroup structure of classical groups \cite{ref:Aschbacher84}, Guralnick, Penttila, Praeger and Saxl, classify the subgroups of $\mathrm{GL}_n(q)$ that contain an element whose order is a primitive prime divisor of $q^m-1$ when $m > \frac{n}{2}$ \cite{ref:GuralnickPenttilaPraegerSaxl97}. With this we deduce that $\mathcal{M}(G,s) = \{ G_U, G_V \}$ where $U$ and $V$ are the obviously stabilised subspaces of dimension $l$ and $n-l$, respectively. The fixed point ratio for classical groups acting on the set of $k$-subspaces of their natural module was studied by Guralnick and Kantor, and by \cite[Proposition~3.1]{ref:GuralnickKantor00}, if $G = \mathrm{PSL}_n(q)$ and $H$ is the stabiliser of a $k$-subspace of $\mathbb{F}_q^n$, then $\frac{|x^G \cap H|}{|x^G|} < \frac{2}{q^k}$. Applying Lemma~\ref{lem:udn_prob} gives $\mathfrak{g}amma_u(G) \leqslant 2n+15$ since
\[
Q(G,s,2n+15) \leqslant \sum_{x \in \mathcal{P}(G)} \left(\sum_{H \in \mathcal{M}(G,s)}\frac{|x^G \cap H|}{|x^G|}\right)^{2n+15} \!\!\leqslant q^{n^2-1} \left( 2 \cdot \frac{2}{q^{\frac{n}{2}-2}} \right)^{2n+15} \!\!< 1.
\]
\varepsilonnd{example}
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\subsection{The spread of a finite group} \label{ss:finite_bgh}
We now look beyond finite simple groups and ask the general question: for which finite groups $G$ is every nontrivial element contained in a generating pair? Brenner and Wiegold's original 1975 paper gives a comprehensive answer for finite soluble groups (see \cite[Theorem~2.01]{ref:BrennerWiegold75} for even more detail).
\begin{theorem} \label{thm:brenner_wiegold}
Let $G$ be a finite soluble group. The following are equivalent:
\begin{enumerate}
\item $s(G) \mathfrak{g}eq 1$
\item $s(G) \mathfrak{g}eq 2$
\item every proper quotient of $G$ is cyclic.
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
The equivalence of (i) and (ii) shows that $s(G)=1$ for no finite soluble groups $G$, so Brenner and Wiegold asked the following question \cite[Problem~1.04]{ref:BrennerWiegold75}.
\begin{question} \label{que:brenner_wiegold}
Which finite groups $G$ satisfy $s(G) = 1$? In particular, are there perhaps only finitely many such groups?
\varepsilonnd{question}
The condition in (iii) is necessary for every nontrivial element of $G$ to be contained in a generating pair, and this is true for an arbitrary group $G$. To see this, assume that every nontrivial element of $G$ is contained in a generating pair. Let $1 \neq N \trianglelefteqslant G$ and let $1 \neq n \in N$. Then there exists $g \in G$ such that $G = \langle n, g \rangle$, so $G/N = \langleNn, Ng\rangle = \langleNg\rangle$, which is cyclic. In 2008, Breuer, Guralnick and Kantor conjectured that this condition is also sufficient for all finite groups \cite[Conjecture~1.8]{ref:BreuerGuralnickKantor08}.
\begin{conjecture} \label{con:breuer_guralnick_kantor}
Let $G$ be a finite group. Then $s(G) \mathfrak{g}eq 1$ if and only if every proper quotient of $G$ is cyclic.
\varepsilonnd{conjecture}
Completing a long line of research in this direction, both Question~\ref{que:brenner_wiegold} and Conjecture~\ref{con:breuer_guralnick_kantor} were settled by Burness, Guralnick and Harper in 2021 \cite{ref:BurnessGuralnickHarper21}.
\begin{theorem} \label{thm:burness_guralnick_harper}
Let $G$ be a finite group. Then $s(G) \mathfrak{g}eq 2$ if and only if every proper quotient of $G$ is cyclic.
\varepsilonnd{theorem}
\begin{corollary} \label{cor:burness_guralnick_harper}
No finite group $G$ satisfies $s(G)=1$.
\varepsilonnd{corollary}
The next example (which is \cite[Remark~2.16]{ref:BurnessGuralnickHarper21}) shows that Theorem~\ref{thm:burness_guralnick_harper} does not hold if spread is replaced with uniform spread (recall Theorem~\ref{thm:spread_sym}).
\begin{example} \label{ex:spread_s6}
Let $G = \mathbb{S}m{n}$ where $n \mathfrak{g}eq 6$ is even. Suppose that $u(G) > 0$ is witnessed by the class $s^G$. Since a conjugate of $s$ generates with $(1 \, 2 \, 3)$, $s$ must be an odd permutation. Since a conjugate of $s$ generates with $(1 \, 2)$, $s$ must have at most two cycles. Since $n$ is even, it follows that $s$ is an $n$-cycle. However, if $n=6$ and $a \in \mathrm{Aut}(G) \setminus G$, then $s^a \in (1 \, 2 \, 3)(4 \, 5)^G$ also witnesses $u(G) > 0$, which is a contradiction. Therefore, $u(\mathbb{S}m{6})=0$.
\varepsilonnd{example}
However, Example~\ref{ex:spread_s6} is essentially the only obstacle to a result for uniform spread analogous to Theorem~\ref{thm:burness_guralnick_harper} on spread. Indeed, Burness, Guralnick and Harper gave the following complete description of the finite groups $G$ with $u(G) < 2$ \cite[Theorem~3]{ref:BurnessGuralnickHarper21}. (Note that the uniform spread of abelian groups $G$ is not interesting: $u(G) = \infty$ if $G$ is cyclic and $u(G) = 0$ otherwise.)
\begin{theorem} \label{thm:burness_guralnick_harper_uniform}
Let $G$ be a nonabelian finite group such that every proper quotient is cyclic. Then
\begin{enumerate}
\item $u(G) = 0$ if and only if $G = \mathbb{S}m{6}$
\item $u(G) = 1$ if and only if the group $G$ has a unique minimal normal subgroup $N = T_1 \times \cdots \times T_k$ where $k \mathfrak{g}eq 2$ and where $T_i = A_6$ and $N_G(T_i)/C_G(T_i) = \mathbb{S}m{6}$ for all $1 \leqslant i \leqslant k$.
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
Theorem~\ref{thm:burness_guralnick_harper_uniform} emphasises the anomalous behaviour of $\mathbb{S}m{6}$: it is the only almost simple group $G$ where every proper quotient of $G$ is cyclic but $u(G) < 2$.
\textbf{Methods. A reduction theorem and Shintani descent. } We now outline the proof of Theorems~\ref{thm:burness_guralnick_harper} and~\ref{thm:burness_guralnick_harper_uniform} in \cite{ref:BurnessGuralnickHarper21}. We need to consider the finite groups $G$ all of whose proper quotients are cyclic. In light of Theorem~\ref{thm:brenner_wiegold}, we will assume that $G$ is insoluble, so $G$ has a unique minimal normal subgroup $T^k$ for a nonabelian simple group $T$, and we can assume that $G = \langle T^k, s \rangle$ where $s = (a, 1, \dots, 1)\sigma \in \mathrm{Aut}(T^k)$ for $a \in \mathrm{Aut}(T)$ and $\sigma = (1 \, 2 \, \dots \, k) \in \mathbb{S}m{k}$. Let us ignore $T = A_6$ due to the complications we have already seen in this case.
The first major step in the proof of Theorems~\ref{thm:burness_guralnick_harper} and~\ref{thm:burness_guralnick_harper_uniform} is the following reduction theorem \cite[Theorem~2.13]{ref:BurnessGuralnickHarper21}.
\begin{theorem} \label{thm:bgh_reduction}
Fix a nonabelian finite simple group $T$ and assume that $T \neq A_6$. Fix $s = (a, 1, \dots, 1)\sigma \in \mathrm{Aut}(T^k)$ with $a \in \mathrm{Aut}(T)$ and $\sigma = (1 \, 2 \, \dots \, k) \in \mathbb{S}m{k}$. Then $s$ witnesses $u(\langleT^k,s\rangle) \mathfrak{g}eq 2$ if the following hold:
\begin{enumerate}
\item $a$ witnesses $u(\langle T, a \rangle) \mathfrak{g}eq 2$
\item $\langle a \rangle \cap T \neq 1$, and if $a$ is square in $\mathrm{Aut}(T)$, then $|\langlea\rangle \cap T|$ does not divide $4$.
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
The second major step is to generalise Breuer, Guralnick and Kantor's result Theorem~\ref{thm:breuer_guralnick_kantor} that $u(T) \mathfrak{g}eq 2$ for all nonabelian finite simple groups $T$ to all almost simple groups $A = \langle T, a \rangle$. This long line of research was initiated by Burness and Guest for $T = \mathrm{PSL}_n(q)$ \cite{ref:BurnessGuest13}, continued by Harper for the remaining classical groups $T$ \cite{ref:Harper17,ref:HarperLNM} and completed by Burness, Guralnick and Harper for exceptional groups $T$ \cite{ref:BurnessGuralnickHarper21}. (We have already noted that the proof of $u(S\!_n) \mathfrak{g}eq 2$ for $n \neq 6$ was completed by Burness and Harper in \cite{ref:BurnessHarper20}, and the result for sporadic groups follows from computational work in \cite{ref:BreuerGuralnickKantor08}).
We conclude by highlighting the major obstacle that this body of work faced and then outlining the technique that overcame this obstacle.
Let $G = \langle T, g \rangle$ where $T$ is a finite simple group of Lie type and $g \in \mathrm{Aut}(T)$. Suppose $s^G$ witnesses $u(G) \mathfrak{g}eq 2$. A conjugate of $s$ generates with any element of $T$, so $G = \langle T, s \rangle$. By replacing $s$ with a power if necessary, we may assume that $s \in Tg$. How do we describe elements of $Tg$ and their overgroups? Suppose that $T = \mathrm{PSL}_n(q)$ with $q=p^f$. If $g \in {\rm PGL}_n(q)$, then there are geometric techniques available, but what if $g$ is, say, the field automorphism $(a_{ij}) \mapsto (a_{ij}^p)$?
The technique that was used to answer these questions is known as \varepsilonmph{Shintani descent}. This was introduced by Shintani in 1976 \cite{ref:Shintani76} (and generalised by Kawanaka in \cite{ref:Kawanaka77}) to study irreducible characters of almost simple groups. However, as first exploited by Fulman and Guralnick in their work on the Boston--Shalev conjecture \cite{ref:FulmanGuralnick12}, Shintani descent also provides a fruitful way of studying the conjugacy classes of almost simple groups. The main theorem is the following, and we follow Desphande's proof \cite{ref:Deshpande16}. (Here $\sigma_i$ is considered as an element of $\langle X, \sigma_i \rangle$ for $i \in \{1,2\}$.)
\begin{theorem} \label{thm:shintani}
Let $X$ be a connected algebraic group, and let $\sigma_1, \sigma_2\colon X \to X$ be commuting Steinberg endomorphisms. Then there is a bijection
\[
F\colon \{ \text{$X_{\sigma_1}$-classes in $X_{\sigma_1}\sigma_2$} \} \to \{ \text{$X_{\sigma_2}$-classes in $X_{\sigma_2}\sigma_1$} \}.
\]
\varepsilonnd{theorem}
\begin{proof}
Let $S$ be the orbits of $\{ (g,h) \in X\sigma_2 \times X\sigma_1 \mid [g,h]=1 \}$ under the conjugation action of $X$. By the Lang--Steinberg theorem, $S$ is in bijection with the orbits of $\{ (x\sigma_2, \sigma_1) \mid x \in X_{\sigma_1} \}$ under conjugation by $X_{\sigma_1}$ and also with the orbits of $\{ (\sigma_2, y\sigma_1) \mid x \in X_{\sigma_2} \}$ under conjugation by $X_{\sigma_2}$. This provides a bijection between the $X_{\sigma_1}$-classes in $X_{\sigma_1}\sigma_2$ and the $X_{\sigma_2}$-classes in $X_{\sigma_2}\sigma_1$.
\varepsilonnd{proof}
The bijection in the proof of Theorem~\ref{thm:shintani}, known as the \varepsilonmph{Shintani map} of $(X,\sigma_1,\sigma_2)$, has desirable properties. For instance, if $\sigma_1 = \sigma_2^e$ for $e \mathfrak{g}eq 1$, then
\begin{equation} \label{eq:shintani}
F((x\sigma_2)^{X_{\sigma_1}}) = (a^{-1}(x\sigma_1)^{-e}a)^{X_{\sigma_2}}
\varepsilonnd{equation}
for some $a \in X$. Moreover, if $F(g^{X_{\sigma_1}}) = h^{X_{\sigma_2}}$, then it is easy to show that $C_{X_{\sigma_1}}(g) \cong C_{X_{\sigma_2}}(h)$, and, by now, extensive information is also available about how maximal overgroups of $g$ in $\langle X_{\sigma_1}, \sigma_2 \rangle$ relate to the maximal overgroups of $h$ in $\langle X_{\sigma_2}, \sigma_1 \rangle$. These latter results are crucial to proving Theorem~\ref{thm:burness_guralnick_harper} for almost simple groups of Lie type. The first result in this direction is due to Burness and Guest \cite[Corollary~2.15]{ref:BurnessGuest13}, and the subsequent developments are unified by Harper in \cite{ref:Harper21}, to which we refer the reader for further detail.
\begin{example} \label{ex:bgh}
We sketch how $u(G) \mathfrak{g}eq 2$ was proved for $G = \langle T, g \rangle$ when $T = \Omega^+_{2m}(q)$ with $q=2^f$ and $g$ is the field automorphism $\varphi\colon(a_{ij}) \mapsto (a_{ij}^2)$. We will not make further assumptions on $q$, but we will assume that $m$ is large.
Let $X$ be the simple algebraic group ${\rm SO}_{2m}(\overline{\mathbb{F}}_2)$ and let $F$ be the Shintani map
\noindent of $(X,\varphi^f,\varphi)$. Then $G = \langle X_{\varphi^f}, \varphi \rangle$, and writing $G_0 = X_{\varphi} = \Omega^+_{2m}(2)$, we observe that $F$ gives a bijection between the conjugacy classes in $Tg$ and those in $G_0$.
We define $s \in Tg$ such that $F(s^G) = s_0^{G_0}$ for a well chosen element $s_0 \in G_0$. In particular, \varepsilonqref{eq:shintani} implies that $s_0$ is $X$-conjugate to a power of $s$. To define $s_0$, fix $k$ such that $m-k$ is even and $\frac{\sqrt{2m}}{4} < 2k < \frac{\sqrt{2m}}{2}$, fix $A \in \Omega^-_{2k}(2)$ and $B \in \Omega^-_{2m-2k}(2)$ of order $2^k+1$ and $2^{m-k}+1$ and let $s_0 = \text{\scalebox{0.75}{$\left( \begin{array}{cc} A & 0 \\ 0 & B \varepsilonnd{array} \right)$}} \in \Omega^+_{2m}(2)$.
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The technique that was used to answer these questions is known as \varepsilonmph{Shintani descent}. This was introduced by Shintani in 1976 \cite{ref:Shintani76} (and generalised by Kawanaka in \cite{ref:Kawanaka77}) to study irreducible characters of almost simple groups. However, as first exploited by Fulman and Guralnick in their work on the Boston--Shalev conjecture \cite{ref:FulmanGuralnick12}, Shintani descent also provides a fruitful way of studying the conjugacy classes of almost simple groups. The main theorem is the following, and we follow Desphande's proof \cite{ref:Deshpande16}. (Here $\sigma_i$ is considered as an element of $\langle X, \sigma_i \rangle$ for $i \in \{1,2\}$.)
\begin{theorem} \label{thm:shintani}
Let $X$ be a connected algebraic group, and let $\sigma_1, \sigma_2\colon X \to X$ be commuting Steinberg endomorphisms. Then there is a bijection
\[
F\colon \{ \text{$X_{\sigma_1}$-classes in $X_{\sigma_1}\sigma_2$} \} \to \{ \text{$X_{\sigma_2}$-classes in $X_{\sigma_2}\sigma_1$} \}.
\]
\varepsilonnd{theorem}
\begin{proof}
Let $S$ be the orbits of $\{ (g,h) \in X\sigma_2 \times X\sigma_1 \mid [g,h]=1 \}$ under the conjugation action of $X$. By the Lang--Steinberg theorem, $S$ is in bijection with the orbits of $\{ (x\sigma_2, \sigma_1) \mid x \in X_{\sigma_1} \}$ under conjugation by $X_{\sigma_1}$ and also with the orbits of $\{ (\sigma_2, y\sigma_1) \mid x \in X_{\sigma_2} \}$ under conjugation by $X_{\sigma_2}$. This provides a bijection between the $X_{\sigma_1}$-classes in $X_{\sigma_1}\sigma_2$ and the $X_{\sigma_2}$-classes in $X_{\sigma_2}\sigma_1$.
\varepsilonnd{proof}
The bijection in the proof of Theorem~\ref{thm:shintani}, known as the \varepsilonmph{Shintani map} of $(X,\sigma_1,\sigma_2)$, has desirable properties. For instance, if $\sigma_1 = \sigma_2^e$ for $e \mathfrak{g}eq 1$, then
\begin{equation} \label{eq:shintani}
F((x\sigma_2)^{X_{\sigma_1}}) = (a^{-1}(x\sigma_1)^{-e}a)^{X_{\sigma_2}}
\varepsilonnd{equation}
for some $a \in X$. Moreover, if $F(g^{X_{\sigma_1}}) = h^{X_{\sigma_2}}$, then it is easy to show that $C_{X_{\sigma_1}}(g) \cong C_{X_{\sigma_2}}(h)$, and, by now, extensive information is also available about how maximal overgroups of $g$ in $\langle X_{\sigma_1}, \sigma_2 \rangle$ relate to the maximal overgroups of $h$ in $\langle X_{\sigma_2}, \sigma_1 \rangle$. These latter results are crucial to proving Theorem~\ref{thm:burness_guralnick_harper} for almost simple groups of Lie type. The first result in this direction is due to Burness and Guest \cite[Corollary~2.15]{ref:BurnessGuest13}, and the subsequent developments are unified by Harper in \cite{ref:Harper21}, to which we refer the reader for further detail.
\begin{example} \label{ex:bgh}
We sketch how $u(G) \mathfrak{g}eq 2$ was proved for $G = \langle T, g \rangle$ when $T = \Omega^+_{2m}(q)$ with $q=2^f$ and $g$ is the field automorphism $\varphi\colon(a_{ij}) \mapsto (a_{ij}^2)$. We will not make further assumptions on $q$, but we will assume that $m$ is large.
Let $X$ be the simple algebraic group ${\rm SO}_{2m}(\overline{\mathbb{F}}_2)$ and let $F$ be the Shintani map
\noindent of $(X,\varphi^f,\varphi)$. Then $G = \langle X_{\varphi^f}, \varphi \rangle$, and writing $G_0 = X_{\varphi} = \Omega^+_{2m}(2)$, we observe that $F$ gives a bijection between the conjugacy classes in $Tg$ and those in $G_0$.
We define $s \in Tg$ such that $F(s^G) = s_0^{G_0}$ for a well chosen element $s_0 \in G_0$. In particular, \varepsilonqref{eq:shintani} implies that $s_0$ is $X$-conjugate to a power of $s$. To define $s_0$, fix $k$ such that $m-k$ is even and $\frac{\sqrt{2m}}{4} < 2k < \frac{\sqrt{2m}}{2}$, fix $A \in \Omega^-_{2k}(2)$ and $B \in \Omega^-_{2m-2k}(2)$ of order $2^k+1$ and $2^{m-k}+1$ and let $s_0 = \text{\scalebox{0.75}{$\left( \begin{array}{cc} A & 0 \\ 0 & B \varepsilonnd{array} \right)$}} \in \Omega^+_{2m}(2)$.
We now study $\mathcal{M}(G,s)$. First, a power of $s_0$ (and hence $s$) has a $1$-eigenspace of codimension $2k < \frac{\sqrt{2m}}{2}$, so \cite[Theorem~7.1]{ref:GuralnickSaxl03} implies that $s$ is not contained in any local or almost simple maximal subgroup of $G$.
Next, a power of $s_0$ has order $2^{m-k}+1$, which is divisible by the \varepsilonmph{primitive part} of $2^{2m-2k}-1$ (the largest divisor of $2^{2m-2k}-1$ that is prime to $2^l-1$ for all $0 < l < 2m-2k$). For sufficiently large $m$, we can apply the main theorem of \cite{ref:GuralnickPenttilaPraegerSaxl97} to deduce that all of the maximal overgroups of $s_0$ in $G_0 = \Omega^+_{2m}(2)$ are reducible. In particular, the only maximal overgroup of $s_0$ in $G_0$ arising as the set of fixed points of a closed positive-dimensional $\varphi$-stable subgroup of $X$ is the obvious reducible subgroup of type $({\rm O}^-_{2k}(2) \times {\rm O}^-_{2m-2k}(2)) \cap G_0$. Now the theory of Shintani descent \cite[Theorem~4]{ref:Harper21} implies that the only such maximal overgroup of $s$ in $G$ is one subgroup $H$ of type $({\rm O}^\varphim_{2k}(q) \times {\rm O}^\varphim_{2m-2k}(q)) \cap G$.
Drawing these observations together and using Aschbacher's subgroup structure theorem \cite{ref:Aschbacher84}, we deduce that $\mathcal{M}(G,s) = \{H\} \cup \mathcal{M}'$, where $\mathcal{M}'$ consists of subfield subgroups. There are at most $\log_2{f}+1 = \log_2\log_2 q + 1$ classes of maximal subfield subgroups of $G$, and by \cite[Lemma~2.19]{ref:BurnessGuest13}, $s$ is contained in at most $|C_{G_0}(s_0)| = (2^k+1)(2^{2m-2k}+1)$ conjugates of a fixed maximal subgroup.
Using the fixed point ratio bound proved for reducible subgroups in \cite{ref:GuralnickKantor00} and irreducible subgroups in \cite{ref:Burness071,ref:Burness072,ref:Burness073,ref:Burness074}, for all nontrivial $x \in G$ we have
\[
Q(x,s) < \sum_{H \in \mathcal{M}(G,s)} \frac{|x^G \cap H|}{|x^G|} < \frac{5}{q^{\sqrt{n}/4}} + (\log_2\log_2{q}+1)(2^k+1)(2^{2m-2k}+1) \frac{2}{q^{n-3}}.
\]
In particular, for sufficiently large $m$, $Q(x,s) < \frac{1}{2}$ and $u(G) \mathfrak{g}eq 2$. Moreover, $Q(x,s) \to 0$ and $u(G) \to \infty$, as $m \to \infty$ or, for sufficiently large $m$, as $q \to \infty$.
\varepsilonnd{example}
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\subsection{The generating graph} \label{ss:finite_graph}
In this section, we introduce a combinatorial object that gives a way to visualise the concepts we have introduced so far. The \varepsilonmph{generating graph} of a group $G$, denoted $\Gamma(G)$, is the graph whose vertex set is $G \setminus 1$ and where two vertices $g, h \in G$ are adjacent if $\langle g, h \rangle = G$. See Figure~\ref{fig:graph} for examples.
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=0.42]
\node[white] (0) at (270:8cm) {};
\node[draw,circle,minimum size=0.8cm] (1) at (45 :5.5cm) {$ab$};
\node[draw,circle,minimum size=0.8cm] (2) at (135 :5.5cm) {$b$};
\node[draw,circle,minimum size=0.8cm] (3) at (225 :5.5cm) {$a^3b$};
\node[draw,circle,minimum size=0.8cm] (4) at (315 :5.5cm) {$a^2b$};
\node[draw,circle,minimum size=0.8cm] (5) at (0:1.5cm) {$a^3$};
\node[draw,circle,minimum size=0.8cm] (6) at (180:1.5cm) {$a$};
\node[draw,circle,minimum size=0.8cm] (8) at (270:6.5cm) {$a^2$};
\varphiath (1) edge[-] (2);
\varphiath (2) edge[-] (3);
\varphiath (3) edge[-] (4);
\varphiath (4) edge[-] (1);
\varphiath (1) edge[-] (5);
\varphiath (2) edge[-] (5);
\varphiath (3) edge[-] (5);
\varphiath (4) edge[-] (5);
\varphiath (1) edge[-] (6);
\varphiath (2) edge[-] (6);
\varphiath (3) edge[-] (6);
\varphiath (4) edge[-] (6);
\varepsilonnd{tikzpicture} \hspace{1cm}
\begin{tikzpicture}[scale=0.5,circle,inner sep=0.01cm,minimum size=0.66cm]
\node[draw] (1) at (90-360/11 :5cm) {\tiny{(1\,2)(3\,4)}};
\node[draw] (2) at (90 :5cm) {\tiny{(1\,3)(2\,4)}};
\node[draw] (3) at (90+360/11 :5cm) {\tiny{(1\,4)(2\,3)}};
\node[draw] (4) at (90+360/11*2: 5cm) {\tiny{(1\,2\,3)}};
\node[draw] (5) at (90+360/11*3: 5cm) {\tiny{(1\,3\,2)}};
\node[draw] (6) at (90+360/11*4: 5cm) {\tiny{(1\,2\,4)}};
\node[draw] (7) at (90+360/11*5: 5cm) {\tiny{(1\,4\,2)}};
\node[draw] (8) at (90+360/11*6: 5cm) {\tiny{(1\,3\,4)}};
\node[draw] (9) at (90+360/11*7: 5cm) {\tiny{(1\,4\,3)}};
\node[draw] (10) at (90+360/11*8: 5cm) {\tiny{(2\,3\,4)}};
\node[draw] (11) at (90+360/11*9: 5cm) {\tiny{(2\,4\,3)}};
\foreach \x in {1,...,3}
\foreach \y in {4,...,11}
{\varphiath (\x) edge[-] (\y);}
\varphiath (5) edge[-] (6);
\varphiath (7) edge[-] (8);
\varphiath (9) edge[-] (10);
\varphiath (11) edge[-] (4);
\foreach \x in {4,...,11}
\foreach \y in {4,...,11}
{
\varphigfmathtruncatemacro{\a}{\x-\y};
\ifnum\a=0{}
\varepsilonlse
{
\ifnum\a=1{}
\varepsilonlse
{
\ifnum\a=-1{}
\varepsilonlse
{
\varphiath (\x) edge[-] (\y);
}
\fi
}
\fi
}
\fi
}
\varepsilonnd{tikzpicture}
\varepsilonnd{center}
\caption{The generating graphs of $D_8 = \langle a, b \mid a^4 = 1, b^2 = 1, a^b = a^{-1} \rangle$ and the alternating group $A_4$.} \label{fig:graph}
\varepsilonnd{figure}
A question that immediately comes to mind is: when is $\Gamma(G)$ is connected? This question has a remarkably straightforward answer.
\begin{theorem} \label{thm:generating_graph}
Let $G$ be a finite group. Then the following are equivalent:
\begin{enumerate}
\item $\Gamma(G)$ has no isolated vertices
\item $\Gamma(G)$ is connected
\item $\Gamma(G)$ has diameter at most two
\item every proper quotient of $G$ is cyclic.
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
Of course, Theorem~\ref{thm:generating_graph} is simply a reformulation of Theorem~\ref{thm:burness_guralnick_harper} due to Burness, Guralnick and Harper. Indeed, the generating graph gives an enlightening perspective on the concepts introduced so far:
\begin{enumerate}
\item $G$ is $\frac{3}{2}$-generated if and only if $\Gamma(G)$ has no isolated vertices
\item $s(G)$ is the greatest $k$ such that any $k$ vertices of $\Gamma(G)$ have a common neighbour, which means that $s(G) \mathfrak{g}eq 2$ if and only if $\mathrm{diam}(\Gamma(G)) \leqslant 2$
\item $\mathfrak{g}amma_t(G)$ is the total domination number of $\Gamma(G)$: the least size of a set of vertices whose neighbours cover $\Gamma(G)$.
\varepsilonnd{enumerate}
Regarding (iii), the total domination number is a well studied graph invariant and was the inspiration for the group theoretic term (and the symbol $\mathfrak{g}amma_t$ is the graph theoretic notation). Regarding (ii), as far as the author is aware, the graph invariant corresponding to the spread of a group, while natural, does not have a canonical name, but, inspired by the recent work on the spread of finite groups, some authors have started to use the term \varepsilonmph{spread} for this graph invariant (that is, the \varepsilonmph{spread} of a graph is the greatest $k$ such that any $k$ vertices have a common neighbour), see for example \cite[Section~2.5]{ref:Cameron22}.
What are the connected components of $\Gamma(G)$ are in general? The following conjecture (first posed as a question in \cite{ref:CrestaniLucchini13-Israel}) proposes a straightforward answer.
\begin{conjecture} \label{con:generating_graph}
Let $G$ be a finite group. Then the graph obtained from $\Gamma(G)$ by removing the isolated vertices is connected.
\varepsilonnd{conjecture}
Conjecture~\ref{con:generating_graph} is known to be true if $G$ is soluble or characteristically simple by work of Crestani and Lucchini \cite{ref:CrestaniLucchini13-Israel,ref:CrestaniLucchini13-JAlgCombin} or if every proper quotient of $G$ is cyclic as a consequence of Theorem~\ref{thm:burness_guralnick_harper}. Otherwise, Conjecture~\ref{con:generating_graph} remains an intriguing open question about the generating sets of finite groups.
There is now a vast literature on the generating graph and surveying it is beyond the scope of this survey article, so we will make only a few remarks. The first paper to study $\Gamma(G)$ in its own right, and call it the \varepsilonmph{generating graph}, was \cite{ref:LucchiniMaroti09Ischia} by Lucchini and Mar\'oti, who then studied various aspects of this graph in subsequent papers (for example, \cite{ref:BreuerGuralnickLucchiniMarotiNagy10,ref:LucchiniMaroti09}). However, $\Gamma(G)$ first appeared in the literature, indirectly, as a construction in a proof of Liebeck and Shalev in \cite{ref:LiebeckShalev96JAlg}. A major result of that paper is that there exist constants $c_1, c_2 > 0$ such that for all finite simple groups $G$, the probability that two randomly chosen elements generate $G$, denoted $P(G)$, satisfies
\begin{equation} \label{eq:prob}
1 - \frac{c_1}{m(G)} \leqslant P(G) \leqslant 1 - \frac{c_2}{m(G)}
\varepsilonnd{equation}
where $m(G)$ is the smallest index of a subgroup of $G$ (for example, $m(A_n)=n$). From this, Liebeck and Shalev deduce that there is a constant $c > 0$ such that every finite simple group $G$ contains at least $c \cdot m(G)$ elements that pairwise generate $G$. The proof of this corollary simply involves applying Tur\'an's theorem to $\Gamma(G)$, exploiting \varepsilonqref{eq:prob}. A couple of subsequent papers \cite{ref:Blackburn06,ref:BritnellEvseevGuralnickHolmesMaroti08} continued the study of cliques in $\Gamma(G)$, partly motivated by the observation that the largest size of a clique in $\Gamma(G)$, denoted $\mu(G)$, is a lower bound for the smallest number of proper subgroups whose union is $G$, denoted $\sigma(G)$. Returning to spread, it is easy to see that $s(G) < \mu(G) \leqslant \sigma(G)$, so these results give upper bounds on spread, which are otherwise difficult to find. Indeed, the best upper bounds for the smaller 14 sporadic groups (including the two where the spread is known exactly) were found in \cite{ref:BradleyHolmes07} by a clever refinement of this bound (for the larger 12 sporadic groups, different methods were used \cite{ref:Fairbarin12CA}).
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\subsection{Applications} \label{ss:finite_app}
We conclude Section~\ref{s:finite} with three applications of spread. The first shows how $\frac{3}{2}$-generation naturally arises in a completely different context. The second highlights the benefit of studying spread, not just $\frac{3}{2}$-generation. The third moves beyond simple groups and applies the classification of finite $\frac{3}{2}$-generated groups.
\textbf{Application 1. Word maps. } From a word $w = w(x,y)$ in the free group $F_2$, we obtain a \varepsilonmph{word map} $w\colon G \times G \to G$ and we write $w(G) = \{ w(g,h) \mid g,h \in G\}$.
Let $G$ be a nonabelian finite simple group. For natural choices of $w$, there has been substantial recent progress showing that $w(G) = G$. For example, solving the Ore Conjecture, Liebeck, O'Brien, Shalev and Tiep \cite{ref:LiebeckOBrienShalevTiep10} proved that $w = x^{-1}y^{-1}xy$ is surjective (that is, every element is a commutator).
Now consider the converse question: which subsets $S \subseteq G$ arise as images of some word $w \in F_2$? Such a subset $S$ must satisfy $1 \in S$ (as $w(1,1)=1$) and $S^a = S$ for all $a \in \mathrm{Aut}(G)$ (as $w(x^a,y^a) = w(x,y)^a$), and Lubotzky \cite{ref:Lubotzky14} proved that these conditions are sufficient.
\begin{theorem} \label{thm:lubotzky}
Let $G$ be a finite simple group and let $S \subseteq G$. Then $S$ is the image of a word $w \in F_2$ if and only if $1 \in S$ and $S^a = S$ for all $a \in \mathrm{Aut}(G)$.
\varepsilonnd{theorem}
The proof of Theorem~\ref{thm:lubotzky} is short and fairly elementary, except it uses Theorem~\ref{thm:guralnick_kantor}. Indeed, Theorem~\ref{thm:guralnick_kantor} is the only way it depends on the CFSG.
\begin{proof}[Proof outline of Theorem~\ref{thm:lubotzky}]
Let $G^2 = \{ (a_i,b_i) \mid 1 \leqslant i \leqslant |G|^2 \}$ be ordered such that $\langle a_i, b_i \rangle = G$ if and only if $i \leqslant \varepsilonll$. Fix the free group $F_2 = \langle x, y \rangle$ and let $\varphi\colon F_2 \to G^{|G|^2}$ be defined as $\varphi(x) = (a_1, \dots, a_{|G|^2})$ and $\varphi(y) = (b_1, \dots, b_{|G|^2})$. Let $z = (z_1, \dots, z_{|G|^2})$ where $z_i = a_i$ if $i \leqslant \varepsilonll$ and $a_i \in S$ and $z_i = 1$ otherwise.
By Theorem~\ref{thm:guralnick_kantor}, every nontrivial element of $G$ is contained in a generating pair, so, in particular, $\{ z_i \mid 1 \leqslant i \leqslant |G|^2 \} = S \cup \{1\} = S$.
By an elementary argument, Lubotzky shows that $\varphi(F_2) = H \times K$ where $H$ and $K$ are the projections of $\varphi(F_2)$ onto the first $\varepsilonll$ factors of $G^{|G|^2}$ and the remaining $|G|^2-\varepsilonll$ factors, respectively. Moreover, using a theorem of Hall \cite{ref:Hall36}, $H$ is the subgroup of $G^\varepsilonll$ isomorphic to $G^{\varepsilonll/|\mathrm{Aut}(G)|}$ with the defining property that $(g_1, \dots, g_\varepsilonll) \in H$ if and only if for all $a \in \mathrm{Aut}(G)$ and $1 \leqslant i,j \leqslant \varepsilonll$ we have $g_i = g_j^a$ whenever $\varphi_i = \varphi_{\!j}\,a$. In particular, since $S^a = S$ for all $a \in \mathrm{Aut}(G)$, we deduce that $z \in \varphi(F_2)$. Therefore, there exists $w \in F_2$ such that for all $1 \leqslant i \leqslant |G|^2$ we have $w(a_i,b_i) = \varphi(w)_i = z_i$.
Combining the conclusions of the previous two paragraphs, we deduce that $w(G) = \{ w(a_i,b_i) \mid 1 \leqslant i \leqslant |G|^2 \} = \{ z_i \mid 1 \leqslant i \leqslant |G|^2 \} = S$.
\varepsilonnd{proof}
\textbf{Application 2. The product replacement graph. } For a positive integer $k$, the vertices of the product replacement graph $\Gamma_k(G)$ are the generating $k$-tuples of $G$, and the neighbours of $(x_1, \dots, x_i, \dots, x_k)$ in $\Gamma_k(G)$ are $(x_1, \dots, x_ix_j^\varphim, \dots, x_k)$ and $(x_1, \dots, x_j^\varphim x_i, \dots, x_k)$ for each $1 \leqslant i \neq j \leqslant k$.
The product replacement graph arises in a number of contexts, most notably, the product replacement algorithm for computing random elements of $G$, which involves a random walk on $\Gamma_k(G)$, see \cite{ref:CellerLeedhamGreenMurrayNiemeyerOBrien95}. Thus, the connectedness of $\Gamma_k(G)$ is of particular interest. Specifically, Pak \cite[Question~2.1.33]{ref:Pak01} asked whether $\Gamma_k(G)$ is connected whenever $k$ is strictly greater than $d(G)$, the smallest size of a generating set for $G$. This question is open, even for finite simple groups where Wiegold conjectured that the answer is true. Nevertheless, the following lemma of Evans \cite[Lemma~2.8]{ref:Evans93} shows the usefulness of spread. (Here a generating tuple of $G$ is said to be \varepsilonmph{redundant} if some proper subtuple also generates $G$.)
\begin{lemma}\label{lem:evans}
Let $k \mathfrak{g}eq 3$ and let $G$ be a group such that $s(G) \mathfrak{g}eq 2$. Then all of the redundant generating $k$-tuples are connected in $\Gamma_k(G)$.
\varepsilonnd{lemma}
\begin{proof}
Let $x = (x_1,\dots,x_k)$ and $y = (y_1,\dots,y_k)$ be two redundant generating $k$-tuples. By an elementary observation of Pak, it is sufficient to show that $x$ and $y$ are connected after permuting of the entries of $x$ and $y$. In particular, since $x$ and $y$ are redundant, we may assume that $\langlex_1,\dots,x_{k-1}\rangle = \langley_1,\dots,y_{k-1}\rangle = G$ and also that $x_1 \neq 1 \neq y_2$. Since $s(G) \mathfrak{g}eq 2$, there exists $z \in G$ such that $\langle x_1, z \rangle = \langle y_2, z \rangle = G$. We now make a series of connections. First, $x=x^{(1)}$ is connected to $x^{(2)} = (x_1,\dots,x_{k-1},z)$ as $\langlex_1,\dots,x_{k-1}\rangle = G$. Next, $x^{(2)}$ is connected to $x^{(3)} = (x_1,y_2,\dots,y_{k-1},z)$ as $\langlex_1,z\rangle = G$. Now, $x^{(3)}$ is connected to $x^{(4)} = (y_1,\dots,y_{k-1},z)$ as $\langley_2,z\rangle = G$. Finally, $x^{(4)}$ is connected to $(y_1,\dots,y_k) = y$ as $\langley_1,\dots,y_{k-1}\rangle = G$. This shows that $x$ is connected to $y$.
\varepsilonnd{proof}
Combining Lemma~\ref{lem:evans} with Theorem~\ref{thm:breuer_guralnick_kantor} shows that to prove Wiegold's conjecture, it suffices to show that for each finite simple group $G$ every irredundant generating $k$-tuple is connected in $\Gamma_k(G)$ to a redundant one.
As further evidence for the relevance of spread in this area, we note that Wiegold's original conjecture was (the a priori weaker claim) that a related graph $\mathbb{S}igma_k(G)$ is connected for all finite simple groups $G$ and $k > d(G)$ (see \cite[Conjecture~2.5.4]{ref:Pak01}), but a short argument of Pak \cite[Proposition~2.5.13]{ref:Pak01} shows that $\mathbb{S}igma_k(G)$ is connected if and only if $\Gamma_k(G)$ is connected, for all groups $G$ such that $s(G) \mathfrak{g}eq 2$, which by Theorem~\ref{thm:breuer_guralnick_kantor} includes all finite simple groups $G$.
\textbf{Application 3. The $\boldsymbol{\mathcal{X}}$-radical of a group. } In 1968, Thompson proved that a finite group is soluble if and only if all of its 2-generated subgroups are soluble \cite[Corollary~2]{ref:Thompson68}. The result follows from Thompson's classification of the finite insoluble groups all of whose proper subgroups are soluble, but, in 1995, Flavell gave a direct proof of this result \cite{ref:Flavell95}. Confirming a conjecture of Flavell \cite[Conjecture~B]{ref:Flavell01}, Guralnick, Kunyavsk\u{\i}i, Plotkin and Shalev proved the following result about the \varepsilonmph{soluble radical} of $G$, written $R(G)$, which is the largest normal soluble subgroup of $G$ \cite[Theorem~1.1]{ref:GuralnickKunyavskiiPlotkinShalev06}.
\begin{theorem} \label{thm:soluble_radical}
Let $G$ be a finite group. Then
\[
R(G) = \{ x \in G \mid \text{$\langlex,y\rangle$ is soluble for all $y \in G$} \}.
\]
\varepsilonnd{theorem}
The key (and only CFSG-dependent) element of the proof of Theorem~\ref{thm:soluble_radical} is a strong version Theorem~\ref{thm:guralnick_kantor} on the $\frac{3}{2}$-generation of finite simple groups. To paint a picture of how the $\frac{3}{2}$-generation of simple groups plays the starring role, we first present the analogue for simple Lie algebras \cite[Theorem~2.1]{ref:GuralnickKunyavskiiPlotkinShalev06}. Recall that the \varepsilonmph{radical} of a Lie algebra $\mathfrak{g}$, denoted $R(\mathfrak{g})$, is the largest soluble ideal of $\mathfrak{g}$.
\begin{theorem} \label{thm:soluble_radical_lie_algebras}
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over $\mathbb{C}$. Then
\[
R(\mathfrak{g}) = \{ x \in \mathfrak{g} \mid \text{$\langlex,y\rangle$ is soluble for all $y \in \mathfrak{g}$} \}.
\]
\varepsilonnd{theorem}
\begin{proof}
Let $x \in \mathfrak{g}$. First assume that $x \in R(\mathfrak{g})$. Let $y \in G$ and let $\mathfrak{h}$ be the smallest ideal of $\langlex,y\rangle$ containing $x$. Now $\mathfrak{h}$ is soluble as it is a Lie subalgebra of $R(\mathfrak{g})$, and $\langlex,y\rangle/\mathfrak{h}$ is soluble as it is $1$-dimensional, so $\langlex,y\rangle$ is soluble.
Now assume that $\langle x, y \rangle$ is soluble for all $y \in G$. We will prove that $x \in R(\mathfrak{g})$. For a contradiction, assume that $\mathfrak{g}$ is a minimal counterexample (by dimension). Consider $\overline{\mathfrak{g}} = \mathfrak{g}/R(\mathfrak{g})$. Then $\langle \overline{x}, \overline{y} \rangle$ is soluble for all $y \in \mathfrak{g}$, and $R(\overline{\mathfrak{g}})$ is trivial, so $\overline{x} \not\in R(\overline{\mathfrak{g}})$. Therefore, $R(\mathfrak{g})=0$, by the minimality of $\mathfrak{g}$. This means that $\mathfrak{g}$ is semisimple, so we may write $\mathfrak{g} = \mathfrak{g}_1 \oplus \cdots \oplus \mathfrak{g}_k$ where $\mathfrak{g}_1, \dots, \mathfrak{g}_k$ are simple. Writing $x = (x_1, \dots, x_k)$, fix $1 \leqslant i \leqslant k$ such that $x_i \neq 0$. Then by Theorem~\ref{thm:ionescu}, there exists $y \in \mathfrak{g}_i$ such that $\langlex_i,y\rangle = \mathfrak{g}_i$. In particular, $\mathfrak{g}_i$ is a quotient of $\langlex,y\rangle$, so $\langlex,y\rangle$ is not soluble, which is a contradiction.
\varepsilonnd{proof}
Returning to groups, again using variants of Theorem~\ref{thm:guralnick_kantor}, Guralnick, Plotkin and Shalev set Theorem~\ref{thm:soluble_radical} in a more general context \cite[Theorem~6.1]{ref:GuralnickPlotkinShalev07}. Here we give a short proof of their result by applying Theorem~\ref{thm:burness_guralnick_harper}.
Let $\mathcal{X}$ be a class of finite groups that is closed under subgroups, quotients and extensions. The $\mathcal{X}$-radical of a group $G$, denoted $\mathcal{X}(G)$, is the largest normal $\mathcal{X}$-subgroup of $G$. For instance, $\mathcal{X}(G) = R(G)$ if $\mathcal{X}$ is the class of soluble groups.
\begin{theorem} \label{thm:x_radical}
Let $\mathcal{X}$ be a class of finite groups that is closed under subgroups, quotients and extensions. Then
\[
\mathcal{X}(G) = \{ x \in G \mid \text{$\langlex^{\langley\rangle}\rangle$ is an $\mathcal{X}$-group for all $y \in G$} \}.
\]
\varepsilonnd{theorem}
\begin{corollary} \label{cor:x_radical}
Let $\mathcal{X}$ be a class of finite groups that is closed under subgroups, quotients and extensions. Then $G$ is an $\mathcal{X}$-group if and only if every $2$-generated subgroup of $G$ is an $\mathcal{X}$-group.
\varepsilonnd{corollary}
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\textbf{Application 3. The $\boldsymbol{\mathcal{X}}$-radical of a group. } In 1968, Thompson proved that a finite group is soluble if and only if all of its 2-generated subgroups are soluble \cite[Corollary~2]{ref:Thompson68}. The result follows from Thompson's classification of the finite insoluble groups all of whose proper subgroups are soluble, but, in 1995, Flavell gave a direct proof of this result \cite{ref:Flavell95}. Confirming a conjecture of Flavell \cite[Conjecture~B]{ref:Flavell01}, Guralnick, Kunyavsk\u{\i}i, Plotkin and Shalev proved the following result about the \varepsilonmph{soluble radical} of $G$, written $R(G)$, which is the largest normal soluble subgroup of $G$ \cite[Theorem~1.1]{ref:GuralnickKunyavskiiPlotkinShalev06}.
\begin{theorem} \label{thm:soluble_radical}
Let $G$ be a finite group. Then
\[
R(G) = \{ x \in G \mid \text{$\langlex,y\rangle$ is soluble for all $y \in G$} \}.
\]
\varepsilonnd{theorem}
The key (and only CFSG-dependent) element of the proof of Theorem~\ref{thm:soluble_radical} is a strong version Theorem~\ref{thm:guralnick_kantor} on the $\frac{3}{2}$-generation of finite simple groups. To paint a picture of how the $\frac{3}{2}$-generation of simple groups plays the starring role, we first present the analogue for simple Lie algebras \cite[Theorem~2.1]{ref:GuralnickKunyavskiiPlotkinShalev06}. Recall that the \varepsilonmph{radical} of a Lie algebra $\mathfrak{g}$, denoted $R(\mathfrak{g})$, is the largest soluble ideal of $\mathfrak{g}$.
\begin{theorem} \label{thm:soluble_radical_lie_algebras}
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over $\mathbb{C}$. Then
\[
R(\mathfrak{g}) = \{ x \in \mathfrak{g} \mid \text{$\langlex,y\rangle$ is soluble for all $y \in \mathfrak{g}$} \}.
\]
\varepsilonnd{theorem}
\begin{proof}
Let $x \in \mathfrak{g}$. First assume that $x \in R(\mathfrak{g})$. Let $y \in G$ and let $\mathfrak{h}$ be the smallest ideal of $\langlex,y\rangle$ containing $x$. Now $\mathfrak{h}$ is soluble as it is a Lie subalgebra of $R(\mathfrak{g})$, and $\langlex,y\rangle/\mathfrak{h}$ is soluble as it is $1$-dimensional, so $\langlex,y\rangle$ is soluble.
Now assume that $\langle x, y \rangle$ is soluble for all $y \in G$. We will prove that $x \in R(\mathfrak{g})$. For a contradiction, assume that $\mathfrak{g}$ is a minimal counterexample (by dimension). Consider $\overline{\mathfrak{g}} = \mathfrak{g}/R(\mathfrak{g})$. Then $\langle \overline{x}, \overline{y} \rangle$ is soluble for all $y \in \mathfrak{g}$, and $R(\overline{\mathfrak{g}})$ is trivial, so $\overline{x} \not\in R(\overline{\mathfrak{g}})$. Therefore, $R(\mathfrak{g})=0$, by the minimality of $\mathfrak{g}$. This means that $\mathfrak{g}$ is semisimple, so we may write $\mathfrak{g} = \mathfrak{g}_1 \oplus \cdots \oplus \mathfrak{g}_k$ where $\mathfrak{g}_1, \dots, \mathfrak{g}_k$ are simple. Writing $x = (x_1, \dots, x_k)$, fix $1 \leqslant i \leqslant k$ such that $x_i \neq 0$. Then by Theorem~\ref{thm:ionescu}, there exists $y \in \mathfrak{g}_i$ such that $\langlex_i,y\rangle = \mathfrak{g}_i$. In particular, $\mathfrak{g}_i$ is a quotient of $\langlex,y\rangle$, so $\langlex,y\rangle$ is not soluble, which is a contradiction.
\varepsilonnd{proof}
Returning to groups, again using variants of Theorem~\ref{thm:guralnick_kantor}, Guralnick, Plotkin and Shalev set Theorem~\ref{thm:soluble_radical} in a more general context \cite[Theorem~6.1]{ref:GuralnickPlotkinShalev07}. Here we give a short proof of their result by applying Theorem~\ref{thm:burness_guralnick_harper}.
Let $\mathcal{X}$ be a class of finite groups that is closed under subgroups, quotients and extensions. The $\mathcal{X}$-radical of a group $G$, denoted $\mathcal{X}(G)$, is the largest normal $\mathcal{X}$-subgroup of $G$. For instance, $\mathcal{X}(G) = R(G)$ if $\mathcal{X}$ is the class of soluble groups.
\begin{theorem} \label{thm:x_radical}
Let $\mathcal{X}$ be a class of finite groups that is closed under subgroups, quotients and extensions. Then
\[
\mathcal{X}(G) = \{ x \in G \mid \text{$\langlex^{\langley\rangle}\rangle$ is an $\mathcal{X}$-group for all $y \in G$} \}.
\]
\varepsilonnd{theorem}
\begin{corollary} \label{cor:x_radical}
Let $\mathcal{X}$ be a class of finite groups that is closed under subgroups, quotients and extensions. Then $G$ is an $\mathcal{X}$-group if and only if every $2$-generated subgroup of $G$ is an $\mathcal{X}$-group.
\varepsilonnd{corollary}
\begin{proof}
If $G$ is an $\mathcal{X}$-group, then every $2$-generated subgroup is. Conversely, if for all $x,y \in G$ the subgroup $\langlex,y\rangle$ is an $\mathcal{X}$-group, then so is $\langlex^{\langley\rangle}\rangle$, so, by Theorem~\ref{thm:x_radical}, $x \in \mathcal{X}(G)$, which shows $G = \mathcal{X}(G)$, which is an $\mathcal{X}$-group.
\varepsilonnd{proof}
\begin{corollary} \label{cor:x_radical_soluble}
Let $\mathcal{X}$ be a class of finite groups that is closed under subgroups, quotients and extensions. Assume that $\mathcal{X}$ contains all soluble groups. Then
\[
\mathcal{X}(G) = \{ x \in G \mid \text{$\langlex, y\rangle$ is an $\mathcal{X}$-group for all $y \in G$} \}.
\]
\varepsilonnd{corollary}
\begin{proof}
Let $x \in G$. If for all $y \in G$, $\langlex,y\rangle$ is an $\mathcal{X}$-group, then so is $\langlex^{\langley\rangle}\rangle$, so, by Theorem~\ref{thm:x_radical}, $x \in \mathcal{X}(G)$. Conversely, if $x \in \mathcal{X}(G)$, then $\langlex^{\langley\rangle}\rangle \leqslant \langlex^G\rangle \leqslant \mathcal{X}(G)$ is an $\mathcal{X}$-group, so $\langlex,y\rangle$, an extension of $\langlex^{\langley\rangle}\rangle$ by a cyclic group, is an $\mathcal{X}$-group.
\varepsilonnd{proof}
\begin{proof}[Proof of Theorem~\ref{thm:x_radical}]
Let $x \in G$. First assume that $x \in \mathcal{X}(G)$. For all $y \in G$, we have $\langlex^{\langley\rangle}\rangle \leqslant \langlex^G\rangle \leqslant \mathcal{X}(G)$, so $\langlex^{\langley\rangle}\rangle$ is an $\mathcal{X}$-group.
Now assume that $\langle x^{\langley\rangle} \rangle$ is an $\mathcal{X}$-group for all $y \in G$. We will prove that $x \in \mathcal{X}(G)$. For a contradiction, assume that $G$ is a minimal counterexample. Consider $\overline{G} = G/\mathcal{X}(G)$. Then $\langle \overline{x}^{\langle\overline{y}\rangle} \rangle$ is an $\mathcal{X}$-group for all $y \in G$ (as $\mathcal{X}$ is closed under quotients), and $\mathcal{X}(\overline{G})$ is trivial (as $\mathcal{X}$ is closed under extensions), so $\overline{x} \not\in \mathcal{X}(\overline{G})$. Therefore, $\mathcal{X}(G)=1$, by the minimality of $G$. Now consider $H = \langlex^G\rangle$. Then $\langlex^{\langleh\rangle}\rangle$ is an $\mathcal{X}$-group for all $h \in H$, and $\mathcal{X}(H) \leqslant \mathcal{X}(G)$ (as $\mathcal{X}(H)$ is characteristic in $H$ so normal in $G$), so $x \not\in \mathcal{X}(H)$. Therefore, $\langlex^G\rangle = G$, by the minimality of $G$. Consider a power $x'$ of $x$ of prime order. Then $\langle x'^{\langley\rangle} \rangle$ is an $\mathcal{X}$-group for all $y \in G$ (as $\mathcal{X}$ is closed under subgroups), and $x' \not\in 1 = \mathcal{X}(G)$. Therefore, it suffices to consider the case where $x$ has prime order.
Let $N$ be a minimal normal subgroup of $G$ and write $N = T^k$ where $T$ is simple. Observe that $N$, or equivalently $T$, is not an $\mathcal{X}$-group, since $\mathcal{X}(G)=1$.
Suppose that $x \in N$. Then $G = N$ since $\langlex^G\rangle = G$. In particular, $k=1$, so, by Theorem~\ref{thm:guralnick_kantor}, there exists $y \in G$ such that $\langlex,y\rangle=G$, which is not an $\mathcal{X}$-group, so $\langlex^{\langley\rangle}\rangle$ is not an $\mathcal{X}$-group either: a contradiction. Therefore, $x \not\in N$.
Suppose that $x$ centralises $N$. Then $N$ is central since $\langlex^G\rangle = G$, so $N \cong C_p$ for a prime $p$. In $\widetilde{G} = G/N$, $\widetilde{x}$ is nontrivial as $x \not\in N$ and $\langle\widetilde{x}^{\langle\widetilde{y}\rangle}\rangle$ is an $\mathcal{X}$-group for all $y \in G$ (as $\langlex^{\langley\rangle}\rangle$ is). The minimality of $G$ means $\widetilde{x} \in \mathcal{X}(\widetilde{G})$, but $\langle\widetilde{x}^{\widetilde{G}}\rangle = \widetilde{G}$ (as $\langlex^G\rangle = G$), so $\widetilde{G}$ is an $\mathcal{X}$-group. Since $N \cong C_p$ is not an $\mathcal{X}$-group, $p$ does not divide $|\widetilde{G}|$. By the Schur--Zassenhaus Theorem, $G = N \times H$ for some $H \cong \widetilde{G}$. Since $N \cong C_p$ is not an $\mathcal{X}$-group and $\langlex\rangle$ is an $\mathcal{X}$-group, $p$ does not divide $|x|$, so $x \in H$, contradicting $\langle x^G \rangle = G$. Therefore, $x$ acts nontrivially on $N$.
Suppose that $N$ is abelian. As $x$ acts nontrivially on $N$, there is $n \in N$ with $[x,n] \neq 1$. Now $\langle [x,n] \rangle$ is isomorphic to $T$, as $[x,n] \in N$, so it is not an $\mathcal{X}$-group, implying that $\langle x^{\langlen\rangle} \rangle$ is not an $\mathcal{X}$-group either. Therefore, $N$ is nonabelian.
Now $x$ permutes the $k$ factors of $N \cong T^k$ and let $M \cong T^l$ be a nontrivial subgroup of $N$ whose factors are permuted transitively by $x$. If $l=1$, then $\langle M, x \rangle$ is almost simple, and if $l > 1$, then, recalling that $x$ has prime order, $\langlex\rangle \cong C_l$ acts regularly on the factors of $M$. In either case, $M$ is the unique minimal normal subgroup of $\langleM,x\rangle$, so every proper quotient of $\langleM,x\rangle$ is cyclic. Therefore, by Theorem~\ref{thm:burness_guralnick_harper}, there exists $m \in \langle M, x \rangle$ such that $\langle m, x \rangle = \langle M, x\rangle$. Moreover, $\langlex^{\langlem\rangle}\rangle$, being a normal subgroup of $\langle M, x \rangle$ containing $x$, is also $\langle M, x\rangle$. However, $\langlex^{\langlem\rangle}\rangle = \langle M, x \rangle$ is not an $\mathcal{X}$-group since the subgroup $T$ is not an $\mathcal{X}$-group. This contradiction completes the proof.
\varepsilonnd{proof}
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\section{Infinite Groups} \label{s:infinite}
\subsection{Generating infinite groups} \label{ss:infinite_intro}
We now turn to infinite groups and their generating pairs. Do the results from Section~\ref{s:finite} on the spread of finite groups extend to infinite groups? Let us recall that the motivating theorem for finite groups is the landmark result that every finite simple group is $2$-generated. This result is easily seen to be false when the assumption of finiteness is removed. For example, the alternating group $\mathrm{Alt}(\mathbb{Z})$ is simple but is not finitely generated since every finite subset of $\mathrm{Alt}(\mathbb{Z})$ is supported on finitely many points and therefore generates a finite subgroup. The problem persists even if we restrict to finitely generated simple groups. Answering a question of Wiegold in the Kourovka Notebook \cite[Problem~6.44]{ref:Kourovka22}, in 1982, Guba constructed a finitely generated infinite simple group that is not 2-generated (in fact, the group constructed has the property that every 2-generated subgroup is free) \cite{ref:Guba82}. More recently, Osin and Thom, by studying the $\varepsilonll^2$-Betti number of groups, proved that for every $k \mathfrak{g}eq 2$ there exists an infinite simple group that is $k$-generated but not $(k-1)$-generated \cite[Corollary~1.2]{ref:OsinThom13}.
With these results in mind, it makes sense to focus on $2$-generated groups and ask whether the results about the spread of finite $2$-generated groups extend to general $2$-generated groups. Recall that Theorem~\ref{thm:burness_guralnick_harper} gives a characterisation of the finite $\frac{3}{2}$-generated groups: a finite group $G$ is $\frac{3}{2}$-generated if and only if every proper quotient of $G$ is cyclic. In particular, every finite simple group is $\frac{3}{2}$-generated. The following example due to Cox in 2022 \cite{ref:Cox22}, highlights that this characterisation does not extend to general $2$-generated groups (that is, there exists a infinite $2$-generated group $G$ that is not $\frac{3}{2}$-generated but for which every proper quotient of $G$ is cyclic).
\begin{example} \label{ex:cox}
For each positive integer $n$, let $G_n$ be the subgroup of $\mathrm{Sym}(\mathbb{Z})$ defined as $\langle \mathrm{Alt}(\mathbb{Z}), t^n \rangle$ where $t\colon \mathbb{Z} \to \mathbb{Z}$ is the translation $x \mapsto x+1$.
It is straightforward to show that $\mathrm{Alt}(\mathbb{Z})$ is the unique minimal normal subgroup of $G_n$, so every proper quotient of $G$ is cyclic.
In addition, $G_n$ is $2$-generated. Indeed, by \cite[Lemma~3.7]{ref:Cox22}, $G_n = \langle a_n, t^n \rangle$ for $a_n = \varphirod_{i=0}^{m+1} x_i^{t^{3ni}}$ where $x_0, \dots, x_{m+1} \in A_{3n}$ satisfy $x_0 = (1 \, 3)$, $x_{m+1} = (2 \, 3)$ and $\{ (1 \, 2 \, 3)^{x_1}, \dots, (1 \, 2 \, 3)^{x_m} \} = (1 \, 2 \, 3)^{A_{3n}}$. To see this, we note that $[a_n^{t^{-3n(m+1)}},a_n] = [x_{m+1},x_0] = (1 \, 2 \, 3)$ and $(1 \, 2 \, 3)^{a_n^{t^{-3ni}}} = (1 \, 2 \, 3)^{x_i}$, so $\langle a_n, t^n \rangle \mathfrak{g}eq \langle (1 \, 2 \, 3)^{A_{3n}}, t^n \rangle = \langle A_{3n}, t^n \rangle$, which is simply $\langle \mathrm{Alt}(\mathbb{Z}), t^n \rangle$ (compare with Lemma~\ref{lem:covering_symmetric} below).
However, in \cite[Theorem~4.1]{ref:Cox22}, Cox proves that if $n \mathfrak{g}eq 3$, then $(1 \, 2 \, 3)$ is not contained in a generating pair for $G_n$, so $G_n$ gives an example of a $2$-generated group all of whose proper quotients are cyclic but which is not $\frac{3}{2}$-generated. To simplify the proof, we will assume that $n \mathfrak{g}eq 4$. Let $g \in G_n$. If $g \in \mathrm{Alt}(\mathbb{Z})$, then $\langle (1 \, 2 \, 3), g \rangle \leqslant \mathrm{Alt}(\mathbb{Z}) < G_n$. Now assume that $g \not\in \mathrm{Alt}(\mathbb{Z})$, so $\langleg\rangle = \langleht^k\rangle$ where $h \in \mathrm{Alt}(\mathbb{Z})$ and $k \mathfrak{g}eq 4$. It is easy to show that $g$ has exactly $k$ infinite orbits $O_1, \dots, O_k$ (indeed, if $\mathrm{supp}(h) \subseteq [a,b]$, then we quickly see that for a suitable permutation $\varphii \in \mathbb{S}m{k}$, we can find $k$ orbits $O_1, \dots, O_k$ of $g$ satisfying $O_i \setminus [a,b] = \{ x > b \mid x \varepsilonquiv i \mod{k} \} \cup \{ x < a \mid x \varepsilonquiv i\varphii \mod{k} \}$). Since $k \mathfrak{g}eq 4$, we can fix $i$ such that $O_i \cap \{1, 2, 3\} = \varepsilonmptyset$, so $O_i$ is an orbit of $\langle (1 \, 2 \, 3), g \rangle$, which implies that $\langle (1 \, 2 \, 3), g \rangle \neq G_n$ in this case too.
In contrast, in \cite[Theorem~6.1]{ref:Cox22}, Cox shows that $G_1$ and $G_2$ are $\frac{3}{2}$-generated, and, in fact, $2 \leqslant u(G_i) \leqslant s(G_i) \leqslant 9$ for $i \in \{1,2\}$.
\varepsilonnd{example}
The groups in Example~\ref{ex:cox} are not simple, so the following question remains.
\begin{question} \label{que:infinte_0}
Does there exist a $2$-generated simple group $G$ with $s(G)=0$?
\varepsilonnd{question}
Recall that for finite groups $G$, Theorem~\ref{thm:burness_guralnick_harper} also establishes that $s(G) \mathfrak{g}eq 1$ if and only if $s(G) \mathfrak{g}eq 2$, so there are no finite groups $G$ satisfying $s(G)=1$. This raises the following question.
\begin{question} \label{que:infinite_1}
Does there exist a $2$-generated simple group $G$ with $s(G)=1$?
\varepsilonnd{question}
There is a clear difference between generating finite and infinite groups, and straightforward analogues of the theorems for finite groups do not hold for infinite groups. Nevertheless, do the results on the spread of finite simple groups extend to important classes of infinite simple groups? The investigation of the infinite simple groups of Richard Thompson (and their many generalisations) in Sections~\ref{ss:infinite_thompson_introduction}--\ref{ss:infinite_thompson_t} demonstrates that the answer is a resounding yes! However, before turning to these infinite simple groups, in Section~\ref{ss:infinite_soluble}, we look at the other important special case we considered for finite groups: soluble groups.
\subsection{Soluble groups} \label{ss:infinite_soluble}
In the opening to Section~\ref{ss:finite_bgh}, we noted that when Brenner and Wiegold introduced the notion of spread, they proved that for a finite soluble group $G$, we have $s(G) \mathfrak{g}eq 1$ if and only if $s(G) \mathfrak{g}eq 2$ if and only if every proper quotient of $G$ is cyclic (see Theorem~\ref{thm:brenner_wiegold}). By Theorem~\ref{thm:breuer_guralnick_kantor}, ``soluble'' can be removed from the hypothesis (while keeping ``finite''). The following theorem establishes that ``finite'' can be removed from the hypothesis (while keeping ``soluble''), in a very strong sense. (Theorem~\ref{thm:infinite_soluble} is due to the author, and this is the first appearance of it in the literature.)
\begin{theorem} \label{thm:infinite_soluble}
Let $G$ be an infinite soluble group such that every proper quotient is cyclic. Then $G$ is cyclic.
\varepsilonnd{theorem}
\begin{proof}
It suffices to show that $G$ is abelian, because an infinite abelian group where every proper quotient is cyclic is itself cyclic. For a contradiction, suppose that $G$ is nonabelian. If $1 \neq N \trianglelefteqslant G$, then $G/N$ is cyclic, so $G' \leqslant N$. Therefore, $G'$ is the unique minimal normal subgroup of $G$. In particular, $G''$ is $G'$ or $1$, but $G$ is soluble, so $G'' = 1$, which implies that $G'$ is abelian. Therefore, $G'$ is an abelian characteristically simple group, so it is isomorphic to the additive group of a vector space $V$ over a field $F$, and we may assume that $F = \mathbb{F}_p$ or $F=\mathbb{Q}$.
Let $g \in G$ such that $G/V = \langleVg\rangle$ and write $H = \langleg\rangle$, so $G = VH$. Observe that $Z(G)=1$, for otherwise $G/Z(G)$ is cyclic, so $G$ is abelian, a contradiction. Now, if $g^i \in V$, then $g^i \in Z(G) = 1$, so $V \cap H = 1$. Hence, $G$ is a semidirect product $V{:}H$. For all nontrivial $v \in V$, we have $V = \langle v^G \rangle$ since $V$ is a minimal normal subgroup and $\langle v^G \rangle = \langle v^H \rangle$ since $V$ is abelian. Therefore, $V$ is an irreducible $FH$-module, so $V$ is finite-dimensional since $H$ is cyclic. (To see this, suppose that $V$ is infinite-dimensional, so $H$ is infinite. We give a proper nonzero submodule $U$, contradicting the irreducibility of $V$. For $0 \neq v \in V$, either $\{ vg^i \mid i \in \mathbb{Z} \}$ is linearly independent, and $U$ is the kernel of $\sum_{i \in \mathbb{Z}} a_ivg^i \mapsto \sum_{i \in \mathbb{Z}} a_i$, or for some $u = vg^i$ we have $a_0 u + a_1 ug + \cdots + a_k ug^k = 0$ and $U = \langle u, ug, \dots, ug^{k-1} \rangle$.) If $g^i \in C_G(V)$, then $g^i \in Z(G) = 1$, so $V$ is a faithful $FH$-module. In particular, if $F$ is finite, then so is $G = F^n{:}H \leqslant F^n{:}\mathrm{GL}_n(F)$, so we must have $F = \mathbb{Q}$.
Let $\chi = X^n + a_{n-1}X^{n-1} + \cdots + a_1X + a_0 \in \mathbb{Q}[X]$ be the characteristic polynomial of $g$, and let $(e_1,\dots,e_n)$ be a basis for $V$ with respect to which the matrix $A$ of $g$ is the companion matrix of $\chi$. Let $P$ be the set of prime divisors appearing in the reduced forms of $a_0, \dots, a_{n-1}$ and note that $P$ is finite. For all $i \in \mathbb{Z}$, write $e_1A^i$ as a linear combination $\lambda_{i1}e_1 + \cdots + \lambda_{in}e_n$. Any prime that divides the denominator of the reduced form of one of the $\lambda_{ij}$ is contained in $P$. Hence, only finitely many primes appear in the denominators of the reduced forms of any element in the subgroup $N$ generated by $\{ e_1A^i \mid i \in \mathbb{Z} \}$. Since $N$ is $\langlee_1^G\rangle$, it is a proper nontrivial subgroup of $V$ that is normal in $G$, which contradicts $V$ being a minimal normal subgroup of $G$. Therefore, $G$ is abelian and so cyclic.
\varepsilonnd{proof}
With a much shorter proof, one can obtain an analogous result for the class of residually finite groups (this was observed by Cox in \cite[Lemma~1.1]{ref:Cox22}).
\begin{theorem} \label{thm:infinite_residually_finite}
Let $G$ be an infinite residually finite group such that every proper quotient is cyclic. Then $G$ is cyclic.
\varepsilonnd{theorem}
\begin{proof}
Suppose that $G$ is nonabelian. Fix $x,y \in G$ with $[x,y] \neq 1$. Since $G$ is residually finite, $G$ has a finite index normal subgroup $N$ such that $[Nx,Ny]$ is nontrivial in $G/N$ (so, $Nx$ and $Ny$ are nontrivial in $G/N$). Since $G$ is infinite and $N$ has finite index, we know that $N$ is nontrivial, so $G/N$ is cyclic, which contradicts $G/N$ being nonabelian. Therefore, $G$ is abelian and hence cyclic.
\varepsilonnd{proof}
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\subsection{Thompson's groups: an introduction} \label{ss:infinite_thompson_introduction}
In 1965, Richard Thompson introduced three finitely generated infinite groups $F < T < V$ \cite{ref:Thompson65}. Among other interesting properties of these groups, $V$ and $T$ were the first known examples of finitely presented infinite simple groups and for 35 years (until the work of Burger and Mozes \cite{ref:BurgerMozes00}) all known examples of such groups were closely related to $T$ and $V$. For an indication of other interesting properties of these groups, we record that $F$ is finitely presented yet it contains a copy of $F \times F$ and is an HNN extension of itself. Moreover, $F$ has exponential growth but contains no nonabelian free groups, and one of the most famous open questions in geometric group theory is whether $F$ is amenable \cite{ref:Cleary17}. However, these three groups not only raise interesting group theoretic questions, but they have also played a role in a whole range of mathematical areas such as the word problem for groups, homotopy theory and dynamical systems (see \cite{ref:Dydak77_2,ref:GhysSergiescu87,ref:Thompson80} for example). We refer the reader to Canon, Floyd and Parry's introduction to these groups \cite{ref:CannonFloydParry96}.
An appealing feature of Thompson's groups is that they admit concrete representations as transformation groups, which we outline now. Let $X = \{0,1\}^*$ be the set of all finite words over $\{0,1\}$, and let $\mathfrak{C} = \{0,1\}^\mathbb{N}$ be \varepsilonmph{Cantor space}, the set of all infinite sequences over $\{0,1\}$ with the usual topology. For $u \in X$, we write $u\mathfrak{C} = \{uw \mid w \in \mathfrak{C} \}$, and we say a finite set $A \subseteq X$ is a \varepsilonmph{basis} of $\mathfrak{C}$ if $\{ u\mathfrak{C} \mid u \in A \}$ is a partition of $\mathfrak{C}$. Thompson's group $V$ is the group of homeomorphisms $g \in \mathrm{Homeo}(\mathfrak{C})$ for which there exists a \varepsilonmph{basis pair}, namely a bijection $\sigma\colon A \mapsto B$ between two bases $A$ and $B$ of $\mathfrak{C}$ such that $(uw)g = (u\sigma)w$ for all $u \in A$ and all $w \in \mathfrak{C}$. In other words, $V$ is the group of homeomorphisms of $\mathfrak{C}$ that act by prefix substitutions. For instance, $c\colon(00,01,1) \mapsto (11,0,10)$ is an element of $V$ that, for example, maps $01010101\dots$ to $0010101\dots$. The selfsimilarity of $\mathfrak{C}$ means that there is not a unique choice of basis pair; indeed, by subdividing $\mathfrak{C}$ further $c$ is also represented by $(000, 001, 01, 1) \mapsto (110, 111, 0, 10)$. By identifying elements of $X$ with vertices of the infinite binary rooted tree, we can represent bases as binary rooted trees and elements of $V$ by the familiar \varepsilonmph{tree pairs}, as shown in Figure~\ref{fig:elements}.
\begin{figure}
\begin{minipage}{0.42\textwidth}
\begin{gather*}
\text{\footnotesize $a = (000,001,01,10,110,111)$} \\
\text{\footnotesize \qquad $\mapsto (001,000,01,110,111,10)$} \\[1pt]
\begin{tikzpicture}[
scale=0.5,
font=\footnotesize,
inner sep=0pt,
baseline=-30pt,
level distance=20pt,
level 1/.style={sibling distance=60pt},
level 2/.style={sibling distance=30pt},
level 3/.style={sibling distance=15pt}
]
\node (root) [circle,fill] {}
child {node (0) [circle,fill] {}
child {node (00) [circle,fill] {}
child {node (000) {1}}
child {node (001) {2}}}
child {node (01) {3}}}
child {node (1) [circle,fill] {}
child {node (10) {4}}
child {node (11) [circle,fill] {}
child {node (110) {5}}
child {node (111) {6}}}};
\varepsilonnd{tikzpicture}
\raisebox{3mm}{ $\longrightarrow$ }
\begin{tikzpicture}[
scale=0.5,
font=\footnotesize,
inner sep=0pt,
baseline=-30pt,
level distance=20pt,
level 1/.style={sibling distance=60pt},
level 2/.style={sibling distance=30pt},
level 3/.style={sibling distance=15pt}
]
\node (root) [circle,fill] {}
child {node (0) [circle,fill] {}
child {node (00) [circle,fill] {}
child {node (000) {2}}
child {node (001) {1}}}
child {node (01) {3}}}
child {node (1) [circle,fill] {}
child {node (10) {6}}
child {node (11) [circle,fill] {}
child {node (110) {4}}
child {node (111) {5}}}};
\varepsilonnd{tikzpicture}
\varepsilonnd{gather*}
\varepsilonnd{minipage}
\begin{minipage}{0.48\textwidth}
\begin{gather*}
\text{\footnotesize $b = (000,001,010,011,100,101,110,111)$} \\
\text{\footnotesize \quad $\mapsto (000,010,011,100,101,110,111,001)$} \\[1pt]
\begin{tikzpicture}[
scale=0.5,
font=\footnotesize,
inner sep=0pt,
baseline=-30pt,
level distance=20pt,
level 1/.style={sibling distance=60pt},
level 2/.style={sibling distance=30pt},
level 3/.style={sibling distance=15pt}
]
\node (root) [circle,fill] {}
child {node (0) [circle,fill] {}
child {node (00) [circle,fill] {}
child {node (000) {1}}
child {node (001) {2}}}
child {node (01) [circle,fill] {}
child {node (010) {3}}
child {node (011) {4}}}}
child {node (1) [circle,fill] {}
child {node (10) [circle,fill] {}
child {node (100) {5}}
child {node (101) {6}}}
child {node (11) [circle,fill] {}
child {node (110) {7}}
child {node (111) {8}}}};
\varepsilonnd{tikzpicture}
\raisebox{3mm}{ $\longrightarrow$ }
\begin{tikzpicture}[
scale=0.5,
font=\footnotesize,
inner sep=0pt,
baseline=-30pt,
level distance=20pt,
level 1/.style={sibling distance=60pt},
level 2/.style={sibling distance=30pt},
level 3/.style={sibling distance=15pt}
]
\node (root) [circle,fill] {}
child {node (0) [circle,fill] {}
child {node (00) [circle,fill] {}
child {node (000) {1}}
child {node (001) {8}}}
child {node (01) [circle,fill] {}
child {node (010) {2}}
child {node (011) {3}}}}
child {node (1) [circle,fill] {}
child {node (10) [circle,fill] {}
child {node (100) {4}}
child {node (101) {5}}}
child {node (11) [circle,fill] {}
child {node (110) {6}}
child {node (111) {7}}}};
\varepsilonnd{tikzpicture}
\varepsilonnd{gather*}
\varepsilonnd{minipage}
\begin{minipage}{0.42\textwidth}
\begin{gather*}
\text{\footnotesize $c = (00,01,1) \mapsto (11,0,10)$} \\[1pt]
\begin{tikzpicture}[
scale=0.5,
font=\footnotesize,
inner sep=0pt,
baseline=-30pt,
level distance=20pt,
level 1/.style={sibling distance=60pt},
level 2/.style={sibling distance=30pt}
]
\node (root) [circle,fill] {}
child {node (0) [circle,fill] {}
child {node (00) {1}}
child {node (01) {2}}}
child {node (1) {3}};
\varepsilonnd{tikzpicture}
\raisebox{5mm}{ \ $\longrightarrow$ \ }
\begin{tikzpicture}[
scale=0.5,
font=\footnotesize,
inner sep=0pt,
baseline=-30pt,
level distance=20pt,
level 1/.style={sibling distance=60pt},
level 2/.style={sibling distance=30pt}
]
\node (root) [circle,fill] {}
child {node (0) {2}}
child {node (1) [circle,fill] {}
child {node (10) {3}}
child {node (11) {1}}};
\varepsilonnd{tikzpicture} \\[-2mm]
\begin{tikzpicture}[scale=2,thick]
\draw [fill=white] ( 0:0.3) -- ( 0:0.2) arc ( 0: 90:0.2) -- ( 90:0.3) arc ( 90: 0:0.3);
\draw [fill=black!40] ( 90:0.3) -- ( 90:0.2) arc ( 90:180:0.2) -- (180:0.3) arc (180: 90:0.3);
\draw [fill=black] (180:0.3) -- (180:0.2) arc (180:360:0.2) -- (360:0.3) arc (360:180:0.3);
\varepsilonnd{tikzpicture}
\raisebox{5mm}{ $\longrightarrow$ }
\begin{tikzpicture}[scale=2,thick,draw=blue!40!black]
\draw [fill=black!40] ( 0:0.3) -- ( 0:0.2) arc ( 0:180:0.2) -- (180:0.3) arc (180: 0:0.3);
\draw [fill=black] (180:0.3) -- (180:0.2) arc (180:270:0.2) -- (270:0.3) arc (270:180:0.3);
\draw [fill=white] (270:0.3) -- (270:0.2) arc (270:360:0.2) -- (360:0.3) arc (360:270:0.3);
\varepsilonnd{tikzpicture}
\varepsilonnd{gather*}
\varepsilonnd{minipage}
\begin{minipage}{0.48\textwidth}
\begin{gather*}
\text{\footnotesize $s = (00,01,10,11) \mapsto (0,100,101,11)$} \\[1pt]
\begin{tikzpicture}[
scale=0.5,
inner sep=0pt,
baseline=-30pt,
level distance=20pt,
level 1/.style={sibling distance=60pt},
level 2/.style={sibling distance=30pt},
level 3/.style={sibling distance=15pt}
]
\node (root) [circle,fill] {}
child {node (0) [circle,fill] {}
child {node (00) {1}}
child {node (01) {2}}}
child {node (1) [circle,fill] {}
child {node (10) {3}}
child {node (11) {4}}};
\varepsilonnd{tikzpicture}
\raisebox{5mm}{ \ $\longrightarrow$ \ }
\begin{tikzpicture}[
scale=0.5,
font=\footnotesize,
inner sep=0pt,
baseline=-30pt,
level distance=20pt,
level 1/.style={sibling distance=60pt},
level 2/.style={sibling distance=30pt},
level 3/.style={sibling distance=15pt}
]
\node (root) [circle,fill] {}
child {node (0) {1}}
child {node (1) [circle,fill] {}
child {node (10) [circle,fill] {}
child {node (100) {2}}
child {node (101) {3}}}
child {node (11) {4}}};
\varepsilonnd{tikzpicture} \\[3.1mm]
\begin{tikzpicture}[scale=2,thick]
\draw (0, 0) rectangle (0.25,0.1);
\draw (0.25,0) rectangle (0.5, 0.1);
\draw (0.5, 0) rectangle (0.75,0.1);
\draw (0.75,0) rectangle (1, 0.1);
\varepsilonnd{tikzpicture}
\longrightarrow
\begin{tikzpicture}[scale=2,thick]
\draw (0, 0) rectangle (0.5, 0.1);
\draw (0.5, 0) rectangle (0.625,0.1);
\draw (0.625,0) rectangle (0.75, 0.1);
\draw (0.75, 0) rectangle (1, 0.1);
\varepsilonnd{tikzpicture}
\varepsilonnd{gather*}
\varepsilonnd{minipage}
\caption{Four elements of Thompson's group $V$.} \label{fig:elements}
\varepsilonnd{figure}
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\varepsilonnd{minipage}
\caption{Four elements of Thompson's group $V$.} \label{fig:elements}
\varepsilonnd{figure}
A motivating perspective in the study of generating sets of Thompson's groups is that $V$ combines the selfsimilarity of the Cantor space with permutations from the symmetric group. Indeed, to $g \in V$ we may associate (not uniquely) a permutation as follows. Let $\sigma\colonA \to B$ be a basis pair for $g$ and write $A = \{ a_1, \dots, a_n \}$ and $B = \{ b_1, \dots, b_n \}$ where $a_1 < \dots < a_n$ and $b_1 < \dots < b_n$ in the lexicographic order. Then the permutation associated to $g$ is the element $\varphii_g \in \mathbb{S}m{n}$ satisfying $a_ig = b_{i\varphii_g}$. For the elements in Figure~\ref{fig:elements}, for instance,
\[
\varphii_a = (1 \, 2)(4 \, 5 \, 6), \quad \varphii_b = (2 \, 3 \, 4 \, 5 \, 6 \, 7 \, 8), \quad \varphii_c = (1 \, 2 \, 3), \quad \varphii_s = 1.
\]
This perspective gives an easy way to define $F$ and $T$. Thompson's groups $F$ and $T$ are the subgroups of $V$ of elements whose associated permutation is trivial and cyclic, respectively (this is well defined). Clearly $F < T < V$ and, referring to Figure~\ref{fig:elements}, we see that $s \in F$, $c \in T \setminus F$ and $a,b \in V \setminus T$.
Given a binary word $u$, we can associate a subset $I_u$ of the unit interval $[0,1]$ (or unit circle $\mathbb{S}^1$) inductively as follows: the empty word corresponds to $(0,1)$ and for any binary word $u$, the words $u0$ and $u1$ correspond to the open left and right halves of $u$. In this way, a basis for $\mathfrak{C}$ can be interpreted as a sequence of disjoint open intervals whose closures cover $[0,1]$ (or $\mathbb{S}^1$), and an element of $V$, as a basis pair, $\sigma\colon\{a_1, \dots, a_n\}\to\{b_1, \dots, b_n\}$ defines a bijection $g\colon[0,1] \to [0,1]$ (or $g\colon\mathbb{S}^1 \to \mathbb{S}^1$) by specifying that $I_{a_i}g = I_{b_i}$ and $g|_{I_{a_i}}$ is affine for all $1 \leqslant i \leqslant n$. Under this correspondence, $F$ is a group of piecewise linear homeomorphisms of $[0,1]$ and $T$ is a group of piecewise linear homeomorphisms of $\mathbb{S}^1$. See Figure~\ref{fig:elements} for some examples.
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\subsection{\boldmath Thompson's group $V$} \label{ss:infinite_thompson_v}
The final three sections of this survey address generating sets for Thompson's groups, which have seen lots of very recent progress. We begin with $V$.
The group $V$ is $2$-generated. Indeed, referring to Figure~\ref{fig:elements}, $V = \langle a,b \rangle$. Given this infinite $2$-generated simple group $V$, we are naturally led to ask: is $V$ $\frac{3}{2}$-generated? An answer was given by Donoven and Harper in 2020 \cite{ref:DonovenHarper20}.
\begin{theorem} \label{thm:donoven_harper}
Thompson's group $V$ is $\frac{3}{2}$-generated.
\varepsilonnd{theorem}
Theorem~\ref{thm:donoven_harper} gave the first example of a noncyclic infinite $\frac{3}{2}$-generated group, other than the pathological \varepsilonmph{Tarski monsters}: the infinite groups whose only proper nontrivial subgroups have order $p$ for a fixed prime $p$, which are clearly simple and $\frac{3}{2}$-generated and were proved to exist for all $p > 10^{75}$ by Olshanskii in \cite{ref:Olshanskii80}. (Note that the groups $G_1$ and $G_2$ in Example~\ref{ex:cox} were found later by Cox in \cite{ref:Cox22}, motivated by a question posed in \cite{ref:DonovenHarper20}.) In particular, $V$ was the first finitely presented example of a noncyclic infinite $\frac{3}{2}$-generated group.
\textbf{Methods. A parallel with symmetric groups. } Let us write $G = \mathbb{S}m{n}$ and $\Omega = \{ 1, \dots, n \}$. It is well known that $G$ is generated by the set of transpositions $\{ (i \, j) \mid \text{distinct $i, j \in \Omega$} \}$ which yields a natural presentation for $G$, namely
\begin{equation} \label{eq:presentation_symmetric}
\langle t_{i,j} \mid t_{i,j}^2, \ t_{i,j}^{t_{k,l}} = t_{i (k \, l),j(k \, l)} \rangle
\varepsilonnd{equation}
Moreover, using the fact that $G$ is generated by transpositions, we obtain the following \varepsilonmph{covering lemma} (here $G_{[A]}$ is the subgroup of $G$ supported on $A \subseteq \Omega$).
\begin{lemma} \label{lem:covering_symmetric}
Let $A_1, \dots, A_k \subseteq \Omega$ satisfy $A_i \cap A_{i+1} \neq \varepsilonmptyset$ and $\bigcup_{i=1}^k{A_i} = \Omega$. Then $G_{[A_i]} \cong \mathbb{S}m{|A_i|}$ for all $i$, and $G = \langle G_{[A_1]}, \dots, G_{[A_k]} \rangle$.
\varepsilonnd{lemma}
These results for $G = \mathbb{S}m{n}$ have analogues for $V$. Here, \varepsilonmph{transpositions} are elements of the following form: for $u, v \in X$ such that the corresponding subsets of $\mathfrak{C}$ are disjoint, we write $(u \, v)$ for the element given as $(u,v,w_1, \dots, w_k) \mapsto (v,u,w_1, \dots, w_k)$ where $\{ u, v, w_1, \dots, w_k\}$ is a basis for $\mathfrak{C}$. Brin \cite{ref:Brin04} proved that $V$ is generated by $\{ (u \, v) \mid \text{disjoint $u,v \in X$} \}$.
As an aside, let us point out why the subgroup of elements that are a product of an even number of transpositions is not a proper nontrivial normal subgroup of $V$ (as with the symmetric and alternating groups): this subgroup is not proper. Indeed, every element of $V$ is a product of an even number of transpositions as the selfsimilarity of $\mathfrak{C}$ shows that $(u \, v)$ can be rewritten as $(u0 \, v0)(u1 \, v1)$.
Bleak and Quick \cite[Theorem~1.1]{ref:BleakQuick17} demonstrated how this generating set gives a presentation for $V$ combining the corresponding presentation for the symmetric group in \varepsilonqref{eq:presentation_symmetric} with the selfsimilarity of $\mathfrak{C}$, namely
\begin{equation} \label{eq:presentation_v}
\langle t_{u,v} \mid t_{u,v}^2, \ t_{u,w}^{t_{x,y}} = t_{u(x \, y),v(x \, y)}, \ t_{u,v} = t_{u0,v0}t_{u1,v1} \rangle
\varepsilonnd{equation}
(see \cite[(1.1)]{ref:BleakQuick17} for a full explanation of the notation used in the relations).
We will say no more about presentations, save that Bleak and Quick found a presentation for $V$ with 2 generators and 7 relations \cite[Theorem~1.3]{ref:BleakQuick17}, which they derived from another, more intuitive, presentation based on the analogy with $S_n$ which has 3 generators and 8 relations \cite[Theorem~1.2]{ref:BleakQuick17}.
As with the symmetric group, the fact that $V$ is generated by transpositions yields an easy proof of the following.
\begin{lemma} \label{lem:covering_v}
Let $U_1, \dots, U_k \subseteq \mathfrak{C}$ be clopen subsets satisfying $U_i \cap U_{i+1} \neq \varepsilonmptyset$ and $\bigcup_{i=1}^k U_i = \mathfrak{C}$. Then $V_{[U_i]} \cong V$ for all $i$, and $V = \langle V_{[U_1]}, \dots, V_{[U_k]} \rangle$.
\varepsilonnd{lemma}
With Lemma~\ref{lem:covering_v} in place, we now highlight the main ideas in the proof of Theorem~\ref{thm:donoven_harper} by way of an example (this is \cite[Example~4.1]{ref:DonovenHarper20}). We will see an alternative approach in Theorem~\ref{thm:donoven_harper_hyde}
\begin{example} \label{ex:donoven_harper}
Let $x = (00 \ \ 01) \in V$. We will construct $y \in V$ such that $\langlex,y\rangle = V$. Let $y_1 = a_{[00]}$ and $y_2 = b_{[01]}$, where for $g \in V$ and clopen $A \subseteq \mathfrak{C}$ we write $g_{[A]}$ for the image of $g$ under the canonical isomorphism $V \to V_{[A]}$. Let $y_3 = (00 \ \ 01 \ \ 10 \ \ 11)_{[0^310]} \cdot (0^310^3 \ \ 010^3)$ and $y_4 = (0000 \ \ 0001 \ \ \cdots \ \ 1010)_{[0^31^2]} \cdot (0^31^20^4 \ \ 1)$, and define $y = y_1y_2y_3y_4$. Note that $y_1$, $y_2$, $y_3$ and $y_4$ have coprime orders (6, 7, 5 and 11, respectively). Moreover, these elements have disjoint support, so they commute. Consequently, all four elements are suitable powers of $y$ and are, thus, contained in $\langlex,y\rangle$. We claim that $\langlex,y\rangle = V$. Recall that $V = \langle a,b\rangle$, so $V_{[00]} = \langlea_{[00]},b_{[00]}\rangle = \langley_1, y_2^x\rangle \leqslant \langlex,y\rangle$. In addition, $V_{[01]} = (V_{[00]})^x \leqslant \langlex,y\rangle$. Using appropriate elements from $V_{[00]}$ and $V_{[01]}$ we can show that $(000 \ \ 01) \in \langle V_{[00]}, V_{[01]}, y_3 \rangle \leqslant \langlex,y\rangle$ and $(000 \ \ 1) \in \langle V_{[00]}, y_4 \rangle \leqslant \langlex,y\rangle$. Therefore, $\langlex,y\rangle \mathfrak{g}eq \langle V_{[00]}, V_{[00]}^{(000 \ \ 01)}, V_{[00]}^{(000 \ \ 1)} \rangle = \langle V_{[000 \cup 001]}, V_{[01 \cup 001]}, V_{[1 \cup 001]} \rangle$. Now applying Lemma~\ref{lem:covering_v} twice gives $\langle x,y \rangle = V$.
\varepsilonnd{example}
\subsection{\boldmath Generalisations of $V$} \label{ss:infinite_thompson_general}
There are numerous variations on $V$, and these are the focus of this section.
The \varepsilonmph{Higman--Thompson group} $V_n$, for $n \mathfrak{g}eq 2$, is an infinite finitely presented group, introduced by Higman in \cite{ref:Higman74}. There is a natural action of $V_n$ on $n$-ary Cantor space $\mathfrak{C}_n = \{0,1,\dots,n-1\}^{\mathbb{N}}$, and $V_2$ is nothing other than $V$. The derived subgroup of $V_n$ equals $V_n$ for even $n$ and has index two for odd $n$. In both cases, $V_n'$ is simple and both $V_n$ and $V_n'$ are $2$-generated \cite{ref:Mason77}.
The \varepsilonmph{Brin--Thompson group} $nV$, for $n \mathfrak{g}eq 1$, acts on $\mathfrak{C}^n$ and was defined by Brin in \cite{ref:Brin04}. The groups $V=1V, 2V, 3V, \dots$ are pairwise nonisomorphic \cite{ref:BleakLanoue10}, simple \cite{ref:Brin10} and $2$-generated \cite[Corollary~1.3]{ref:Quick19}.
The results about generating $V$ by transpositions have analogues for $V_n$ and $nV$ (see \cite[Section~3]{ref:DonovenHarper20}), and, in \cite[Theorem~1.1]{ref:Quick19}, Quick gives a presentation for $nV$ analogous to the one for $V$ in \varepsilonqref{eq:presentation_v}. Theorem~\ref{thm:donoven_harper} extends to all of these groups too \cite[Theorems~1 \& 2]{ref:DonovenHarper20}.
\begin{theorem} \label{thm:donoven_harper_generalisation}
For all $n \mathfrak{g}eq 2$, the Higman--Thompson groups $V_n$ and $V_n'$ are $\frac{3}{2}$-generated, and for all $n \mathfrak{g}eq 1$, the Brin--Thompson group $nV$ is $\frac{3}{2}$-generated.
\varepsilonnd{theorem}
In particular, the groups $V_n$ when $n$ is odd give infinitely many examples of infinite $\frac{3}{2}$-generated groups that are not simple.
As we introduced them, the Higman--Thompson group $V_n'$ is a simple subgroup of $\mathrm{Homeo}(\mathfrak{C}_n)$ and the Brin--Thompson group $nV$ is a simple subgroup of $\mathrm{Homeo}(\mathfrak{C}^n)$. Since $\mathfrak{C}^n$ ($n$th power of $\mathfrak{C}$) and $\mathfrak{C}_n$ ($n$-ary Cantor space) are both homeomorphic to $\mathfrak{C}$, all of these groups can be viewed as subgroups of $\mathrm{Homeo}(\mathfrak{C})$. Recent work of Bleak, Elliott and Hyde \cite{ref:BleakElliottHyde}, highlights that these groups, and numerous others (such as Nekrashevych's simple groups of dynamical origin), can be viewed within one unified dynamical framework.
A group $G \leqslant \mathrm{Homeo}(\mathfrak{C})$ is said to be \varepsilonmph{vigorous} if for any clopen subsets $\varepsilonmptyset \subsetneq B,C \subsetneq A \subseteq \mathfrak{C}$ there exists $g \in G$ supported on $A$ such that $Bg \subseteq C$. In \cite{ref:BleakElliottHyde}, Bleak, Elliot and Hyde study vigorous groups and, among much else, prove that a perfect vigorous group $G \leqslant \mathrm{Homeo}(\mathfrak{C})$ is simple if and only if it is generated by its elements of \varepsilonmph{small support} (namely, elements supported on a proper clopen subset of $\mathfrak{C}$). To give a flavour of how these dynamical properties suitably capture the ideas we have seen in this section, compare the following, which is \cite[Lemma~2.18 \& Proposition~2.19]{ref:BleakElliottHyde}, with Lemma~\ref{lem:covering_v}.
\begin{lemma} \label{lem:covering_vigorous}
Let $G$ be a vigorous group that is generated by its elements of small support. Let $U_1, \dots, U_k \subseteq \mathfrak{C}$ be clopen subsets satisfying $U_i \cap U_{i+1} \neq \varepsilonmptyset$ and $\bigcup_{i=1}^k U_i = \mathfrak{C}$. Then $G = \langle G_{[U_1]}, \dots, G_{[U_k]} \rangle$. Moreover, if $G$ is simple, then for each $i$ the group $G_{[U_i]}$ is a simple vigorous group.
\varepsilonnd{lemma}
Bleak, Elliott and Hyde go on to prove that every finitely generated simple vigorous group is $2$-generated \cite[Theorem~1.12]{ref:BleakElliottHyde}. Are all such groups $\frac{3}{2}$-generated? Bleak, Donoven, Harper and Hyde \cite{ref:BleakDonovenHarperHyde} recently proved that $u(G) \mathfrak{g}eq 1$.
\begin{theorem} \label{thm:donoven_harper_hyde}
Let $G \leqslant \mathrm{Homeo}(\mathfrak{C})$ be a finitely generated simple vigorous group. Then there exists an element $s \in G$ of small support and order 30 such that for every nontrivial $x \in G$ there exists $y \in s^G$ such that $\langle x, y \rangle = G$.
\varepsilonnd{theorem}
Theorem~\ref{thm:donoven_harper_hyde} gives $u(G) \mathfrak{g}eq 1$ for all the simple groups $G$ in Theorem~\ref{thm:donoven_harper_generalisation}. In particular, we obtain a strong version of Theorem~\ref{thm:donoven_harper} on Thompson's group $V$, improving $s(V) \mathfrak{g}eq 1$ to $u(V) \mathfrak{g}eq 1$. It is possible to obtain stronger results on the (uniform) spread of $V$ and its generalisations (and $T$, discussed below), and this is the subject of current work of the author and others (e.g. \cite{ref:BleakDonovenHarperHyde}).
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\subsection{\boldmath Thompson's groups $T$ and $F$} \label{ss:infinite_thompson_t}
In this final section, we discuss generating sets of Thompson's groups $T$ and $F$. We begin with $T$, which is a simple $2$-generated group, so it is natural to study its (uniform) spread. In 2022, Bleak, Harper and Skipper \cite{ref:BleakHarperSkipper} proved $u(T) \mathfrak{g}eq 1$.
\begin{theorem} \label{thm:bleak_harper_skipper}
There exists an element $s \in T$ such that for every nontrivial $x \in T$ there exists $y \in s^T$ such that $\langle x, y \rangle = T$.
\varepsilonnd{theorem}
\begin{corollary} \label{cor:bleak_harper_skipper}
Thompson's group $T$ is $\frac{3}{2}$-generated.
\varepsilonnd{corollary}
The element $s$ in Theorem~\ref{thm:bleak_harper_skipper} can be chosen as the one in Figure~\ref{fig:elements}. Moreover, in \cite[Proposition~3.1]{ref:BleakHarperSkipper}, it is shown that if we restrict to elements $x$ of infinite order, then we can choose $s$ to be any infinite order element, that is to say, for any two infinite order elements $x,s \in T$ there exists $g \in T$ such that $\langle x, s^g \rangle = T$. This naturally raises the question of whether an arbitrary infinite order element can be chosen for $s$ in Theorem~\ref{thm:bleak_harper_skipper} (see \cite[Question~1]{ref:BleakHarperSkipper}).
We now turn to Thompson's group $F$. This is $2$-generated since if we write $x_0 = (00, 01, 1) \mapsto (0, 10, 11)$ and $x_1 = (0, 100, 101, 1) \mapsto (0, 10, 110, 111)$, then, by \cite[Theorem~3.4]{ref:CannonFloydParry96} for example, $F = \langle x_0, x_1 \rangle$. Moreover, if we inductively define $x_{i+1} = x_i^{x_0}$ for all $i \mathfrak{g}eq 1$, then the elements $x_0, x_1, x_2, \dots$ witness the following well-known presentation
\[
F = \langle x_0, x_1, x_2, \dots \mid \text{$x_j^{x_i} = x_{j+1}$ for $i < j$} \rangle.
\]
However, $F$ is not a simple group. Considering $F$ in its natural action on $[0,1]$, the homomorphism $\varphii\colonF \to \mathbb{Z}^2$ defined as $f \mapsto (\log_2{f'(0^+)}, \log_2{f'(1^-)})$ is surjective and the kernel of $\varphii$ is the derived subgroup $F'$, which is simple. Moreover, $F'$ is the unique minimal normal subgroup of $F$, so the nontrivial normal subgroups of $F$ are in bijection with normal subgroups of $F/F' = \mathbb{Z}^2$ (see \cite[Section~4]{ref:CannonFloydParry96} for proofs of these claims). In particular, $F$ is not $\frac{3}{2}$-generated since it has a proper noncyclic quotient.
Now $F'$ is not $\frac{3}{2}$-generated for a different reason: it is not finitely generated. Indeed, for any nontrivial normal subgroup $N = \varphii^{-1}(\langle (a_0,a_1), (b_0,b_1) \rangle)$, if $\{ a_0, b_0 \} = \{ 0 \}$ or $\{ a_1, b_1 \} = \{ 0 \}$, then $N$ is not finitely generated. To see this in the former case, for finitely many elements each of which acts as the identity on an interval containing $0$, there exists an interval containing $0$ on which they all act as the identity, so they generate a proper subgroup of $N$ (for the latter case, replace $0$ with $1$). However, the following recent theorem of Golan \cite[Theorem~2]{ref:GolanGen} shows that these are the only obstructions to $\frac{3}{2}$-generation.
\begin{theorem} \label{thm:golan}
Let $(a_0,a_1), (b_0,b_1) \in \mathbb{Z}^2$ with $\{a_0,b_0\} \neq \{0\}$ and $\{a_1,b_1\} \neq \{0\}$. Let $x \in F$ be a nontrivial element such that $\varphii(x) = (a_0,a_1)$. Then there exists $y \in F$ such that $\varphii(y) = (b_0,b_1)$ and $\langlex,y\rangle = \varphii^{-1}(\langle(a_0,a_1),(b_0,b_1)\rangle$.
\varepsilonnd{theorem}
Theorem~\ref{thm:golan} has the following consequence, which asserts that $F$ is almost $\frac{3}{2}$-generated \cite[Theorem~1]{ref:GolanGen}.
\begin{corollary} \label{cor:golan}
Let $f \in F$ and assume that $\varphii(f)$ is contained in a generating pair of $\varphii(F)$. Then $f$ is contained in a generating pair of $F$.
\varepsilonnd{corollary}
Theorem~\ref{thm:golan} also implies that every finitely generated normal subgroup of $F$ is $2$-generated. In particular, every finite index subgroup of $F$ is $2$-generated.
\textbf{Methods. Covering lemmas and a generation criterion. } We conclude the survey by discussing how Theorems~\ref{thm:bleak_harper_skipper} and~\ref{thm:golan} are proved in \cite{ref:BleakHarperSkipper} and \cite{ref:GolanGen}, respectively. Covering lemmas (analogues of Lemma~\ref{lem:covering_symmetric}), again, play a role. For $F$ and $T$, these results are well known, see \cite[Corollary~2.6 \& Lemma~2.7]{ref:BleakHarperSkipper} for example. (We call an interval $[a,b]$ \varepsilonmph{dyadic} if $a,b \in \mathbb{Z}[\frac{1}{2}]$.)
\begin{lemma} \label{lem:covering_f}
Let $[a_1,b_1], \dots, [a_k,b_k] \subseteq [0,1]$ be dyadic intervals satisfying $\bigcup_{i=1}^k (a_i,b_i) = (0,1)$. Then $F_{[a_i,b_i]} \cong F$ for all $i$, and $F = \langle F_{[a_1,b_1]}, \dots, F_{[a_k,b_k]} \rangle$.
\varepsilonnd{lemma}
\begin{lemma} \label{lem:covering_t}
Let $[a_1,b_1], \dots, [a_k,b_k] \subseteq \mathbb{S}^1$ be dyadic intervals satisfying $\bigcup_{i=1}^k (a_i,b_i) = \mathbb{S}^1$. Then $T_{[a_i,b_i]} \cong F$ for all $i$, and $T = \langle T_{[a_1,b_1]}, \dots, T_{[a_k,b_k]} \rangle$.
\varepsilonnd{lemma}
Another key ingredient is a criterion due to Golan, for which we need some further notation. Fix a subgroup $H \leqslant F$. An element $f \in F \leqslant \mathrm{Homeo}([0,1])$ is \varepsilonmph{piecewise-$H$} if there is a finite subdivision of $[0,1]$ such that on each interval in the subdivision, $f$ coincides with an element of $H$. The closure of $H$, written ${\rm Cl}(H)$, is the subgroup of $F$ containing all elements that are piecewise-$H$. The following result combines \cite[Theorem~1.3]{ref:GolanMAMS} with \cite[Theorem~1.3]{ref:GolanMax}.
\begin{theorem} \label{thm:golan_criterion}
Let $H \leqslant F$. Then the following hold:
\begin{enumerate}
\item $H \mathfrak{g}eq F'$ if and only if ${\rm Cl}(H) \mathfrak{g}eq F'$ and there exist $f \in H$ and a dyadic $\omega \in (0,1)$ such that $f'(\omega^+)=2$ and $f'(\omega^-)=1$
\item $H = F$ if and only if ${\rm Cl}(H) \mathfrak{g}eq F'$ and there exist $f,g \in H$ such that $f'(0^+)=g'(1^-)=2$ and $f'(1^-)=g'(0^+)=1$.
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
We now discuss the proof of Theorem~\ref{thm:bleak_harper_skipper} on $T$ given by Bleak, Harper and Skipper \cite{ref:BleakHarperSkipper}. By Lemma~\ref{lem:covering_t}, for each nontrivial $x \in T$ it suffices to find a dyadic interval $[a,b] \subseteq \mathbb{S}^1$ and $y \in s^T$ such that $\bigcup_{g \in \langle x,y \rangle} (a,b)g = \mathbb{S}^1$ and $T_{[a,b]} \leqslant \langle x, y \rangle$. If $|x|$ is infinite, a dynamical argument is used (for any infinite order element $s$), see \cite[Proposition~3.1]{ref:DonovenHarper20}. The key ingredients for finite $|x|$ are highlighted in the following example.
\begin{example} \label{ex:bleak_harper_skipper}
Let $x \in T$ be a nontrivial torsion element. We will prove that there exists $y \in s^T$ (for $s$ as in Figure~\ref{fig:elements}) such that $\langle x, y \rangle = T$. By replacing $x$ by a power if necessary, $x$ has rotation number $\frac{1}{p}$ for prime $p$. For exposition, we only discuss the case $p \mathfrak{g}eq 5$. Since any two torsion elements of $T$ with the same rotation number are conjugate, by replacing $x$ by a conjugate if necessary, $x = (00, 01, 10, 110, \dots, 1^{p-3}0, 1^{p-2}) \mapsto (01, 10, 110, \dots, 1^{p-3}0, 1^{p-2}, 00)$.
We claim that $T = \langle x, s \rangle$. By Lemma~\ref{lem:covering_t}, since $\mathbb{S}^1 = \bigcup_{i \in \mathbb{Z}} (0,\frac{7}{8})x^i$, it suffices to prove that $T_{[0,\frac{7}{8}]} \leqslant \langle x, s \rangle$. Indeed, we claim that $T_{[0,\frac{7}{8}]} = \langley_0,y_1\rangle$ for $y_0 = s$ and $y_1 = s^x$. Defining $t\colon(0,\frac{7}{8}) \to (0,1)$ as $\omega t = \omega$ if $\omega \leqslant \frac{3}{4}$ and $\omega t = 2\omega-\frac{3}{4}$ if $\omega > \frac{3}{4}$, it suffices to prove that $\langle y_0^t, y_1^t \rangle = (T_{[0,\frac{7}{8}]})^t = F$.
To do this, we apply Theorem~\ref{thm:golan_criterion}(ii). To verify the second condition, choose $f = y_0^t$ and $g = (y_1^t)^{-1}$, so $f'(0^+)=g'(1^-)=2$ and $f'(1^-)=g'(0^+)=1$. It remains to prove that ${\rm Cl}(\langle y_0^t, y_1^t \rangle) \mathfrak{g}eq F'$. Here we apply another criterion: for $g_1, \dots, g_k \in F$ we have $\langle g_1, \dots, g_k \rangle \mathfrak{g}eq F'$ if and only if the Stallings $2$-core of $\langleg_1, \dots, g_k\rangle$ equals the Stallings $2$-core of $F$ \cite[Lemma~7.1 \& Remark~7.2]{ref:GolanMAMS}. The \varepsilonmph{Stallings $2$-core} is a directed graph associated to a diagram group introduced by Guba and Sapir \cite{ref:GubaSapir97}. Given elements $g_1, \dots, g_k \in F$ represented as tree pairs, there is a short combinatorial algorithm to find the Stallings 2-core of $\langle g_1, \dots, g_k \rangle$, and it is straightforward to compute the Stallings 2-core of $\langle y_0^t, y_1^t \rangle$ and note that it is the Stallings 2-core of $F$ (see the proof of \cite[Proposition~3.2]{ref:BleakHarperSkipper}). Therefore, $F = \langle y_0^t, y_1^t \rangle$, completing the proof that $T = \langle x, s \rangle$.
\varepsilonnd{example}
We conclude by briefly outlining the proof of Theorem~\ref{thm:golan} on $F$ given by Golan \cite{ref:GolanGen}, which uses similar methods to those in \cite{ref:BleakHarperSkipper} on $T$ and \cite{ref:DonovenHarper20} on $V$. Let $(a_0,a_1), (b_0,b_1) \in \mathbb{Z}^2$ with $\{a_0,b_0\} \neq \{0\}$ and $\{a_1,b_1\} \neq \{0\}$, and let $x \in F \setminus 1$ with $\varphii(x) = (a_0,a_1)$. Observe that it suffices to find an element $y$ such that $\varphii(y) = (b_0,b_1)$ and $\langle x,y \rangle \mathfrak{g}eq F'$. In \cite{ref:GolanGen}, an explicit choice of $y$, based on $x$, is given and the condition $\langle x,y \rangle \mathfrak{g}eq F'$ is verified via Theorem~\ref{thm:golan_criterion}(i).
\begin{thebibliography}{999}
\bibitem{ref:Aschbacher84}
M.~Aschbacher, \varepsilonmph{On the maximal subgroups of the finite classical groups},
Invent. Math. \textbf{76} (1984), 469--514.
\bibitem{ref:AschbacherGuralnick84}
M.~Aschbacher and R.~Guralnick, \varepsilonmph{Some applications of the first cohomology
group}, J. Algebra \textbf{90} (1984), 446--460.
\bibitem{ref:Binder68}
G.~J. Binder, \varepsilonmph{The bases of the symmetric group}, Izv. Vyssh. Uchebn.
Zaved. Mat. \textbf{78} (1968), 19--25.
\bibitem{ref:Binder70MZ}
G.~J. Binder, \varepsilonmph{Certain complete sets of complementary elements of the
symmetric and the alternating group of the nth degree}, Mat. Zametiki
\textbf{7} (1970), 173--180.
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\begin{example} \label{ex:bleak_harper_skipper}
Let $x \in T$ be a nontrivial torsion element. We will prove that there exists $y \in s^T$ (for $s$ as in Figure~\ref{fig:elements}) such that $\langle x, y \rangle = T$. By replacing $x$ by a power if necessary, $x$ has rotation number $\frac{1}{p}$ for prime $p$. For exposition, we only discuss the case $p \mathfrak{g}eq 5$. Since any two torsion elements of $T$ with the same rotation number are conjugate, by replacing $x$ by a conjugate if necessary, $x = (00, 01, 10, 110, \dots, 1^{p-3}0, 1^{p-2}) \mapsto (01, 10, 110, \dots, 1^{p-3}0, 1^{p-2}, 00)$.
We claim that $T = \langle x, s \rangle$. By Lemma~\ref{lem:covering_t}, since $\mathbb{S}^1 = \bigcup_{i \in \mathbb{Z}} (0,\frac{7}{8})x^i$, it suffices to prove that $T_{[0,\frac{7}{8}]} \leqslant \langle x, s \rangle$. Indeed, we claim that $T_{[0,\frac{7}{8}]} = \langley_0,y_1\rangle$ for $y_0 = s$ and $y_1 = s^x$. Defining $t\colon(0,\frac{7}{8}) \to (0,1)$ as $\omega t = \omega$ if $\omega \leqslant \frac{3}{4}$ and $\omega t = 2\omega-\frac{3}{4}$ if $\omega > \frac{3}{4}$, it suffices to prove that $\langle y_0^t, y_1^t \rangle = (T_{[0,\frac{7}{8}]})^t = F$.
To do this, we apply Theorem~\ref{thm:golan_criterion}(ii). To verify the second condition, choose $f = y_0^t$ and $g = (y_1^t)^{-1}$, so $f'(0^+)=g'(1^-)=2$ and $f'(1^-)=g'(0^+)=1$. It remains to prove that ${\rm Cl}(\langle y_0^t, y_1^t \rangle) \mathfrak{g}eq F'$. Here we apply another criterion: for $g_1, \dots, g_k \in F$ we have $\langle g_1, \dots, g_k \rangle \mathfrak{g}eq F'$ if and only if the Stallings $2$-core of $\langleg_1, \dots, g_k\rangle$ equals the Stallings $2$-core of $F$ \cite[Lemma~7.1 \& Remark~7.2]{ref:GolanMAMS}. The \varepsilonmph{Stallings $2$-core} is a directed graph associated to a diagram group introduced by Guba and Sapir \cite{ref:GubaSapir97}. Given elements $g_1, \dots, g_k \in F$ represented as tree pairs, there is a short combinatorial algorithm to find the Stallings 2-core of $\langle g_1, \dots, g_k \rangle$, and it is straightforward to compute the Stallings 2-core of $\langle y_0^t, y_1^t \rangle$ and note that it is the Stallings 2-core of $F$ (see the proof of \cite[Proposition~3.2]{ref:BleakHarperSkipper}). Therefore, $F = \langle y_0^t, y_1^t \rangle$, completing the proof that $T = \langle x, s \rangle$.
\varepsilonnd{example}
We conclude by briefly outlining the proof of Theorem~\ref{thm:golan} on $F$ given by Golan \cite{ref:GolanGen}, which uses similar methods to those in \cite{ref:BleakHarperSkipper} on $T$ and \cite{ref:DonovenHarper20} on $V$. Let $(a_0,a_1), (b_0,b_1) \in \mathbb{Z}^2$ with $\{a_0,b_0\} \neq \{0\}$ and $\{a_1,b_1\} \neq \{0\}$, and let $x \in F \setminus 1$ with $\varphii(x) = (a_0,a_1)$. Observe that it suffices to find an element $y$ such that $\varphii(y) = (b_0,b_1)$ and $\langle x,y \rangle \mathfrak{g}eq F'$. In \cite{ref:GolanGen}, an explicit choice of $y$, based on $x$, is given and the condition $\langle x,y \rangle \mathfrak{g}eq F'$ is verified via Theorem~\ref{thm:golan_criterion}(i).
\begin{thebibliography}{999}
\bibitem{ref:Aschbacher84}
M.~Aschbacher, \varepsilonmph{On the maximal subgroups of the finite classical groups},
Invent. Math. \textbf{76} (1984), 469--514.
\bibitem{ref:AschbacherGuralnick84}
M.~Aschbacher and R.~Guralnick, \varepsilonmph{Some applications of the first cohomology
group}, J. Algebra \textbf{90} (1984), 446--460.
\bibitem{ref:Binder68}
G.~J. Binder, \varepsilonmph{The bases of the symmetric group}, Izv. Vyssh. Uchebn.
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T.~C. Burness, R.~M. Guralnick and S.~Harper, \varepsilonmph{The spread of a finite
group}, Ann. of Math. \textbf{193} (2021), 619--687.
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T.~C. Burness and S.~Harper, \varepsilonmph{On the uniform domination number of a finite
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T.~C. Burness and S.~Harper, \varepsilonmph{Finite groups, $2$-generation and the
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T.~C. Burness, M.~.W. Liebeck and A.~Shalev, \varepsilonmph{Base sizes for simple groups
and a conjecture of {C}ameron}, Proc. Lond. Math. Soc. \textbf{98} (2009),
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T.~C. Burness and A.~R. Thomas, \varepsilonmph{Normalisers of maximal tori and a
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| 4,046 | 63,985 |
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0.128.17
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\bibitem{ref:Brin04}
M.~G. Brin, \varepsilonmph{Higher dimensional {T}hompson groups}, Geom. Dedicata
\textbf{108} (2004), 163--192.
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M.~G. Brin, \varepsilonmph{On the baker's maps and the simplicity of the higher
dimensional {T}hompson's groups {$nV$}}, Publ. Mat. \textbf{54} (2010),
433--439.
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J.~R. Britnell, A.~Evseev, R.~M. Guralnick, P.~E. Holmes and A.~Mar\'oti,
\varepsilonmph{Sets of elements that pairwise generate a linear group}, J. Combin.
Theory Ser. A \textbf{115} (2008), 442--465.
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M.~Burger and S.~Mozes, \varepsilonmph{Lattices in product of trees}, Inst. Hautes
\'Etudes Sci. Publ. Math. \textbf{92} (2000), 151--194.
\bibitem{ref:Burness071}
T.~C. Burness, \varepsilonmph{Fixed point ratios in actions of finite classical groups,
{I}}, J. Algebra \textbf{309} (2007), 69--79.
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T.~C. Burness, \varepsilonmph{Fixed point ratios in actions of finite classical groups,
{II}}, J. Algebra \textbf{309} (2007), 80--138.
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T.~C. Burness, \varepsilonmph{Fixed point ratios in actions of finite classical groups,
{III}}, J. Algebra \textbf{314} (2007), 693--748.
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T.~C. Burness, \varepsilonmph{Fixed point ratios in actions of finite classical groups,
{IV}}, J. Algebra \textbf{314} (2007), 749--788.
\bibitem{ref:Burness18}
T.~C. Burness, \varepsilonmph{Simple groups, fixed point ratios and applications}, in
\varepsilonmph{Local Representation Theory and Simple Groups}, {EMS} Series of
Lectures in Mathematics, European Mathematical Society, 2018, 267--322.
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T.~C. Burness, \varepsilonmph{Simple groups, generation and probabilistic methods}, in
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Series, vol. 455, Cambridge University Press, 2019, 200--229.
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T.~C. Burness and S.~Guest, \varepsilonmph{On the uniform spread of almost simple linear
groups}, Nagoya Math. J. \textbf{209} (2013), 35--109.
\bibitem{ref:BurnessGuralnickHarper21}
T.~C. Burness, R.~M. Guralnick and S.~Harper, \varepsilonmph{The spread of a finite
group}, Ann. of Math. \textbf{193} (2021), 619--687.
\bibitem{ref:BurnessHarper19}
T.~C. Burness and S.~Harper, \varepsilonmph{On the uniform domination number of a finite
simple group}, Trans. Amer. Math. Soc. \textbf{372} (2019), 545--583.
\bibitem{ref:BurnessHarper20}
T.~C. Burness and S.~Harper, \varepsilonmph{Finite groups, $2$-generation and the
uniform domination number}, Israel J. Math. \textbf{239} (2020), 271--367.
\bibitem{ref:BurnessLiebeckShalev09}
T.~C. Burness, M.~.W. Liebeck and A.~Shalev, \varepsilonmph{Base sizes for simple groups
and a conjecture of {C}ameron}, Proc. Lond. Math. Soc. \textbf{98} (2009),
116--162.
\bibitem{ref:BurnessThomas}
T.~C. Burness and A.~R. Thomas, \varepsilonmph{Normalisers of maximal tori and a
conjecture of {V}dovin}, J. Algebra \textbf{619} (2023), 459--504.
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P.~J. Cameron, \varepsilonmph{Graphs defined on groups}, Int. J. Group Theory
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| 3,990 | 63,985 |
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0.128.18
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\bibitem{ref:Flavell95}
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D.~Goldstein and R.~M. Guralnick, \varepsilonmph{Generation of {J}ordan algebras and
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\varepsilonnd{thebibliography}
\varepsilonnd{document}
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\begin{document}
\title[Nonhyperbolic ergodic measures: positive entropy and full support]{Robust existence of nonhyperbolic ergodic measures with positive entropy and full support}
\date{\today}
\author[Ch.~Bonatti]{Christian Bonatti}
\address{Institut de Math\'ematiques de Bourgogne}
\email{[email protected]}
\author[L.~J.~D\'\i az]{Lorenzo J.~D\'\i az}
\address{Departamento de Matem\'atica, Pontif\'{\i}cia Universidade Cat\'olica do Rio de Janeiro}
\email{[email protected]}
\author[D.~Kwietniak]{Dominik Kwietniak}
\address{Faculty of Mathematics and Computer Science, Jagiellonian University in Krak\'ow}
\urladdr{http://www.im.uj.edu.pl/DominikKwietniak/}
\email{[email protected]}
\begin{abstract}
We prove that for some manifolds $M$ the set of robustly transitive partially hyperbolic diffeomorphisms of $M$ with one-dimensional nonhyperbolic centre direction contains a $C^1$-open and dense subset of diffeomorphisms with nonhyperbolic measures which are ergodic, fully supported and have positive entropy. To do so, we formulate abstract conditions sufficient for the construction of an ergodic, fully supported measure $\mu$ which has positive entropy and is such that for a continuous function $\varphi\colon X\to\mathbb{R}$ the integral $\int\varphi\,d\mu$ vanishes. The criterion is an extended version of the \emph{control at any scale with a long and sparse tail} technique coming from the previous works.
\end{abstract}
\begin{thanks}{This research has been supported [in part] by CAPES - Ci\^encia sem fronteiras,
CNE-Faperj, and CNPq-grants (Brazil), the research of DK was supported by the National Science Centre (NCN) grant
2013/08/A/ST1/00275 and his stay in Rio de Janeiro, where he joined this project was possible thanks to
the CAPES/Brazil grant no. 88881.064927/2014-01.
The authors acknowledge the hospitality of PUC-Rio and IM-UFRJ.
LJD thanks the hospitality and support of
ICERM - Brown University during the thematic semester
``Fractal Geometry, Hyperbolic Dynamics, and Thermodynamical Formalism''.}
\end{thanks}
\keywords{Birkhoff average,
entropy,
ergodic measure,
Lyapunov exponent,
nonhyperbolic measure,
partial hyperbolicity,
transitivity}
\subjclass[2000]{
37D25,
37D35,
37D30,
28D99
}
\maketitle
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\section{Introduction}
Our motivation is the following problem:
\emph{To what extent does ergodic theory detect the nonhyperbolicity of a dynamical system? Do nonhyperbolic dynamical systems always have a nonhyperbolic ergodic measure?}
These questions originated with a construction presented in \cite{GIKN} and inspired many papers exploring the properties of nonhyperbolic ergodic measures.
We emphasise that the answer to the second question be ``no'' in general: there are examples of nonhyperbolic diffeomorphisms
whose all ergodic measures are hyperbolic (even with all Lyapunov exponents uniformly bounded away from $0$), see \cite{BBS}. However, these examples are ``fragile'': ergodic nonhyperbolic measures reappear after an arbitrarily small perturbation of these diffeomorphisms.
Thanks to the works of Abraham and Smale \cite{AbSm} and Newhouse \cite{New} it is known since late sixties that there exist open sets of nonhyperbolic systems. On the other hand, the first examples of open sets of diffeomorphisms with nonhyperbolic ergodic measures appeared just recently, in 2005 (see \cite{KN}). The construction in \cite{KN} uses the \emph{method of periodic approximations} introduced in \cite{GIKN} (we will outline this method later). Note that this technique works in the specific setting of partially hyperbolic diffeomorphisms of the three-dimensional torus $\mathbb{T}^3$ with compact centre leaves.
The existence of nonhyperbolic ergodic measures for some nonhyperbolic systems raises immediately further questions, which we address here: \emph{Which nonhyperbolic dynamical systems have ergodic nonhyperbolic measures? What is the support of these measures?
What is their entropy? How many zero Lyapunov exponents do they have? How about other ergodic-theoretic properties of these measures?}
The following theorem is a simplified version of our main result. For the full statement see Theorems \ref{t.openanddense} and \ref{t.average} below.
\begin{theorem*}\lambdabel{thm:one}
For every $n\ge 3$ there are a closed manifold $M$ of dimension
$n$, a nonempty open set $\mathcal{U}$ in the space $\operatorname{Diff}^1(M)$ of $C^1$-diffeomorphisms defined
on $M$, and a constant $C>0$
such that every $f\in \mathcal{U}$ has a nonhyperbolic invariant measure $\mu$ (i.e., with some zero Lyapunov exponent) satisfying:
\begin{enumerate}
\item \lambdabel{tp1} $\mu$ is ergodic,
\item \lambdabel{tp2} the support of $\mu$ is the whole manifold $M$,
\item \lambdabel{tp3} the entropy of $\mu$ is larger than $C$.
\end{enumerate}
\end{theorem*}
The Theorem applies to any manifold allowing a robustly transitive diffeomorphism which has partially hyperbolic splitting with one-dimensional centre. In particular, by \cite{BD-robtran} it applies to
\begin{itemize}
\item the $n$-dimensional torus ($n\ge 3$),
\item (more generally) any manifold carrying a transitive Anosov flow.
\end{itemize}
The nonhyperbolic measure $\mu$ as in the Theorem above apart from being ergodic, fully supported and having positive entropy, is also \emph{robust}. Namely, our proof shows that the three former properties appear \emph{robustly} in the space $\operatorname{Diff}^1(M)$ (i.e. we provide an open set of diffeomorphisms with an ergodic nonhyperbolic fully supported measure of positive entropy). Several previous works established the existence of
nonhyperbolic measures in similar settings as in Theorem above and these measures have some (but not all) of the properties listed above. These references present two approaches to the construction of ergodic nonhyperbolic measures: the first one is the already mentioned \emph{method of periodic approximations} \cite{GIKN}, the second is the method of generating a nonhyperbolic measure by a \emph{controlled point} \cite{BBD:16,BDB:}. Note that \cite{BZ} combines these two methods. Both schemes are detailed in Section \ref{ss.compar}. A discussion of previous results on nonhyperbolic ergodic measures see Section \ref{ss.history} or \cite{D-ICM} for a more comprehensive survey on that topic.
Here we further extend the method of construction of a controlled point presented in \cite{BDB:} and complement it by an abstract result in ergodic theory (see the theorem about entropy control and the discussion in Section \ref{ss.control}). These allows us to address simultaneously all four properties listed above.
\subsection{Precise results for robustly transitive diffeomorphisms}\lambdabel{ss.precise}
In what follows $M$ denotes a closed compact manifold, $\operatorname{Diff}^1(M)$ is the space of $C^1$-diffeo\-morphisms
of $M$ endowed with the usual $C^1$-topology, and $f\in \operatorname{Diff}^1(M)$. For $\Lambdambda\subset M$ we write
$\cM_f(\Lambdambda)$ for the set of all $f$-invariant measures with support contained in $\Lambdambda$.
A $Df$-invariant splitting $T_\Lambda M=E\oplus F$ is \emph{dominated} if
there are constants $C>0$ and $\lambdambda<1$ such that
$\| Df^{n} E_{x}\|\cdot \| Df^{-n} F_{f^{n}(x)}\| < C \lambdambda^n$
for every $x\in\Lambdambda$ and $n\in \mathbb{N}N$.
We say that a compact $f$-invariant set $\Lambdambda$
is \emph{partially hyperbolic with one-di\-men\-sio\-nal center}
if there is a $Df$-invariant
splitting with three nontrivial bundles\begin{equation}\lambdabel{e.ph}
T_\Lambdambda M = E^{{\mathrm{s}}s} \oplus E^{{\mathrm{c}}} \oplus E^{{\mathrm{u}}u}
\end{equation}
such that
$E^\mathrm{ss}$ is uniformly contracting and
$E^\mathrm{uu}$ is uniformly expanding, $\operatorname{dim} E^{{\mathrm{c}}}=1$, and the $Df$-invariant
splittings
$E^{\mathrm{cs} } \oplus E^{{\mathrm{u}}u}$
and
$E^{{\mathrm{s}}s} \oplus E^{\mathrm{cu}}$ are both dominated,
where
$E^\mathrm{cs}= E^{{\mathrm{s}}s} \oplus E^{{\mathrm{c}}}$ and $E^\mathrm{cu}= E^{{\mathrm{c}}} \oplus E^{{\mathrm{u}}u}$.
The bundles
$E^{{\mathrm{u}}u}$ and $E^{{\mathrm{s}}s}$ are called \emph{strong stable} and \emph{strong unstable},
respectively, and
$E^{\mathrm{c}}$ is the \emph{center bundle}. We abuse a bit the terminology and say that the splitting given by \eqref{e.ph} is also \emph{dominated}.
If $\Lambdambda$ is a partially hyperbolic set with one-dimensional center, then
the bundles $E^{{\mathrm{s}}s}, E^{{\mathrm{c}}}, E^{{\mathrm{u}}u}$ depend continuously on the point $x\in \Lambdambda$. Hence the
\emph{logarithm of the center derivative}
\begin{equation}\lambdabel{e.logmap}
\mathrm{J}_f^{{\mathrm{c}}}(x) \eqdef \log | Df_x |_{E^{\mathrm{c}} (x)\setminus\{0\}}|
\end{equation}
is a continuous map. If, in addition, $\mu\in \cM_f(\Lambdambda)$ is ergodic, then the Oseledets Theorem
implies that there is a
number $\chi^{{\mathrm{c}}} (\mu)$, called the {\emph{central Lyapunov exponent of $\mu$,}}
such that for $\mu$-almost every point $x\in \Lambdambda$ it holds
$$
\lim_{n \to \pm \infty} \frac{\log |Df^n_x (v)|}{n} = \int \mathrm{J}_f^{\mathrm{c}} \, d\mu=\chi^{{\mathrm{c}}}(\mu), \qquad
\mbox{for every $v\in E^{{\mathrm{c}}} \setminus \{0\}$}.
$$
In particular, the function $\mu\mapsto\chi^{{\mathrm{c}}}(\mu)$ is continuous with respect to the weak$^*$ topology on $\cM_f(\Lambdambda)$.
A diffeomorphism $f\in \operatorname{Diff}^1(M)$ is
\emph{transitive} if it has a dense orbit. The diffeomorphism $f$ is
\emph{$C^1$-robustly transitive}
if it belongs to the $C^1$-interior of the set of transitive diffeomorphisms
(i.e., all $C^1$-nearby diffeomorphisms are also transitive).
We denote by $\cR\cT(M)$ the $C^1$-open set of such diffeomorphisms $f\in\operatorname{Diff}^1(M)$ that:
\begin{itemize}
\item
$f$ is robustly transitive,
\item
$f$ has a pair of hyperbolic periodic points with different indices,
\item $M$ is a partially hyperbolic set for $f$ with one-dimensional center.
\end{itemize}
These assumptions imply that
$\operatorname{dim}(M)\ge 3$, because in dimension two robustly transitive diffeomorphisms are
always hyperbolic, see \cite{PuSa}.
The set $\cR\cT(M)$ contains
well studied and interesting examples of diffeomorphisms. Among them there are:
different types of skew product diffeomorphisms, see
\cite{BD-robtran,Sh};
derived from Anosov diffeomorphisms, see \cite{Mda};
and perturbations of time-one maps of transitive Anosov flows, see
\cite{BD-robtran}.
Our main theorem provides a measure $\mu$ which is ergodic \emph{and} has full support \emph{and} positive entropy, \emph{and} is such that
the integral $\int J_f^{{\mathrm{c}}}d\mu$ has a prescribed value ($=0$).
\begin{thm}
\lambdabel{t.openanddense}
There is a $C^1$-open and dense subset $\cZ(M)$ of $\cR\cT(M)$ such that
every $f\in \cZ(M)$ has an ergodic nonhyperbolic measure
with positive entropy and full support. Furthermore, for each
$f\in \cZ(M)$ there is a neighbourhood $\cV_f$ of $f$ and a constant $c_f>0$ such that
for every $g\in\cV_f\cap\cZ(M)$ we have $h(\mu_g)\ge c_f$.
\end{thm}
In what follows, given a periodic point $p$ of a diffeomorphism $f$ we denote by
$\mu_{\cO(p)}$ the unique $f$-invariant measure supported on the orbit ${\cO(p)}$ of $p$.
The next result is a reformulation and an extension of Corollary 6 in \cite{BDB:} to our context:
\begin{thm}\lambdabel{t.average}
Consider an open subset $\cU$ of $\cR\cT(M)$
such that there is a continuous map defined on $\cU$
\[
f\mapsto (p_f,q_f)
\]
that associates to each
$f\in \cU$ a pair of hyperbolic periodic points with $\cO(p_f)\cap\cO(q_f)=\emptyset$.
Let $\varphi\colon M\to\mathbf{R}$ be a continuous function
satisfying
$$
\int \varphi\, d\mu_{\cO(p_f)}<0<\int \varphi \,d\mu_{\cO(q_f)}, \quad \mbox{for every $f\in \cU$}.
$$
Then there is a $C^1$-open
and dense
set $\cV$ of $ \cU$ such that every $f\in\cV$ has an ergodic measure
$\mu_f$ whose support is $M$, satisfies
$\int \varphi\, d\mu_f=0$, and has positive entropy. Furthermore, for each
$f\in \cV$ there is a neighbourhood $\cV_f$ of $f$ and a constant $c_f>0$ such that
for every $g\in\cV_f\cap\cV$ we have $h(\mu_g)\ge c_f$.
\end{thm}
\begin{rem}
Note that in Theorem~\ref{t.average} there is no condition on the indices of the periodic points.
Observe also that Theorem~\ref{t.openanddense} is not a particular case of
Theorem~\ref{t.average}:
in Theorem~\ref{t.openanddense} there is no fixed map $\varphi$,
the considered map is the logarithm of the center derivative $J_f^{{\mathrm{c}}}$ and therefore it changes with $f$ (i.e.,
there is a family of maps $\varphi_f$ depending on $f$).
\end{rem}
\begin{rem}
We could state a version of Theorem \ref{t.openanddense} replacing the logarithm of the center derivative $J_f^{{\mathrm{c}}}$ by any
continuous map $\varphi\colon M\to \mathbb{R}$ and picking any value $t$ satisfying
\[
\inf\left\{ \int \varphi\, d\mu\,: \, \mu \in \cM_f(M)\right\}<\, t\, <
\sup\left\{ \int \varphi\, d\mu\,:\, \mu \in \cM_f(M)\right\}.
\]
\end{rem}
\begin{rem}
Our methods imply that
the sentence ``{\emph{Moreover, the measure $\mu$ can be taken with positive entropy.}}''
can be added to Theorems 5, 7, and 8 and Proposition 4b in \cite{BDB:}.
Furthermore, the entropy will have locally a uniform lower bound.
The details are left to the reader.
\end{rem}
\subsection{Previous results on nonhyperbolic ergodic measures}\lambdabel{ss.history}
By \cite{CCGWY} nonhyperbolic homoclinic classes of $C^1$-generic diffeomorphisms always support nonhyperbolic ergodic measures. The proof uses the periodic approximation method and extends \cite{DG}. In some settings the results of \cite{BDG} imply that these measures have full support in the homoclinic class.
Specific examples of open sets of diffeomorphisms of the three torus with nonhyperbolic ergodic measures were first obtained in \cite{KN} using the periodic approximation method. In \cite{BBD:16} there are general results guaranteeing for an open and dense subset of $C^1$-robustly transitive diffeomorphisms the existence of nonhyperbolic ergodic measures with positive entropy. The latter paper uses the controlled point method.
All results above provide measures with only one zero Lyapunov exponent. In \cite{WZ} yet another adaptation of the method of periodic approximation yields ergodic measures with multiple zero Lyapunov exponents for some $C^1$-generic diffeomorphisms (see also \cite{BBD:14} for results about skew-products).
Some limitations of these previous constructions are already known. By \cite{KL} any measure obtained by the method of periodic approximations has zero entropy. Hence all nonhyperbolic measures defined in \cite{BDG,BZ,CCGWY,DG,GIKN, KN,WZ} have necessarily zero entropy. On the other hand, the measures produced in \cite{BBD:16} cannot have full support for their definition immediately implies that they are supported on a Cantor-like subset of the ambient manifold.
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\subsection{Previous results on nonhyperbolic ergodic measures}\lambdabel{ss.history}
By \cite{CCGWY} nonhyperbolic homoclinic classes of $C^1$-generic diffeomorphisms always support nonhyperbolic ergodic measures. The proof uses the periodic approximation method and extends \cite{DG}. In some settings the results of \cite{BDG} imply that these measures have full support in the homoclinic class.
Specific examples of open sets of diffeomorphisms of the three torus with nonhyperbolic ergodic measures were first obtained in \cite{KN} using the periodic approximation method. In \cite{BBD:16} there are general results guaranteeing for an open and dense subset of $C^1$-robustly transitive diffeomorphisms the existence of nonhyperbolic ergodic measures with positive entropy. The latter paper uses the controlled point method.
All results above provide measures with only one zero Lyapunov exponent. In \cite{WZ} yet another adaptation of the method of periodic approximation yields ergodic measures with multiple zero Lyapunov exponents for some $C^1$-generic diffeomorphisms (see also \cite{BBD:14} for results about skew-products).
Some limitations of these previous constructions are already known. By \cite{KL} any measure obtained by the method of periodic approximations has zero entropy. Hence all nonhyperbolic measures defined in \cite{BDG,BZ,CCGWY,DG,GIKN, KN,WZ} have necessarily zero entropy. On the other hand, the measures produced in \cite{BBD:16} cannot have full support for their definition immediately implies that they are supported on a Cantor-like subset of the ambient manifold.
\subsection{Comparison of constructions of nonhyperbolic ergodic measures}
\lambdabel{ss.compar}
So far there are two ways to construct nonhyperbolic ergodic measures: these measures are either ``approximated by periodic measures'' as in \cite{GIKN} or ``generated by a controlled point'' as in \cite{BBD:16,BDB:}.
Both methods apply to any partially hyperbolic diffeomorphism as in Subsection~\ref{ss.precise} whose domain contains two special subsets:
a centre contracting region
(where the central direction is contracted)
and a
centre expanding region (where the central direction is expanded). Furthermore, the orbits can travel from one of these regions to the other in a controlled way. For example, both methods work for a diffeomorphism with a transitive set (which is persistent to perturbations
if one wants to obtain a robust result) containing two ``heteroclinically related blenders'' of different
indices and a region where the dynamics is partially hyperbolic with one dimensional centre. We discuss blenders in Section \ref{s.robustly}.
Let us briefly describe these two approaches. For simplicity, we will restrict ourselves to the partially hyperbolic setting as above.
As mentioned above the ``approximation by periodic orbits'' construction from \cite{GIKN}
builds a sequence of periodic orbits
$\Gammamma_i$ and periodic measures $\mu_i$ supported on these orbits such that the sequence of central Lyapunov exponents $\lambdambda_i$ of these measures tends to zero as $i$ approaches infinity. Since in our partial hyperbolic setting the central Lyapunov exponent vary continuously with the measure, it vanishes for
every weak$^*$ accumulation point $\mu$ of the sequence of measures $\mu_i$ so we get a nonhyperbolic measure.
The difficulty is to prove that every limit measure is ergodic as ergodicity is not closed property in the weak$^*$ topology.
The arguments in \cite{GIKN} contain a general criterion for the weak$^*$ convergence and ergodicity of the limit of a sequence periodic orbits $\Gammamma_i$. It requires that for some summable sequence $(\gamma_i)$ of positive reals most points on the orbit $\Gammamma_{i+1}$ shadow $\gamma_i$-close a point on the precedent orbit $\Gammamma_i$ for $\card{\Gammamma_i}$ iterates. If the proportion of these shadowing points among all points of $\Gammamma_{i+1}$ tends to $1$
as $i\to +\infty$ sufficiently fast, then $\mu_i$ weak$^*$ converge to an ergodic measure. The ``non-shadowing'' points are used to decrease the absolute value of the central Lyapunov exponent over $\Gammamma_i$ and to spread the support of $\Gammamma_i$ in the ambient space.
This forces the limit measure to have zero central Lyapunov exponent and full support, as in this case the central Lyapunov exponent depends continuously on the measure (see below).
It turns out that the repetitive nature of the method of periodic approximation forces the resulting measure to be close (Kakutani equivalent\footnote{Two measure preserving systems are \emph{Kakutani equivalent} if they have a common derivative. A \emph{derivative} of a measure preserving system is an another measure preserving system isomorphic with a system induced by the first one. See Nadkarni's book \cite{Nadkarni}, Chapter 7.}) to a group rotation. Actually, it is proved in \cite{KL} that the periodic measures $\mu_i$ described above converge to $\mu$ in a much stronger sense than weak$^*$ convergence. This new notion of convergence is coined \emph{Feldman--Katok convergence} and it implies that all measures obtained following \cite{GIKN} (thus the measures from \cite{ BDG, BZ,CCGWY, DG, KN, WZ} obtained by this method) are loosely Kronecker measures with zero entropy. For more details we refer to \cite{KL}.
The authors of \cite{BBD:16} devised a new method for constructing nonhyperbolic ergodic measures
using blenders and flip-flop configurations (we review the former in Section \ref{s.robustly} and the latter in Section~\ref{s.flipflop}). Applying these tools one defines a point $x$ such that the Birkhoff averages of the central derivative along
segments of its forward orbit go to zero uniformly. In a bit more precise terms,
there are sequences of positive reals and positive integers, denoted $\varepsilon_n$ and $T_n$, with $\varepsilon_n\to 0^+$
and $T_n\to \infty$ as $n\to \infty$ such that the average of the central derivative
along a segment of the $x$-orbit
$\{f^t(x), \dots,f^{t+T_n}(x)\}$ is less than $\varepsilon_n$ for any $t\geq 0$. We say that such an $x$ is \emph{controlled at any scale}.
Then the $\omega$-limit set $\omega(x)$ of $x$ is an invariant compact set such that for all measures supported on $\omega(x)$ the centre Lyapunov exponent vanishes,
see \cite[Lemma 2.2]{BBD:16}.
Under some mild assumptions one can find a point $x$ such that the compact invariant set $\omega(x)$ is also partially hyperbolic and $f|_{\omega(x)}$ has the full shift over a finite alphabet as a factor, thus it has positive topological entropy. To achieve this one finds a pair of disjoint compact subsets $K_0$ and $K_1$ of $M$ such that for some $k>0$ and $\omega\in\{0,1\}^\mathbb{Z}Z$ which is generic for the Bernoulli measure $\xi_{1/2}$ for every $j\in\mathbb{Z}Z$ we have $f^{jk}(x)\in K_{\omega(j)}$. By the variational principle for topological entropy \cite{Wal:82} the set $\omega(x)$
supports an ergodic nonhyperbolic measure with positive entropy.
Unfortunately, the existence of a semi-conjugacy from $\omega(x)$ to a Cantor set carrying the full shift forces $\omega(x)$ to be a proper subset of $M$, thus these measures cannot be supported on the whole manifold.
In \cite{BDB:} the procedure from \cite{BBD:16} was modified.
More precisely, in \cite{BDB:} the control over orbit of a point $x$ is relaxed:
one splits the orbit of $x$ into a ``regular part'' and a ``tail'', and
one only needs to control the averages over the orbit segment $\{f^t(x), \dots,f^{t+T_n}(x)\}$ of length $T_n$ only for $t$ belonging to the regular part, which is a set of positive density in $\mathbb{N}$ and at the same time the iterates corresponding to the tail part are dense in $X$.
Under quite restrictive conditions on the tail (coined \emph{longness} and \emph{sparseness}), Theorem~1 from \cite{BDB:} claims that if $x$ is
controlled at any scale with a long sparse tail then any measure $\nu$ generated\footnote{A measure $\mu$ is generated by a point $x$ if $\mu$ is a weak$^*$ limit point of the sequence of measures $\frac1n\sum_{i=0}^{n-1}\deltalta_{f^i(x)}$,
where $\deltalta_{f^i(x)}$ is the Dirac measure at the point $f^i(x)\in X$.} by $x$ has vanishing central Lyapunov exponent and full support.
The underlying topological mechanism providing points whose orbits are controlled at any scale with long sparse tail are
the \emph{flip-flop families with sojourns in $X$}.
Since there is no longer a semi-conjugacy to a full shift a different method has to be applied to establish positivity of the entropy.
\subsection{Control of entropy}\lambdabel{ss.control}
In this paper we combine the two methods above. We pick a pair of disjoint compact subsets $K_0$ and $K_1$ of $M$ and we divide the orbit into the regular and tail part as in \cite{BDB:}. We assume that the controlled points visits $K_0$ and $K_1$ following the same pattern as some point generic for the Bernoulli measure $\xi$ (as in \cite{BBD:16}), but we require that this happens \emph{only} for iteration in the regular part of the orbit. We also assume that the tail is even more structured: apart of being long and sparse, the tail intersected with nonnegative integers is a \emph{rational subset of $\mathbb{N}N$}. A set $A\subset\mathbb{N}N$ is \emph{rational} if it can be approximated with arbitrary precision by sets which
are finite union of arithmetic progressions (sets of the form $a+b\mathbb{N}N$, where $a,b\in\mathbb{N}N$, with $b\neq 0$). Here, the ``precision'' is measured in terms of the upper asymptotic density $\bar{d}$ of the symmetric difference of $A$ and a finite union of arithmetic progressions.
To get positive entropy, we show that
the measure $\mu$ is an extension of a loosely Bernoulli system with positive entropy
(a measure preserving system with a subset of positive measure such that the induced system is a Bernoulli process).
More precisely, we have the following general criterion for the positivity of any measure generated by a point (actually, we prove even more general Theorem \ref{thm:mainbis}, but for the full statement we need more notation, see Section \ref{s.measurespositive}).
\begin{theorem*}[Control of entropy]\lambdabel{thm:main}
Let $(X, \rho) $ be a compact metric space and $f\colon X\to X$ be a continuous map.
Assume that $\mathbf{K}=(K_0,K_1)$ is a pair of disjoint compact subsets of $X$ and $J\subset \mathbb{N}$ is
a rational.
If $\bar x\in X$ is such that $f^j(\bar x)\in K_{\bar z(j)}$ for every $j\in J$ where $z$ is a generic point for the Bernoulli measure $\xi_{1/2}$, then the entropy of
any measure $\mu$ generated by $x$ has entropy at least $d(J)\cdot \log 2$,
where $d(J)$ stands for the asymptotic density of $J$.
\end{theorem*}
We apply the above criterion to the controlled point and the rational set $J$ which is the complement (in $\mathbb{N}N$) of the intersection of the rational tail with $\mathbb{N}$. This implies that $J$ is also a rational set (Remark~\ref{r.complement}) and the asymptotic density of $J$ exists and satisfies $0<d(J)<1$.
The underlying topological mechanism we use to find the controlled point with the required behavior
is provided by a new object we call
the
\emph{double flip-flop family for $f^N$ with $f$-sojourns in $X$}, where $N>0$ is some integer.
This mechanism is a variation of the notion of the flip-flop family with sojourns in $X$, indeed
we will see that a flip-flop family with sojourns in $X$ yields
a double flip-flop family for some $f^N$ with $f$-sojourns in $X$, see Proposition~\ref{p.yieldsdouble}.
\subsection*{Organisation of the paper}
This paper is organised as follows. In Section~\ref{s.measurespositive}, we prove Theorem~\ref{thm:main}.
Section~\ref{s.scalesandtails} is devoted to the construction of rational sparsely long tails. In Section~\ref{s.flipflop}, we
study different types of flip-flop families and prove that they generate ergodic measures with full support and positive entropy
with appropriate averages.
Finally, in Section~\ref{s.robustly}
devoted to robustly transitive diffeomorphisms, we complete the proofs of Theorems~\ref{t.openanddense} and \ref{t.average}.
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\section{Measures with positive entropy. Proof of Theorem~\ref{thm:main}}
\lambdabel{s.measurespositive}
The goal of this section is to prove Theorem~\ref{thm:main}.
Throughout what follows, $X$ is a compact metric space, $\rho$ is a metric for $X$, and $f\colon X\to X$ is a continuous map (not necessarily a homeomorphism). First we introduce some notation.
Given a continuous function $\varphi\colon X\to \mathbf{R}$, $n>0$, and $x\in X$ we denote by $\varphi_n(x)$
the Birkhoff average of $\varphi$ over the orbit segment $x,f(x),\ldots,f^{n-1}(x)$, that is,
\begin{equation}\lambdabel{e.averages}
\varphi_n(x)\eqdef \frac 1n\sum_{i=0}^{n-1}\varphi\circ f^i(x).
\end{equation}
\subsection*{Generated measures, generic/ergodic points} Let $\M_f(X)$ denote the set of $f$-invariant Borel probability measures on $X$. Given $x\in X$ and $n\ge 1$ we set
\begin{equation}\lambdabel{e.empiric}
\mu_n(x,f)\eqdef \frac1n\sum_{i=0}^{n-1}\deltalta_{f^i(x)},
\end{equation}
where $\deltalta_{f^i(x)}$ is the Dirac measure at the point $f^i(x)\in X$.
We say that a point $x\in X$ {\emph{generates $\mu\in\M_f(X)$} along $(n_k)_{k\in\mathbb{N}}$} if $(n_k)_{k\in\mathbb{N}}$ is a strictly increasing sequence of integers such that $\lim_{k\to\infty}\mu_{n_k}(x,f)=\mu$ in the weak$^*$ topology on the space of probability measures on $X$. If $\mu$ is a measure generated by a point $x\in X$, then $\mu$ is invariant, that is $\mu\in\cM_f(X)$. If there is no need to specify $(n_k)$ we just say that \emph{$x$ generates $\mu$}. We write
$V(x)$ for the set of all $f$-invariant measures generated by $x$.
We say that $x$ is a \emph{generic point} for $\mu$ if $\mu$ is the unique measure generated by $x$,
i.e., $V(x)=\{\mu\}$.
A point is an \emph{ergodic point} if it is a generic point for an ergodic measure.
We write $h_\mu(f)$ for
the \emph{entropy} of $f$ with respect to $\mu$, see \cite{Wal:82} for its
definition and basic properties.
\subsection*{Sets and their densities}
The \emph{upper asymptotic density} of a set $J\subset \mathbb{Z}$ ($J\subset\mathbb{N}$) is the number
\[
\bar d(J)=\limsup_{N\to\infty}\frac{1}{N}\card{J\cap [0,N-1]}.
\]
Similarly, we define the \emph{lower asymptotic density} $\underline{d}(J)$ of $J$. If $\bar d(J)=\underline{d}(J)$, then we say that $J$ has \emph{asymptotic density} $d(J)=\underline{d}(J)=\bar d (J)$. Note that $\bar d$, $\underline{d}$, and $d$ (if defined) are determined only by $J\cap\mathbb{N}$.
\subsection*{Symbolic dynamics} Let $\Omegaega_M=\{0,1,\ldots,M-1\}^\mathbb{Z}$ be the full shift over $\cA=\{0,1,\ldots,M-1\}$.
For $\alpha\in\cA$ we write $[\alpha]_0$ for the \emph{cylinder set} defined as $\{\omega\in\Omegaega_M:\omega_0=\alpha\}$. Similarly, given a word $u=\alpha_1\dots \alpha_k\in\cA^k$ the set
$[u]_0$ is the cylinder defined as $\{\omega\in\Omegaega_M:\omega_0=\alpha_1, \dots,
\omega_{k-1}=\alpha_k\}$. We will often identify a set $A\subset\mathbb{Z}$ ($A\subset\mathbb{N}$) with its characteristic function $\chi_A\in\Omegaega_2$,
that is $(\chi_A)_i=1$ if and only if $i\in A$. By $\sigma$ we denote the shift homeomorphism on $\Omegaega_M$ given by $(\sigma(\omega))_i=\omega_{i+1}$ for each $i\in \mathbb{Z}$.
For more details on symbolic dynamics we refer the reader to \cite{LM}.
\subsection*{Completely deterministic sequences}
A point $x\in\Omegaega_2$ is \emph{completely deterministic} (or \emph{deterministic} for short) if every measure generated by $x$ has zero entropy, that is, $h(\mu)=0$ for every $\mu\in V(x)$. A set $J\subset \mathbb{Z}$ is \emph{completely deterministic} if its characteristic function is a completely deterministic point in $\Omegaega_2$. This notion is due to B.~Weiss, see \cite{Weiss}.
\subsection*{Bernoulli measure} The {\emph{Bernoulli measure $\xi_{1/2}$}} is the shift invariant measure on $\Omegaega_2$ such that for each $N\in\mathbb{N}$ and $u=u_1\ldots u_N\in\{0,1\}^N$ we have $\xi_{1/2}([u]_0)=1/2^N$.
\subsection*{Itineraries}
Let $\mathbf{K}=(K_0,K_1)$
be disjoint compact subsets of $X$ and $J\subset\mathbb{Z}$.
We say that $\omega\in\Omegaega_2$ is the $\mathbf{K}$-itinerary of $x\in X$ over $J$ if $f^{j}(x)\in K_{\omega(j)}$ for each $j\in J$.
The next result is the main step in the proof of Theorem~\ref{thm:main}.
\begin{thm}[Control of the entropy]\lambdabel{thm:mainbis}
Let $(X, \rho) $ be a compact metric space and $f\colon X\to X$ be a continuous map.
Assume that $\mathbf{K}=(K_0,K_1)$ is a pair of disjoint compact subsets of $X$ and $J\subset \mathbb{N}$ is
completely deterministic with $\underline{d}(J)>0$.
If the $\mathbf{K}$-itinerary of $\bar x\in X$ over $J$ is a generic point for $\xi_{1/2}$, then the entropy of
any measure $\mu\in V(x)$ satisfies
$$
h _{\mu}(f)\ge \underline{d}(J)\cdot \log 2>0.
$$
\end{thm}
Our proof is based on the following property of completely deterministic sets:
\emph{A sequence formed by symbols chosen from a generic point of a Bernoulli measure $\xi_{1/2}$ along a completely deterministic set with positive density is again a generic point for $\xi_{1/2}$}. This is a result of Kamae and Weiss (see \cite{Weiss}) originally formulated in the language of normal numbers and admissible selection rules.
\begin{rem}
When we were finishing writing this paper, {\L}{\c{a}}cka announced in her PhD thesis \cite{Martha} a version of Theorem \ref{thm:mainbis} with relaxed assumptions on the point $\bar x$ and the sequence $J$. Her proof is based on properties of the $\bar{f}$-pseudometric discussed in \cite{KL}.
\end{rem}
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\subsection{Proof of Theorem~\ref{thm:mainbis}}
Let $K_2=X\setminus (K_0\cup K_1)$. Let $\iota_\mathcal{P}\colon X\to\Omegaega_3$ be the coding map with respect to the partition $\mathcal{P}=\{K_0,K_1,K_2\}$. In other words, $\iota_\mathcal{P}(x)=y\in\Omegaega_3$, where
\[
\iota_\mathcal{P}(x)=\begin{cases}
i, & \mbox{if $j\in\mathbb{N}$ and $f^j(x)\in K_i$}, \\
0, & \mbox{if $j<0$}.
\end{cases}
\]
Fix $\nu\in V(\bar x)$. We modify $K_0$ and $K_1$ without changing the $\mathbf{K}$-itinerary of $\bar x$ over $J$ so that the topological boundary of $K_i$ for $i=0,1,2$, denoted $\partial K_i$, is $\nu$-null. To this end, we set
\[
\tilde{c}=\frac{1}{2}\min\{\rho(x_0,x_1):x_0\in K_0,\,x_1\in K_1\}
\]
and for $0<c<\tilde{c}$ we define sets $\partial_c K_i =\{x\in X: \dist (x,K_i)=c\}$ for $i=0,1$. Note that for each $c$ the set $\partial_c K_0\cup\partial_c K_1$ contains (but need not to be equal) the topological boundaries of the sets: $K^c_0$, $K^c_1$, and $K'_2=X\setminus (K^c_0\cup K^c_1)$, where $K^c_i=\{x\in X: \dist (x,K_i)\le c\}$ for $i=0,1$. Consider a family of closed sets $\mathcal{C}=\{\partial_c K_0\cup\partial_c K_1: 0<c<\tilde c\}$.
Since elements of $\mathcal{C}$ are pairwise disjoint, only countably many of them can be of positive $\nu$-measure.
Fix any $0<c<\tilde{c}$ such that $\nu(\partial_c K_0\cup\partial_c K_1)=0$ and replace $K_i$ by $K^c_i$ for $i=0,1$, and $K_2$ by $K'_2$. Note that this does not change the $\mathbf{K}$-itinerary of $x$ over $J$. Furthermore, the elements of our redefined partition, which we will still denote $\mathcal{P}=\{K_0,K_1,K_2\}$ have $\nu$-null boundaries. Let $(n_k)$ be the sequence of integers along which $\bar x$ generates $\nu$. Then $y\eqdef\iota_\mathcal{P}(\bar x)\in\Omegaega_3$ generates a shift-invariant measure $\mu$ on $\Omegaega_3$ which is the push-forward of $\nu$ through the coding map $\iota_\mathcal{P}$. Furthermore, the dynamical entropy of $\nu$ with respect to the partition $\mathcal{P}$ equals $h_\mu(\sigma)$. See \cite[Lemma 2]{KL} for more details. Note that the proof in \cite{KL} is stated for generic points, but it is easily adapted to the measures generated along a sequence as considered here. Now Theorem~\ref{thm:mainbis} follows from the following fact.
\begin{cla}
\lambdabel{t.p.entropy}
If $\mu \in \mathcal{M}_\sigma(\Omegaega_3)$ is a measure
generated by $y$, then
$h_{\mu}(\sigma) \ge \underline{d}(J)\cdot \log 2$.
\end{cla}
\begin{proof}[Proof of the Claim \ref{t.p.entropy}]
Let $\mu\in V(y)$ and $(n_k)_{k\in \mathbb{N}}$ be a strictly increasing sequence such that $\mu_{n_k}(y,\sigma)\to\mu$ as $k\to\infty$.
Let $\chi_J\in\Omegaega_2$ be the characteristic sequence of $J$. Let $\mu_J$ by any measure generated by $\chi_J$ along
$(n_k)_{k\in \mathbb{N}}$, that is, is any limit point of $(\mu_{n_k}(\chi_J,\sigma))_{k\in\mathbb{N}}$. Passing to a subsequence (if necessary) we assume that $\mu_{n_k}(\chi_J,\sigma)$ converges as $k\to\infty$ to a shift-invariant measure $\mu_J$ on $\Omegaega_2$.
Consider the product dynamical system on $\Omegaega_3\times \Omegaega_2$ given by
$$
S \eqdef
\sigma\times \sigma\colon \Omegaega_3\times \Omegaega_2\to \Omegaega_3\times \Omegaega_2.
$$
Again passing to a subsequence if necessary, we may assume that there exists $\mu'\in\mathcal{M}_{S}(\Omegaega_3\times\Omegaega_2)$ such that
\[
\mu_{n_k}((y,\chi_J),S)
\to \mu'\text{ as }k\to\infty.
\]
Recall that a \emph{joining} of $\mu$ and $\mu_J$ is an $S$-invariant measure on $\Omegaega_3\times \Omegaega_2$ which projects $\mu$ in the first coordinate and $\mu_J$ in the second. Observe that $\mu'$ is a joining of $\mu$ and $\mu_J$ (because the marginal distributions of $\mu_{n_k}((y,\chi_J),S)$ converge as $k\to\infty$ to, respectively, $\mu$ and $\mu_J$).
As the entropy of a joining is bounded below by the entropy of any of its marginals and is bounded above by the sum of the entropies of its marginals (see \cite[Fact 4.4.3]{Downarowicz}) we have that
\[
h_\mu(\sigma)\le h_{\mu'}(S)\le h_\mu(\sigma)+h_{\mu_J}(\sigma)=h_\mu(\sigma),
\]
where the equality uses that $J$ is completely deterministic.
Now to complete the proof the Claim \ref{t.p.entropy} it suffices to show the
following fact.
\begin{cla}
\lambdabel{p.l.entropy}
Every $S$-invariant measure $\mu'\in V(y,\chi_J)$ satisfies $h_{\mu'}(S)\ge \und d(J)\cdot \log 2>0$.
\end{cla}
\begin{proof}[Proof of Claim \ref{p.l.entropy}]
Let $\Psi\colon \{0,1,2\}\times\{0,1\}\to\{0,1,2\}\times\{0,1\}$ be the $1$-block map given by
$\Psi(\alpha,1)=(\alpha,1)$ and $\Psi(\alpha,0)=(2,0)$ for $\alpha\in\{0,1,2\}$.
Consider the factor map $\psi\colon\Omegaega_3\times\Omegaega_2\to\Omegaega_3\times\Omegaega_2$ determined by $\Psi$, that is
\[
\psi(\omega)=\big((\psi(\omega))_i\big)_{i\in\mathbb{Z}},\qquad\text{where }(\psi(\omega))_i = \Psi(\omega_i) \,\text{for }i\in\mathbb{Z}.
\]
Observe that we have defined $\psi$ so that if $\omega,\omega'\in \Omegaega_3$ satisfy $\omega|_J=\omega'|_J$, then $(\omega'',\chi_J)\eqdef\psi(\omega,\chi_J)=\psi(\omega',\chi_J)$. Furthermore, $\omega''$ agrees with both, $\omega$ and $\omega'$, over $J$ and $\omega''_j=2$ for all $j\notin J$.
In particular, if $\bar z\in\Omegaega_2$ is a generic point for the Bernoulli measure $\xi_{1/2}$ such that
$\bar z|_J=
y|_J$,
then $(z,\chi_J)\eqdef\psi(y,\chi_J)=\psi(\bar z,\chi_J)$.
Recall that the only joining of a Bernoulli measure $\xi_{1/2}$ and the zero entropy measure $\mu_J$ is the product measure $\xi_{1/2}\times\mu_J$, see \cite[Theorem 18.16]{Glasner}.
As any limit point of $(\mu_{n_k}((\bar z,\chi_J),S))_{k\in\mathbb{N}}$ is a joining of $\xi_{1/2}$ and $\mu_J$, we get that
$(\bar z,\chi_J)$ generates along $(n_k)_{k\in\mathbb{N}}$ the $S$-invariant measure $\xi_{1/2}\times \mu_J$. It follows that $(z,\chi_J)$ generates along $(n_k)_{k\in\mathbb{N}}$ the $S$-invariant measure
$\mu''=\psi_*(\xi_{1/2}\times\mu_J)$.
Note that this shows that all measures in $\mathcal{M}_S(\Omegaega_3\times\Omegaega_2)$ generated by $(y,\chi_J)$ along $(n_k)_{k\in\mathbb{N}}$ are pushed forward by $\psi$ onto $\mu''$.
Therefore to finish the proof of the Claim \ref{p.l.entropy} it is enough to see that
\begin{equation}
\lambdabel{e.toseethat}
h_{\mu''}(S) \ge \underline{d}(J)\cdot \log 2.
\end{equation}
Let $\mathbb{I}_{[1]_0}$ be the characteristic function of the cylinder $[1]_0\subset\Omegaega_2$.
Note that from the definition of $\underline{d}(J)$ and $\bar d(J)$ it follows immediately that
\begin{equation}
\lambdabel{e.dj}
\underline{d}(J) \le
\lim_{k\to\infty} \frac{1}{n_k}\sum_{j=0}^{n_k-1} \mathbb{I}_{[1]_0}(\sigma^{j}(\chi_J))
\le \bar d (J).
\end{equation}
Observe also that the measure $\mu''$, by its definition, is concentrated on the set
\[
\big( [0]_0\times[1]_0 \big) \cup \big( [1]_0 \times [1]_0 \big) \cup \big( [2]_0\times [0]_0 \big)\subset \Omegaega_3\times\Omegaega_2.
\]
Consider the set $E\eqdef \big([0]_0\times[1]_0\big)\cup\big([1]_0\times [1]_0\big)\subset \Omegaega_3\times\Omegaega_2$
and let $\mathbb{I}_E$ be its characteristic function.
\begin{cla}
\lambdabel{cl.theclaim} We have
$0<\underline{d}(J)\le\mu''(E)=\mu_J([1]_0)\le\bar d(J)$.
\end{cla}
\begin{proof}[Proof of Claim \ref{cl.theclaim}]
Note that for $n\in\mathbb{N}$ it holds $S^n(z,\chi_J)\in E$ if and only if $\sigma^n(\chi_J)\in [1]_0$, equivalently, if $n\in J$.
Recall that along $(n_k)$, the point $(z,\chi_J)$ generates $\mu''$ and $\chi_J$ generates $\mu_J$.
Furthermore,
as the topological boundaries of $E$ and $[1]_0$ are empty, it follows from the portmanteau theorem \cite[Thm. 18.3.4]{Garling}, that
\[
\mu''(E)=\lim_{k\to\infty}\frac{1}{n_k}\sum_{j=0}^{n_k-1} \mathbb{I}_E(S^{j}(z,\chi_J))
=\lim_{k\to\infty} \frac{1}{n_k}\sum_{j=0}^{n_k-1} \mathbb{I}_{[1]_0}(\sigma^{j}(\chi_J)).
\]
Equation \eqref{e.dj} implies now that
$0<\underline{d}(J)\le\mu''(E)\le\bar d(J)$, proving Claim \ref{cl.theclaim}.
\end{proof}
By Claim \ref{cl.theclaim} we have that $S$ induces a measure preserving system $(E,\mu''_E,S_E)$ on $E$,
where $\mu''_E(A)=\mu''(A\cap E)/\mu''(E)$ for every Borel set $A\subset \Omegaega_3\times\Omegaega_2$
and $S_E(x) =S^{r(x)} (x)$, where $r(x)=\inf\{ q>0 \colon S^q(x) \in E\}$ is defined for $\mu''$-a.e. point $x\in E$.
\begin{cla}\lambdabel{l.bernoulli}
The measure preserving system $(\Omegaega_2,\xi_{1/2},\sigma)$ is a factor of $(E,\mu''_E,S_E)$.
\end{cla}
Let us assume that Claim \ref{l.bernoulli} holds and conclude the proof of Claim~\ref{p.l.entropy}.
Note that by Claim \ref{l.bernoulli} we have
$h_{\mu''_E}(S_E)\ge \log 2$. Now by Abramov's formula\footnote{The proof that this well-known formula works for transformations which can be not ergodic nor invertible, is due H. Scheller, see \cite{Krengel} or \cite[p. 257]{Petersen}.}
it follows that
\[
h_{\mu''_E}(S_E)=h_{\mu''}(S)/\mu''(E).
\]
By Claim~\ref{cl.theclaim}, this yields $\underline{d}(J)\cdot \log 2\le h_{\mu''}(S)$,
proving \eqref{e.toseethat} and finishing the proof of Claim~\ref{p.l.entropy}.
\end{proof}
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By Claim \ref{cl.theclaim} we have that $S$ induces a measure preserving system $(E,\mu''_E,S_E)$ on $E$,
where $\mu''_E(A)=\mu''(A\cap E)/\mu''(E)$ for every Borel set $A\subset \Omegaega_3\times\Omegaega_2$
and $S_E(x) =S^{r(x)} (x)$, where $r(x)=\inf\{ q>0 \colon S^q(x) \in E\}$ is defined for $\mu''$-a.e. point $x\in E$.
\begin{cla}\lambdabel{l.bernoulli}
The measure preserving system $(\Omegaega_2,\xi_{1/2},\sigma)$ is a factor of $(E,\mu''_E,S_E)$.
\end{cla}
Let us assume that Claim \ref{l.bernoulli} holds and conclude the proof of Claim~\ref{p.l.entropy}.
Note that by Claim \ref{l.bernoulli} we have
$h_{\mu''_E}(S_E)\ge \log 2$. Now by Abramov's formula\footnote{The proof that this well-known formula works for transformations which can be not ergodic nor invertible, is due H. Scheller, see \cite{Krengel} or \cite[p. 257]{Petersen}.}
it follows that
\[
h_{\mu''_E}(S_E)=h_{\mu''}(S)/\mu''(E).
\]
By Claim~\ref{cl.theclaim}, this yields $\underline{d}(J)\cdot \log 2\le h_{\mu''}(S)$,
proving \eqref{e.toseethat} and finishing the proof of Claim~\ref{p.l.entropy}.
\end{proof}
Since Claim~\ref{p.l.entropy} implies Claim \ref{thm:mainbis}, and the latter implies Theorem \ref{t.p.entropy},
it remains to prove Claim \ref{l.bernoulli}.
\begin{proof}[Proof of Claim~\ref{l.bernoulli}]
Consider the partition $\cP_E\eqdef \{P_0,P_1\}$ of $E$, where
$P_0\eqdef [0]_0\times[1]_0$ and $P_1\eqdef [1]_0\times [1]_0$.
Fix $N\in\mathbb{N}$ and $v=v_1\ldots v_N\in\{0,1\}^N$. Let
\[
\mathcal P_v\eqdef P_{v_1}\cap S_E^{-1}(P_{v_2})\cap\ldots\cap S_E^{-N+1}(P_{v_N}).
\]
Our goal is to prove that $\mu''_E(\mathcal P_v) = 1/2^N$, which implies that $(\Omegaega_2,\xi_{1/2},\sigma)$ is a factor $(E,\mu''_E,S_E)$ through the factor map generated by $\cP_E$.
To this end we need some auxiliary notation. Let $\mathcal G_J^N$ be the set of blocks over $\{0,1\}$ which contain exactly $N$ occurrences of $1$, start with $1$, and end with $1$. For $u\in\mathcal G_J^N$ and $1\le j\le N$ we denote by $o(j)$ the position of the $j$-th occurrence of $1$ in $u$
and define
\[
V_{v,u}\eqdef \{(\omega,\bar\omega)\in \operatorname{supp}\mu'': \bar \omega\in [u]_0 \text{ and }\omega_{o(j)}=v_{j}\text { for }j=1,\ldots,N\}.
\]
From the definition of $\mu''$ it follows that
$\mu''(V_{v,u})=(1/2^N)\,\mu_J([u]_0)$.
Furthermore, we set
\[
U_v\eqdef \bigcup_{u\in \mathcal G_J^N}V_{v,u},
\qquad\text{hence}\qquad
\mu''(U_v)=\frac{1}{2^N}\, \sum_{u\in \mathcal G_J^N}\mu_J([u]_0).
\]
Noting that
$\mu_J$-almost every point $\bar\omega\in[1]_0$ belongs to some
$[u]_0$ with $u\in\mathcal G_J^N$
we get that
$$
\sum_{u\in \mathcal G_J^N}\mu_J([u]_0)=\mu_J([1]_0).
$$
Therefore, using Claim~\ref{cl.theclaim}, we get
$$
\mu''(U_v)=\frac{\mu_J([1]_0)}{2^N}= \frac{\mu''(E)}{2^N}.
$$
Note also that $\mathcal P_v=U_v\cap E=U_v$, thus
\[
\mu''_E(\mathcal P_v)=\frac{\mu''(U_v)}{\mu''(E)}= \frac{1}{2^N},
\]
proving Claim \ref{l.bernoulli}.
\end{proof}
The proof of Claim~\ref{t.p.entropy} is now complete. This ends the proof of Theorem \ref{thm:mainbis}.
\end{proof}
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\subsection{Rational sets and proof of Theorem~\ref{thm:main}}\lambdabel{ss.rational}
The notion of a rational set was introduced by Bergelson and Ruzsa \cite{BR1}.
Below, by an {\emph{arithmetic progression}} we mean a set of the form $a\mathbb{Z}+b$ for some $a,b\in\mathbb{N}$, $a\neq 0$.
\begin{defn}\lambdabel{d.rational}
We say that a set $A\subset\mathbb{Z}$ is \emph{rational} if for every $\varepsilon>0$ there is a set $B\subset\mathbb{Z}$ which is a union of finitely many arithmetic progressions and satisfies $\bar d(A\div B)<\varepsilon$, where $A\div B$ stands for the symmetric difference of $A$ and $B$. A subset $B$ of $\mathbb{N}$ is \emph{rational} if $B=C\cap \mathbb{N}$ for some rational set $C\subset \mathbb{Z}$.
\end{defn}
\begin{rem}\lambdabel{r.complement}
If $A \subset \mathbb{Z}$ is rational then its complement $\mathbb{Z}\setminus A$ is also rational. The same holds for rational subsets of $\mathbb{N}$.
Note that, by definition, a rational set has a well defined density.
\end{rem}
Recall here that the formula
\[
\bar d(x,y)\eqdef \limsup_{N\to\infty}\frac{1}{N}
\card{\{0\le n<N:x_n\neq y_n\}},\quad \text{for }x,y\in\Omegaega_M
\]
defines a pseudometric on $\Omegaega_M$.
In the following, we need the following properties of $\bar d$:
\begin{enumerate}
\item\lambdabel{p1} If $(x_n)_{n\in\mathbb{N}}\subset\Omegaega_M$ is a sequence of ergodic points and $x\in\Omegaega_M$ is such that $\bar d(x_n,x)\to 0$ as $n\to \infty$,
then $x$ is also an ergodic point.
\item\lambdabel{p2} Furthermore, if $(x_n)_{n\in\mathbb{N}}\subset\Omegaega_M$ and $x\in\Omegaega_M$ are as above and $V(x_n)=\{\mu_n\}$ and $V(x)=\{\mu\}$, then $h_{\mu_n}(\sigma)\to h_\mu(\sigma)$ as $n\to\infty$.
\end{enumerate}
A proof of \eqref{p1} is sketched in \cite{Weiss}, alternatively it follows from \cite[Theorem 15 and Corollary 5]{KLO}.
To see \eqref{p2} one combines \eqref{p1} with
\cite[Theorem I.9.16]{Shields} and the proof of \cite[Theorem I.9.10]{Shields}.
Corollary~\ref{c:main} (and therefore Theorem~\ref{thm:main}) follows from the following result.
\begin{lem}\lambdabel{lem:genericity}
If $A\subset \mathbb{Z}$ or $A\subset \mathbb{N}$ is a rational set, then its characteristic function $\chi_A\in\Omegaega_2$ is a completely deterministic ergodic point.
\end{lem}
\begin{proof}
Note that a set $B\subset \mathbb{Z}$ is a union of finitely many arithmetic progressions if and only if its characteristic function
$\chi_B\in\Omegaega_2$ is a periodic point for the shift map $\sigma\colon\Omegaega_2\to\Omegaega_2$. Furthermore,
$\bar d(A\div B)<\varepsilon$
is equivalent to $\bar d(\chi_A,\chi_B)<\varepsilon$.
Thus, by definition, for every rational set $A\subset \mathbb{Z}$ there is a sequence $(z_n)_{n=0}^\infty$ of $\sigma$-periodic points in $\Omegaega_2$ such that $\bar d(\chi_A,z_n)\to 0$ as $n\to\infty$.
Since each $z_n$ is a generic point for a zero entropy ergodic measure, the properties \eqref{p1}--\eqref{p2} of $\bar d$ mentioned above allow us to finish the proof if $A\subset \mathbb{Z}$. For $A\subset\mathbb{N}$ it is enough to note that if $C\subset \mathbb{Z}$ is such that $\chi_C\in\Omegaega_2$ is a completely deterministic ergodic point and $A=C\cap\mathbb{N}$, then $\chi_A\in\Omegaega_2$ is also a completely deterministic ergodic points, as these notions depend only on the forward orbit of a point.
\end{proof}
Using Lemma~\ref{lem:genericity} we see that the characteristic function $\chi_A\in\Omegaega_2$
of a rational set $A\subset \mathbb{Z}$ is a completely deterministic ergodic point and has a well defined density. Therefore we get the following corollary of Theorem \ref{thm:mainbis}, which explains why the theorem about control of entropy stated in the introduction follows from Theorem \ref{thm:mainbis}.
\begin{coro}\lambdabel{c:main}
The conclusion of Theorem~\ref{thm:mainbis} holds if $J$ is a rational set of $\mathbb{N}$ with $0<d(J)<1$.
\end{coro}
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\section{Rational long sparse tails}
\lambdabel{s.scalesandtails}
The aim of this section is to find rational subsets of $\mathbb{N}$, which fulfill the requirements needed in the construction of a controlled point from \cite{BDB:}. These sets are coined in \cite{BDB:} \emph{$\cT$-long $\bar\varepsilon$-sparse tails} (associated to a scale $\cT$ and a controlling sequence $\bar\varepsilon$) and are discussed below under the name of \emph{$\cT$-sparsely long tails}. Their elements are times where
the control of the averages of a function is partially lost. The times in a tail allows us to control and
spread the support of measures generated by a controlled point (here longness of a tail is crucial for guaranteeing the full support), while we retain some control on the averages due to sparseness. To control the entropy we define our tail so that it is also a rational set. Hence its complement, that is, the set of times defining the regular part of the orbit, is also rational. It follows that both sets, the tail and the regular part, have nontrivial and well-defined densities. This allows us to apply the criterion for positivity of the entropy (Theorem \ref{thm:mainbis}) to the measures generated by a controlled point.
We first recall from \cite{BDB:} the definitions
of scales and sparsely long tails.
\begin{defn}[Scale]\lambdabel{d.scale}
We say that a sequence of positive integers $\cT=(T_n)_{n\in\mathbb{N}}$ is \emph{a scale} if
there is an integer sequence $\bar \kappa=(\kappa_n)_{n\in\mathbb{N}}$ of \emph{factors of $\cT$}
such that
\begin{itemize}
\item $\kappa_0=3$, and $\kappa_{n}$ is a multiple of $3\kappa_{n-1}$ for every $n\ge 1$,
\item
$T_0$ is a multiple of $3$ and
$T_{n}=\kappa_{n} \, T_{n-1}$ for every $n\ge 1$,
\item
$\kappa_{n+1}/\kappa_n \to \infty$ as $n\to\infty$.
\end{itemize}
\end{defn}
\begin{rem}
We have that
\[
\sum_{n=0}^\infty\frac{1}{\kappa_n}\le\sum\frac{1}{3^n}<1.
\]
\end{rem}
\begin{defn}[$\mathbb{N}$-interval]
An \emph{interval of integers} (or $\mathbb{N}$-interval for short) is a set
$[a,b]_{\mathbb{N}N}\eqdef[a,b]\cap \mathbb{N}N$, where $a,b\in \mathbb{N}N$.
\end{defn}
\begin{defn}[Component of a set ${\mathbb M}} \deltaf\NN{{\mathbb N}} \deltaf\OO{{\mathbb O}} \deltaf\PP{{\mathbb P}\subset \mathbb{N}$]
Given a subset ${\mathbb M}} \deltaf\NN{{\mathbb N}} \deltaf\OO{{\mathbb O}} \deltaf\PP{{\mathbb P}$ of $\mathbb{N}N$
a \emph{component} of ${\mathbb M}} \deltaf\NN{{\mathbb N}} \deltaf\OO{{\mathbb O}} \deltaf\PP{{\mathbb P}$ is any maximal $\mathbb{N}$-interval contained in ${\mathbb M}} \deltaf\NN{{\mathbb N}} \deltaf\OO{{\mathbb O}} \deltaf\PP{{\mathbb P}$, that is,
$[a,b]_{\mathbb{N}N}\subset {\mathbb M}} \deltaf\NN{{\mathbb N}} \deltaf\OO{{\mathbb O}} \deltaf\PP{{\mathbb P}$ is a component of ${\mathbb M}} \deltaf\NN{{\mathbb N}} \deltaf\OO{{\mathbb O}} \deltaf\PP{{\mathbb P}$ if and only if $b+1\notin {\mathbb M}} \deltaf\NN{{\mathbb N}} \deltaf\OO{{\mathbb O}} \deltaf\PP{{\mathbb P}$ and $a-1\notin {\mathbb M}} \deltaf\NN{{\mathbb N}} \deltaf\OO{{\mathbb O}} \deltaf\PP{{\mathbb P}$.
\end{defn}
\begin{defn}[$T$-regular interval]
Let $T$ be a positive integer. We say that an $\mathbb{N}$-interval $I$ is \emph{$T$-regular interval} if
$I=[kT,(k+1)T-1]_\mathbb{N}$ for some $k\ge 0$.
\end{defn}
\begin{defn}[$\cT$-adapted set, $n$-skeleton]
Let $\cT=(T_n)_{n\in \mathbb{N}N}$ be a scale. We say that a set $R_\infty\subset\mathbb{N}$ is \emph{$\cT$-adapted} if every component of $R_\infty$ is a $T_n$-regular interval for some $n\in\mathbb{N}$.
Given $n\in \mathbb{N}$ the \emph{$n$-skeleton} $R_n$ of $R_\infty$ is the union of all components of $R_\infty$ which are $T_k$-regular intervals for some $k\ge n$.
\end{defn}
By definition for any $\cT$-adapted set $R_\infty$ we have $R_\infty=R_0\supset R_1\supset R_2\supset\ldots$.
\begin{defn}[Sparsely long tail]\lambdabel{d.tail}
Consider a scale
$\cT=(T_n)_{n\in \mathbb{N}N}$ with a sequence of factors $\bar\kappa=(\kappa_n)_{n\in \mathbb{N}N}$.
A set $R_\infty\subset \mathbb{N}N$ is a \emph{$\cT$-sparsely long tail} if the following holds:
\begin{enumerate}
\item\lambdabel{i.adapted}
$R_\infty$ is $\cT$-adapted,
\item\lambdabel{i.0} $0\notin R_\infty$, in particular
$[0,T_n-1]_{\mathbb{N}N} \not\subset R_n$, where $R_n$ is the $n$-skeleton of $R_\infty$,
\item\lambdabel{i.center} If a $T_n$-regular interval $I=[a,b]_\mathbb{N}$ is not contained in $R_\infty$,
equivalently if $I\not\subset R_n$, then the $(n-1)$-skeleton $R_{n-1}$ of $R_\infty$ can intersect nontrivially only in the middle third interval of $I$ and is {\emph{$1/\kappa_n$-sparse}} in $I$, that is
\begin{gather*}
I\cap R_{n-1}\subset\left[ \left(a +\nicefrac{T_n}{3}\right),\left(b-\nicefrac{T_n}{3}\right)\right]_\mathbb{N}N, \\
0< \frac{ \card{R_{n-1}\cap I}}{T_n}\le 1/\kappa_n.
\end{gather*}
\end{enumerate}
\end{defn}
\begin{defn}[Rational sparsely long tail]
We say that $R_\infty$ is a \emph{rational $\mathcal{T}$-sparsely long tail} if it satisfies Definitions \ref{d.rational} and \ref{d.tail}.
\end{defn}
Next, we extend \cite[Lemma 2.7]{BDB:} adding rationality of the tail to its conclusion. In fact, the tail constructed in \cite{BDB:} is also a rational tail but this fact is not noted there.
\begin{propo}[Existence of rational sparsely long tails]\lambdabel{p.l.tailexistence}
Let $\cT=(T_n)_{n\in \mathbb{N}N}$ be a scale and $\bar \kappa=(\kappa_n)_{n\in\mathbb{N}}$ be its sequence of factors.
Then there is a rational $\cT$-sparsely long tail $R_\infty$ with $0<d(R_\infty)<1$.
\end{propo}
\begin{proof}
For each $n\in\mathbb{N}$ define $A_{n+1} \eqdef\left[\nicefrac{T_{n+1}}{3}, \nicefrac{T_{n+1}}{3}+T_n-1 \right]_\mathbb{N}N$. Then we set
\[
R^*_\infty = \bigcup_{n\ge 1} (A_n + T_n \mathbb{Z}Z)\qquad\text{and}\qquad R_\infty=R^*_\infty\cap \mathbb{N}.
\]
It is easy to see that the requirements imposed on the growth of the scale $\cT$ imply that the characteristic function $\chi_\infty$ of $R^*_\infty$ is a \emph{regular Toeplitz sequence} (see \cite{Downarowicz-Toeplitz}), which immediately yields that $R_\infty$ is rational. Clearly, $R_\infty\neq \mathbb{N}$, which gives $0<d(R_\infty)<1$, again because $\chi_\infty$ is Toeplitz. But for the convenience of the readers we provide a direct elementary proof of these facts.
Let $\Pi_0=\emptyset$ and for each $n\ge1$ we set $\Pi_n=R_\infty\cap[0,T_n-1]_\mathbb{N}$. It follows that for each $n\in\mathbb{N}$ we have $\Pi_n\subset\Pi_{n+1}$ and
\begin{equation}\lambdabel{e.pin}
\Pi_{n+1}=
\left( \bigcup_{k=0}^{\kappa_{n+1}-1} \Pi_n + k T_n \right) \cup A_{n+1}.
\end{equation}
\begin{cla}
\lambdabel{cl.longsparse}
The set $R_\infty $ is a $\cT$-sparsely long tail.
\end{cla}
\begin{proof}We need to check conditions \eqref{i.adapted}, \eqref{i.0} and \eqref{i.center} from Definition \ref{d.tail}.
First we prove \eqref{i.adapted}, which says that $R_\infty$ is $\mathcal{T}$-adapted. Fix $n\ge 0$ and note that neither $0$, nor $T_{n+1}-1$ belongs to $\Pi_{n+1}$. We claim that the components of $\Pi_{n+1}$ are $T_i$-regular for some $i\le n$. Note that the $T_n$-regular interval $A_{n+1}$ is a component of $\Pi_{n+1}$. The other components of $\Pi_{n+1}$ are components of $\Pi_n$ translated by a number $\ell T_n$, for some $\ell\in \{0,\dots,\kappa_{n+1}-1\}$. Arguing inductively, we get that the components of $R_\infty$ are $T_i$-regular for some $i\in\mathbb{N}$.
Obviously $0\not \in R_\infty$, yielding \eqref{i.0}. It remains to prove \eqref{i.center}. Fix $n\ge 1$. Note that
$
R_{n-1} \cap \Pi_n= A_n,
$ and $A_n$ is contained in the middle third interval of $\Pi_n$.
It follows that
$$
\frac{\card{R_{n-1} \cap \Pi_n}}{T_n}= \frac{\card{A_n}}{T_n}=\frac{T_{n-1}}{T_n}=\frac{1}{\kappa_n}.
$$
This proves that $T_n$-regular interval $[0,T_n-1]\not\subset R_\infty$ containing $\Pi_n$ satisfies \eqref{i.center}.
The same holds for every $T_n$-regular interval $I$ not contained in $R_\infty$, because such $I\cap R_\infty=\Pi_n+j T_n$ for some $j\ge 1$.
\end{proof}
\begin{cla}
\lambdabel{cl.rational}
The set $R_\infty $ is rational.
\end{cla}
\begin{proof}
Define $Q_n\eqdef \bigcup_{i=1}^{n} (A_i + T_i \mathbb{N})$. Then $Q_n$is a finite union of arithmetic sequences. Furthermore, $R_\infty\div Q_n \subset R_n$ and the $n$ skeleton $R_n$ of
$R_\infty$ satisfies
\begin{equation}\lambdabel{e.rn}
R_n =\bigcup_{i=n+1}^\infty (A_i + T_i \mathbb{N}).
\end{equation}
Thus it is enough to see that $\bar d (R_n) \to 0$ as $n\to\infty$. But by \eqref{e.rn} and subadditivity of $\bar d$ we have
\[
\bar d(R_n) =\bar d\bigg(\bigcup_{i=n+1}^\infty (A_i + T_i \mathbb{N})\bigg)\le \sum_{i=n+1}^\infty \bar d(A_i + T_i \mathbb{N}).
\]
It is easy to see that for each $i\ge 1$ we have $\bar d(A_i + T_i \mathbb{N}) = T_{i-1}/T_i=1/\kappa_i$. As a conclusion, we get
$\bar d (R_n) \le \sum_{i=n+1}^\infty \frac{1}{\kappa_n}$,
proving that the tail is rational.
\end{proof}
\begin{cla}\lambdabel{cl.density}
We have $0<d(R_\infty)<1$.
\end{cla}
\begin{proof}
The set is rational and hence
it has a well defined density, see Remark~\ref{r.complement}. We also have
\[
\frac{1}{\kappa_1}=
\bar d(A_1+T_1\mathbb{N})\le \bar d(R_\infty)=\bar d\bigg(\bigcup_{i=1}^\infty (A_i + T_i \mathbb{N})\bigg)
\le\sum_{i=1}^\infty\frac{1}{\kappa_i}<1.\qedhere
\]
\end{proof}
The lemma now follows from Claims~\ref{cl.longsparse}, \ref{cl.rational},
and
\ref{cl.density}.
\end{proof}
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\section{Double flip-flop families}
\lambdabel{s.flipflop}
In this section, we review the definitions of flip-flop families following \cite{BBD:16,BDB:}. We also introduce the notion of a double flip-flop family and
prove that flip-flops families yield double flip-flop families, see Proposition~\ref{p.yieldsdouble}.
Using these families and Theorem~\ref{thm:main} obtain ergodic measures with full support, positive entropy, and zero average for a continuous potential $\varphi\colon X \to \mathbb{R}$, see Theorem~\ref{t.flipfloptailqual}.
In what follows, $(X, \rho)$ is a compact metric space,
$f\colon X \to X$ is a homeomorphism, and $\varphi\colon X \to \mathbb{R}$ is
a continuous function.
\subsection{Flip-flop families}
We begin by recalling the definition of flip-flop families.
\begin{defn}[Flip-flop family]\lambdabel{d.flipflop}
A \emph{flip-flop family} associated to $\varphi$ and $f$ is a family
$\mathfrak{F}=\mathfrak{F}^+\sqcup\mathfrak{F}^-$ of compact subsets of $X$ called {\emph{plaques}}\footnote{We pay special attention to the case when the sets of the flip-flop family
are discs tangent to a strong unstable cone field. This justifies this name.}
such that there are $\alpha>0$ and a
sequence of numbers $(\zeta_n)_n$ and $\zeta_n\to 0^+$ as $n\to \infty$,
satisfying:
\begin{enumerate}
\item\lambdabel{i.flipflop1}
let $F_\mathfrak{F}^+\eqdef \bigcup_{D\in\mathfrak{F}^+} D$ (resp. $F_\mathfrak{F}^-\eqdef \bigcup_{D\in\mathfrak{F}^-} D $), then $\varphi(x)>\alpha$ for every $x\in F_\mathfrak{F}^+$
(resp. $\varphi(x)< -\alpha$ for every $x\in F_\mathfrak{F}^-$);
\item\lambdabel{i.flipflop2}
for every $D\in \mathfrak{F}$, there are sets $D^+\in \mathfrak{F}^+$ and $D^-\in\mathfrak{F}^-$ contained in $f(D)$;
\item\lambdabel{i.flipflop33}
for every $n>0$ and every family of sets $D_i\in \mathfrak{F}$, $i\in\{0,\dots ,n\}$, with $D_{i+1}\subset f(D_i)$
it holds
$$
\mathrm{diam} (f^{-i} (D_n))\le \zeta_i, \quad \mbox{for every $i\in\{0,\dots,n\}$.}
$$
\end{enumerate}
\end{defn}
We now recall the notions of $f$-sojourns. Note that in this definition the flip-flop family may be relative to
a power $f^k$ of $f$, but the sojourns are relative to $f$. Furthermore, $\varphi_k$ stands here for the Birkhoff averages of a map $\varphi$ with respect to $f$ introduced in \eqref{e.averages}.
\begin{defn}[Flip-flop family with $f$-sojourns]\lambdabel{d.flipfloptail}
Consider a flip-flop family
$\mathfrak{F}=\mathfrak{F}^+\sqcup\mathfrak{F}^-$
associated to $\varphi_k$ and $f^k$ for some $k\ge 1$ and
a compact subset $Y$ of $X$.
The \emph{flip-flop family $\mathfrak{F}$ has $f$-sojourns along $Y$}
(or \emph{$\mathfrak{F}$ $f$-sojourns along $Y$}) if
there is a sequence $(\eta_n)_n$ and $\eta_n\to 0^+$, such that
for every $\deltalta>0$ there is an integer $N=N_\deltalta$ so that
every plaque $D\in\mathfrak{F}$ contains subsets $\widehat D^+, \widehat D^-$ satisfying:
\begin{enumerate}
\item\lambdabel{i.defff0}
for every $x\in \widehat D^+\cup \widehat D^-$ the orbit segment $\{x,\dots, f^N(x)\}$ is $\deltalta$-dense in $Y$
(i.e., the $\deltalta$-neighbourhood of the orbit segment contains $Y$);
\item\lambdabel{i.defff1}
$f^N(\widehat D^+)=\widehat D^+_N\in \mathfrak{F}^+$ and $f^N(\widehat D^-)=\widehat D^-_N\in\mathfrak{F}^-$;
\item\lambdabel{i.defff2}
for every $i\in\{0,\dots, N\}$
it holds
$$
\mathrm{diam} (f^{-i} (\widehat D_N^\pm))\le \eta_i.
$$
\end{enumerate}
\end{defn}
We are now in position to define double flip-flop families (with sojourns). Observe that the remark before Definition \ref{d.flipfloptail} applies also to Definition \ref{d.dflipfloptail}.
\begin{defn}[Double flip-flop family]\lambdabel{d.dflipflop}
A \emph{double flip-flop family} associated to $\varphi$ and $f$ is a family
$\mathfrak{D}\eqdef \mathfrak{D}^+_0\sqcup\mathfrak{D}^+_1 \sqcup \mathfrak{D}^-_0\sqcup\mathfrak{D}^-_1 $ of compact subsets of $X$ such that there are $\alpha>0$ and a
sequence of numbers $(\zeta_n)_n$ with $\zeta_n\to 0^+$ as $n\to \infty$,
and with the following properties:
let
$E^+_i\eqdef \bigcup_{D\in\mathfrak{D}^+_i} D$ and
$E^-_i\eqdef \bigcup_{D\in\mathfrak{D}^-_i} D$, where $i=0,1$,
$E^+\eqdef E^+_0 \cup E^+_1$, and
$E^-\eqdef E^-_0 \cup E^-_1$.
\begin{enumerate}
\item\lambdabel{i.dflipflop1} $\varphi(x)\geq\alpha$ for every $x\in E^+$ and $\varphi(x)\leq -\alpha$ for every $x\in E^-$;
\item\lambdabel{i.dflipflop2}
for every $D\in \mathfrak{D}$, there are sets $D^+_0\in \mathfrak{D}^+_0$, $D^+_1\in \mathfrak{D}^+_1$, $D^-_0\in \mathfrak{D}^-_0$, and
$D^-_1\in \mathfrak{D}^-_1$ contained in $f(D)$;
\item\lambdabel{i.dflipflop3}
for every $n>0$ and every family of sets $D_i\in \mathfrak{D}$, $i\in\{0,\dots ,n\}$, with $D_{i+1}\subset f(D_i)$
it holds
$$
\mathrm{diam} (f^{-i} (D_n))\le \zeta_i, \quad \mbox{for every $i\in\{0,\dots,n\}$.}
$$
\item\lambdabel{i.dflipflop4}
The closures of the sets $E^+_0, E^+_1, E^-_0, E^-_1$ are pairwise disjoint\footnote{This condition is straightforward in the flip-flop case since
$\varphi$ is strictly bigger that $\alpha>0$ in $F^+_\cF$ and strictly less than $-\alpha<0$ in $F^-_\mathfrak{F}$.}.
\end{enumerate}
\end{defn}
\begin{defn}[Double flip-flop family with $f$-sojourns along $Y$]\lambdabel{d.dflipfloptail}Let $\mathfrak{D}\eqdef \mathfrak{D}^+_0\sqcup\mathfrak{D}^+_1 \sqcup \mathfrak{D}^-_0\sqcup\mathfrak{D}^-_1 $ be a double flip-flop family associated to $\varphi_k$, $f^k$, $k\ge 1$.
Given a compact subset $Y$ of $X$ we say that
\emph{$\mathfrak{D}$ has $f$-sojourns along $Y$}
(or that \emph{$\mathfrak{D}$ $f$-sojourns along $Y$}) if
there is a sequence $(\eta_n)_n$ and $\eta_n\to 0^+$, such that
for every $\deltalta>0$ there is an integer $N=N_\deltalta$ such that
every plaque $D\in\mathfrak{D}$ contains subsets $\widehat D^+_0, \widehat D^+_1,
\widehat D^-_0, \widehat D^-_1$ such that:
\begin{enumerate}
\item\lambdabel{i.def-dff0}
for every $x\in \widehat D^+_0\cup \widehat D^+_1 \widehat D^-_0 \cup \widehat D^-_1$ the orbit segment $\{x,f(x),\dots, f^N(x)\}$ is $\deltalta$-dense in $Y$;
\item\lambdabel{i.def-dff1}
$f^N(\widehat D^i_j)= D^i_{N,j} \in \mathfrak{D}^i_j$, $i\in \{-,+\}$ and $j\in \{0,1\}$;
\item\lambdabel{i.def-dff2}
for every $i\in\{0,\dots, N\}$
it holds
$$
\mathrm{diam} (f^{-i} (\widehat D_{N,j}^\pm))\le \eta_i.
$$
\end{enumerate}
\end{defn}
\begin{rem} \lambdabel{r.multiple}
In the previous definitions the constant $N$ can be chosen a multiple of $k$.
\end{rem}
\subsection{Existence of double flip-flop families with sojourns}
\lambdabel{ss.double}
We now prove that existence of flip-flop families with sojourns implies the existence of double flip-flop families with sojourns.
\begin{propo}
\lambdabel{p.yieldsdouble}
Consider a flip-flop family
$\mathfrak{F}=\mathfrak{F}^+\sqcup\mathfrak{F}^-$
associated to $\varphi_k$ and $f^k$ for some $k\ge 1$ with $f$-sojourns in
a compact subset $Y$ of $X$. Then there are $r\ge 1$ and a double flip-flop family $\mathfrak{D}= \mathfrak{D}^+_0\sqcup \mathfrak{D}^+_1 \sqcup
\mathfrak{D}^-_0\sqcup \mathfrak{D}^-_1$ associated to $f^r$ and $\varphi_r$ with $f$-sojourns along $Y$.
\end{propo}
\begin{proof}
Given a plaque $D\in \mathfrak{F}$ and $\ell \ge 1$, consider subsets
$D_{+^\ell,+}$ $D_{+^\ell,-}$, $D_{-^\ell,+}$, and $D_{-^\ell,-}$ of $D$ satisfying
\begin{itemize}
\item
$f^{ki} (D_{+^\ell,+})$ is contained in some plaque of $\mathfrak{F}^+$
for every $i\in \{1,\dots,\ell\}$ and $f^{k(\ell+1)} (D_{+^\ell,+})\in \mathfrak{F}^+$,
\item
$f^{ki} (D_{+^\ell,-})$
is contained in some plaque of $\mathfrak{F}^+$
for every $i\in \{1,\dots,\ell\}$ and $f^{k(\ell+1)} (D_{+^\ell,-})\in \mathfrak{F}^-$,
\item
$f^{ki} (D_{-^\ell,-})$ is contained in some plaque of $\mathfrak{F}^-$ for every $i\in \{1,\dots,\ell\}$ and $f^{k(\ell+1)} (D_{+^\ell,-})\in \mathfrak{F}^-$,
\item
$f^{ki} (D_{-^\ell,+}))$ is contained in some plaque of $\mathfrak{F}^-$ for every $i\in \{1,\dots,\ell\}$ and $f^{k(\ell+1)} (D_{-^\ell,+})\in \mathfrak{F}^+$.
\end{itemize}
The existence of these subsets is assured by item \eqref{i.flipflop2} in the definition of a flip-flop family.
Using the continuity of $\varphi$, we have that for every $\ell$ large enough there is $\alpha'>$ such that
\[
\begin{split}
&\varphi_{k(\ell+1)} (x) > \alpha'>0 \quad \mbox{if $x\in D_{+^\ell,\pm}$},\\
&\varphi_{k(\ell+1)} (x) < -\alpha'<0 \quad \mbox{if $x\in D_{-^\ell,\pm}$}.
\end{split}
\]
We use here that the $\ell+1$ Birkhoff averages of $\varphi_k$ with respect to $f^k$ are the same as $\varphi_{k(\ell+1)}$, that is, $k(\ell+1)$ averages of $\varphi$ with respect to $f$.
We fix such a large $\ell$ and define
\[
\begin{split}
\mathfrak{D}^+_0&\eqdef \{D_{+^\ell,+}, \, D\in \mathfrak{F}\}, \quad
\mathfrak{D}^+_1\eqdef \{D_{+^\ell,-}, \, D\in \mathfrak{F}\},\\
\mathfrak{D}^-_0&\eqdef \{D_{-^\ell,+}, \, D\in \mathfrak{F}\}, \quad
\mathfrak{D}^-_1\eqdef \{D_{-^\ell,-}, \, D\in \mathfrak{F}\}.
\end{split}
\]
By construction $\mathfrak{D}=\mathfrak{D}^+_0\sqcup \mathfrak{D}^+_1\sqcup \mathfrak{D}^-_0\sqcup \mathfrak{D}^-_1$ satisfies conditions
\eqref{i.dflipflop1}, \eqref{i.dflipflop2}, and \eqref{i.dflipflop3} in the definition of double flip-flop family for $f^{k(\ell+1)}$ and $\varphi_{k(\ell+1)}$.
To check condition \eqref{i.dflipflop4}, i.e, the closures of the sets $E^+_0, E^+_1, E^-_0, E^-_1$ are pairwise disjoint, just observe that
the value of $\varphi$ on the $k\ell$ and $k(\ell+1)$ iterates of theses sets are uniformly separated.
It remains to get the sojourns property.
Fix small $\deltalta>0$ and consider the number $N=N_\deltalta$ in the definition of sojourn for $\mathfrak{F}$.
Take a set $D\in \mathfrak{D}$ and consider $f^{k(\ell+1)}(D)=\widehat D\in \mathfrak{F}$. The sojourns property for $\mathfrak{F}$ provides a
subset $\widehat D'$ such that $f^N (\widehat D')\in \mathfrak{F}$ and the first $N$ iterates of any point $x\in \widehat D'$
are $\deltalta$-dense in $Y$. Consider now $f^{-k(\ell+1)} (\widehat D')\subset D$. It is enough now to observe that any point in that set is such that its first
$k(\ell+1)+N$ iterates are $\deltalta$-dense in $Y$. We omit the choice of the sequences $\zeta_i$ and $\eta_i$ in the previous construction. We finish the proof by taking $r=k(\ell+1)$.
\end{proof}
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\subsection{Existence of double flip-flop families with sojourns}
\lambdabel{ss.double}
We now prove that existence of flip-flop families with sojourns implies the existence of double flip-flop families with sojourns.
\begin{propo}
\lambdabel{p.yieldsdouble}
Consider a flip-flop family
$\mathfrak{F}=\mathfrak{F}^+\sqcup\mathfrak{F}^-$
associated to $\varphi_k$ and $f^k$ for some $k\ge 1$ with $f$-sojourns in
a compact subset $Y$ of $X$. Then there are $r\ge 1$ and a double flip-flop family $\mathfrak{D}= \mathfrak{D}^+_0\sqcup \mathfrak{D}^+_1 \sqcup
\mathfrak{D}^-_0\sqcup \mathfrak{D}^-_1$ associated to $f^r$ and $\varphi_r$ with $f$-sojourns along $Y$.
\end{propo}
\begin{proof}
Given a plaque $D\in \mathfrak{F}$ and $\ell \ge 1$, consider subsets
$D_{+^\ell,+}$ $D_{+^\ell,-}$, $D_{-^\ell,+}$, and $D_{-^\ell,-}$ of $D$ satisfying
\begin{itemize}
\item
$f^{ki} (D_{+^\ell,+})$ is contained in some plaque of $\mathfrak{F}^+$
for every $i\in \{1,\dots,\ell\}$ and $f^{k(\ell+1)} (D_{+^\ell,+})\in \mathfrak{F}^+$,
\item
$f^{ki} (D_{+^\ell,-})$
is contained in some plaque of $\mathfrak{F}^+$
for every $i\in \{1,\dots,\ell\}$ and $f^{k(\ell+1)} (D_{+^\ell,-})\in \mathfrak{F}^-$,
\item
$f^{ki} (D_{-^\ell,-})$ is contained in some plaque of $\mathfrak{F}^-$ for every $i\in \{1,\dots,\ell\}$ and $f^{k(\ell+1)} (D_{+^\ell,-})\in \mathfrak{F}^-$,
\item
$f^{ki} (D_{-^\ell,+}))$ is contained in some plaque of $\mathfrak{F}^-$ for every $i\in \{1,\dots,\ell\}$ and $f^{k(\ell+1)} (D_{-^\ell,+})\in \mathfrak{F}^+$.
\end{itemize}
The existence of these subsets is assured by item \eqref{i.flipflop2} in the definition of a flip-flop family.
Using the continuity of $\varphi$, we have that for every $\ell$ large enough there is $\alpha'>$ such that
\[
\begin{split}
&\varphi_{k(\ell+1)} (x) > \alpha'>0 \quad \mbox{if $x\in D_{+^\ell,\pm}$},\\
&\varphi_{k(\ell+1)} (x) < -\alpha'<0 \quad \mbox{if $x\in D_{-^\ell,\pm}$}.
\end{split}
\]
We use here that the $\ell+1$ Birkhoff averages of $\varphi_k$ with respect to $f^k$ are the same as $\varphi_{k(\ell+1)}$, that is, $k(\ell+1)$ averages of $\varphi$ with respect to $f$.
We fix such a large $\ell$ and define
\[
\begin{split}
\mathfrak{D}^+_0&\eqdef \{D_{+^\ell,+}, \, D\in \mathfrak{F}\}, \quad
\mathfrak{D}^+_1\eqdef \{D_{+^\ell,-}, \, D\in \mathfrak{F}\},\\
\mathfrak{D}^-_0&\eqdef \{D_{-^\ell,+}, \, D\in \mathfrak{F}\}, \quad
\mathfrak{D}^-_1\eqdef \{D_{-^\ell,-}, \, D\in \mathfrak{F}\}.
\end{split}
\]
By construction $\mathfrak{D}=\mathfrak{D}^+_0\sqcup \mathfrak{D}^+_1\sqcup \mathfrak{D}^-_0\sqcup \mathfrak{D}^-_1$ satisfies conditions
\eqref{i.dflipflop1}, \eqref{i.dflipflop2}, and \eqref{i.dflipflop3} in the definition of double flip-flop family for $f^{k(\ell+1)}$ and $\varphi_{k(\ell+1)}$.
To check condition \eqref{i.dflipflop4}, i.e, the closures of the sets $E^+_0, E^+_1, E^-_0, E^-_1$ are pairwise disjoint, just observe that
the value of $\varphi$ on the $k\ell$ and $k(\ell+1)$ iterates of theses sets are uniformly separated.
It remains to get the sojourns property.
Fix small $\deltalta>0$ and consider the number $N=N_\deltalta$ in the definition of sojourn for $\mathfrak{F}$.
Take a set $D\in \mathfrak{D}$ and consider $f^{k(\ell+1)}(D)=\widehat D\in \mathfrak{F}$. The sojourns property for $\mathfrak{F}$ provides a
subset $\widehat D'$ such that $f^N (\widehat D')\in \mathfrak{F}$ and the first $N$ iterates of any point $x\in \widehat D'$
are $\deltalta$-dense in $Y$. Consider now $f^{-k(\ell+1)} (\widehat D')\subset D$. It is enough now to observe that any point in that set is such that its first
$k(\ell+1)+N$ iterates are $\deltalta$-dense in $Y$. We omit the choice of the sequences $\zeta_i$ and $\eta_i$ in the previous construction. We finish the proof by taking $r=k(\ell+1)$.
\end{proof}
\subsection{Support, average, and entropy}
\lambdabel{ss.support}
We now obtain ergodic measures with full support and positive entropy satisfying $\int \varphi d\mu=0$.
\begin{theo}\lambdabel{t.flipfloptailqual}
Let $(X,\rho)$ be a compact metric space,
$Y$ a compact subset of $X$, $f\colon X \to X$ a homeomorphism, and $\varphi\colon X \to \mathbb{R}$
a continuous function.
Assume that there is a flip-flop family $\mathfrak{F}$ associated to $\varphi_k$ and $f^k$ for some $k\ge 1$ having
$f$-sojourns along $Y$.
Then there is an ergodic measure $\mu$ with positive entropy whose support contains $Y$ and such that $\int \varphi \,d\mu =0$.
\end{theo}
First note that by Proposition~\ref{p.yieldsdouble} we can assume that there is a double flip-flop family
$\mathfrak{D}=\mathfrak{D}^+_0\sqcup \mathfrak{D}^+_1\sqcup \mathfrak{D}^-_0\sqcup \mathfrak{D}^-_1$
relative to
$\varphi_r$, $f^r$, and some $r\ge 1$ with $f$-sojourns along $Y$.
We define the sets
$$
K_0\eqdef
\mathrm{closure} \left(
\bigcup_{D\in \mathfrak{D}^+_0\cup \mathfrak{D}^-_0} D \right)
\quad
\mbox{and}
\quad
K_1\eqdef
\mathrm{closure} \left(
\bigcup_{D\in \mathfrak{D}^+_1\cup \mathfrak{D}^-_1} D \right).
$$
Recall that the sets $K_0$ and $K_1$ are disjoint.
The pair $\mathbf{K}\eqdef (K_0, K_1)$ is the {\emph{division associated to $\mathfrak{D}$.}}
We need to recall some definitions from \cite{BDB:}.
Consider sequences $\bar \deltalta =(\deltalta_n)_{n\in \mathbb{N}N}$,
$\bar \alpha =(\alpha_n)_{n\in \mathbb{N}N}$, and $\bar \varepsilon =(\varepsilon_n)_{n\in \mathbb{N}N}$ of positive numbers converging to $0$ as $n\to \infty$.
Consider a scale $\mathcal{T}=(T_n)_{n\in \mathbb{N}N}$ and a $\mathcal{T}$-long $\bar \varepsilon$-sparse tail $R_\infty$.
\begin{defn}[$\bar \alpha$-control and $\bar\deltalta$-denseness]
A point $x\in X$ is
{\emph{$\bar \alpha$-controlled for $\varphi$
with a tail $R_\infty$}} if for every $n\in\mathbb{N}$ and every $T_n$-regular interval $I$ that is not strictly contained in a component
of $R_\infty$ it holds
$$
\frac{1}{T_n} \sum_{j\in I} \varphi (f^j(x))\in [-\alpha_n, \alpha_n].
$$
The orbit of a point $x\in X$ is {\emph{$\bar\deltalta$-dense in $Y$ along the tail $R_\infty$ }} if for every component $I$ of $R_\infty$ of size $T_n$
the segment of orbit $\{f^j(x), j\in I\}$ is $\deltalta_n$-dense in $Y$.
\end{defn}
We are now ready to state the main technical step of the proof of Theorem~\ref{t.flipfloptailqual}.
This is a reformulation of \cite[Theorem 2]{BDB:} with an additional control of the itine\-ra\-ries. This control leads to
positive entropy.
For the notion of a $\bf K$-itinerary
of a point over a set see Section~\ref{s.measurespositive}.
\begin{propo}\lambdabel{p.maintech}
Let $(X, \rho)$ be a compact metric space, $Y$ a compact subset of $X$, $f\colon X \to X$ be a homeomorphism, and
$\varphi\colon X \to \mathbb{R}$ be a continuous map. Assume that
there is a double flip-flop family
$\mathfrak{D}$ associated to $\varphi_r$, $f^r$ for some $r\ge 1$ with $f$-sojourns along $Y$. Let $\mathbf{K}=(K_0,K_1)$ the division of $\mathfrak{D}$.
Consider sequences $\bar \alpha =(\alpha_n)_{n\in \mathbb{N}N}$ and $\bar \deltalta=(\deltalta_n)_{n\in \mathbb{N}N}$ of positive numbers converging to $0$ and
$\omega \in \Omegaega_2$.
Then there are a scale $\mathcal{T}$ and a rational and $\mathcal{T}$-sparsely long tail $R_\infty$ such that: for every plaque $D\in \mathfrak{D}$ there is a point $x\in D$ satisfying
\begin{enumerate}
\item\lambdabel{i.p.control}
the Birkhoff averages of $\varphi_r$ along the orbit of $x$ with respect to $f^r$ are $\bar\alpha$-controlled
with the tail $R_\infty$,
\item \lambdabel{i.p.density}
the $f$-orbit of $x$ is $\bar\deltalta$-dense in $Y$ along $R_\infty$,
\item \lambdabel{i.p.itinerary} $\omega$ is the $\mathbf{K}$-itinerary of $x$ with respect to $f^r$ over $J\eqdef \mathbb{N}N \setminus R_\infty$.
\end{enumerate}
\end{propo}
\subsubsection{Proposition~\ref{p.maintech} implies Theorem~\ref{t.flipfloptailqual}}
We now deduce Theorem~\ref{t.flipfloptailqual}. Let $x$ be the point given by Proposition~\ref{p.maintech}
associated to $\omega\in \Omegaega_2$, which is a generic point for the Bernoulli measure $\xi_{1/2}$.
By \cite[Proposition 2.17]{BDB:},
if $\widetilde \mu$ is a accumulation point of the sequence of empirical measures $(\mu_n(x,f^r))_{n\in \mathbb{N}N}$ (recall \eqref{e.empiric}), then
for $\widetilde \mu$-almost every point $y$ it holds $\frac{1}{n} \sum_{i=0}^{n-1} \varphi_r (f^{ri}(y))=0$.
This implies that for every $\mu$ generated by $x$ for $f$
it also holds that
\begin{equation}\lambdabel{e.zero-av}
\frac{1}{n} \sum_{i=0}^{n-1} \varphi (f^{i}(y))=0, \quad
\mbox{for $ \mu$-almost every point $y$.}
\end{equation}
Moreover,
by \cite[Proposition 2.2]{BDB:} for every measure $\mu$ generated by $x$
one has the $f$-orbit of $\mu$-almost every point $y$ is dense in $Y$.
Since $R_\infty$ is rational we have that its complement $J=\mathbb{N}\setminus R_\infty$ is also rational and has a density
$d(J)>0$, see Remark~\ref{r.complement}. By Theorem~\ref{thm:main}, every $f^r$-invariant measure $\widetilde \mu$ generated by $f^r$ along the orbit of $x$ satisfies $h(\widetilde\mu) > d(J)\log 2=\lambdambda>0$. Therefore, every $f$-invariant measure $\mu$ generated by $f$ along the orbit of $x$
has entropy at least $\lambdambda/r$. This implies that the ergodic decomposition of $\mu$ has some measure $\nu$ with full support, positive entropy,
and, by \eqref{e.zero-av}, $\int \varphi d\nu=0$. The proof of Theorem~\ref{t.flipfloptailqual} is now complete.
$\square$
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\subsection{Proof of Proposition~\ref{p.maintech}}
Fix sequences $\bar \alpha =(\alpha_n)_{n\in \mathbb{N}N}$ and $\bar \deltalta=(\deltalta_n)_{n\in \mathbb{N}N}$ of positive reals converging to $0$. Take any $\omega\in\Omegaega_2$.
Consider a double flip-flop $\mathfrak{D}=\mathfrak{D}^+_0 \sqcup \mathfrak{D}^+_1 \sqcup \mathfrak{D}^-_0 \sqcup \mathfrak{D}^-_1$ associated to $\varphi_r$ and $f^r$ for some $r\ge 1$ with sojourns along $Y$. Let $\mathbf{K}=(K_0,K_1)$ denote the associated division of $\mathfrak{D}$.
We let $\mathfrak{D}^+\eqdef \mathfrak{D}^+_0 \sqcup \mathfrak{D}^+_1$ and $\mathfrak{D}^-\eqdef\mathfrak{D}^-_0 \sqcup \mathfrak{D}^-_1$ and note that $\mathfrak{D}^+\sqcup\mathfrak{D}^-$ is a flip-flop family for $f^r$. We also consider
$\mathfrak{D}_0\eqdef \mathfrak{D}^+_0 \sqcup \mathfrak{D}^-_0$ and $\mathfrak{D}_1\eqdef \mathfrak{D}^+_1 \sqcup \mathfrak{D}^-_1$.
Following \cite{BDB:} we will use the induction on $n$ to construct a scale $\cT=(T_n)_{n\in\mathbb{N}}$ and a $\cT$-sparsely long tail $R_\infty$ (see Section \ref{s.scalesandtails}) such that there exists a point $x\in X$ satisfying conditions \eqref{i.p.control}, \eqref{i.p.density}, and \eqref{i.p.itinerary} of our proposition.
After $n$ steps of our induction we will have $T_0,\ldots, T_n$ and
$\Pi_{n-1}=R_\infty\cap [0,T_n-1]$.
Assume that all these objects are defined up to the index $n-1$. Note that no parameters beyond $n$ are required to check that some set $R\subset [0,T_n-1]$ satisfies the conditions from the definition of the $\cT$-sparsely long tail. Furthermore, knowing that $\Pi_{n-2}$ satisfies these conditions we can use translates of this set by a multiple of $T_{n-1}$ to get a set which we declare to be $\Pi_{n-1}=R_\infty\cap[0,T_{n}-1]$. The double flip-flop family is used as follows: the partition $\mathfrak{D}^+\sqcup \mathfrak{D}^-$ is used for controlling averages and the partition
$\mathfrak{D}_0 \sqcup \mathfrak{D}_1$ is used to follow a prescribed itinerary.
In the above situation, following the reasoning in \cite{BDB:} we obtain that there is an infinite set $\cS$ of multiples of $T_{n-1}$ such that
for every $S\in\cS$ and every $R\subset [0,S-1]$ following the rules of a tail (up to time $S$) and such that $R\cap[0,T_{n-1}-1]=\Pi_{n-1}$, given any $D\in \mathfrak{D}$
there is a family of plaques
$D_i\in \mathfrak{D}$,
$i\not\in R$,
such that
\begin{itemize}
\item
for every $i,j\in [0,S-1]\setminus R$ with $j>i$ it holds $ D_j \subset f^{r(j-i)} (D_i)$,
\item
for every
$x\in f^{-rS} (D_{S})$ the
orbit segment $\{x,f^r(x),\dots,f^{rS}(x)\}$ is controlled for $\varphi_r$ with parameters $(\alpha_1,\dots,\alpha_n)$ and the tail $R$, that is,
for every $i\le n$ and every $T_i$-regular interval $I$ contained in $[0,S-1]_\mathbb{N}$ that is not contained in $R$ the average of $\varphi_r$ over $I$ is in $[-\alpha_i, -\alpha_i/2] \cup [\alpha_i/2, \alpha_i]$.
\item for every $x\in f^{-rS} (D_{S})$ and every component $I=[a,b]_\mathbb{N}$ of $R$ of size $T_i$
the orbit segment $\{f^{i}(x):i\in [ar,br]_\mathbb{N}\}$ is $\deltalta_i$-dense in $Y$.
\end{itemize}
Actually, exploiting the fact that we deal with double flip-flop family, we can combine the reasoning of \cite[Section 2.5.2]{BDB:} with the one in \cite{BBD:16} to add one more claim: we choose $D_i\in \mathfrak{D}_{\omega_i}$.
Now, given $\cS$ we can choose $T_n$ which is large enough to obtain that $T_{n-1}/T_n$ is sufficiently small (since $\cS$is infinite we can do it).
Furthermore we can extend $\Pi_{n-2}$ to $\Pi_{n-1}$ exactly as in the proof of Proposition \ref{p.l.tailexistence} see formula \eqref{e.pin}.
This completes the induction step of our construction $R_\infty$.
Now observe that the set $\bigcap_{i\in R_\infty}f^{-ri}(D_i)\subset D_{0}$ is a nested intersection of nonempty compact sets with diameters converging to $0$, thus it contains only one point $x$, which by our construction satisfies conditions \eqref{i.p.control}, \eqref{i.p.density}, and \eqref{i.p.itinerary}.
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\section{Robustly transitive diffeomorphisms}
\lambdabel{s.robustly}
In this section, we prove Theorems~\ref{t.openanddense} and \ref{t.average}.
Recall that $\cR\cT(M)$ is the (open) subset of $\operatorname{Diff}^1(M)$
of diffeomorphisms that are
robustly transitive,
have a pair of hyperbolic periodic points of different indices,
and have
a partially hyperbolic splitting
$TM = E^{{\mathrm{u}}u} \oplus E^{{\mathrm{c}}} \oplus E^{{\mathrm{s}}s}$
with one-dimensional center $E^{\mathrm{c}}$,
where
$E^\mathrm{uu}$ is uniformly expanding and
$E^\mathrm{ss}$ is uniformly contracting.
Let $d^\mathrm{uu}$ be the dimension of
$E^\mathrm{uu}$.
Note that the map
$\mathrm{J}_f^{{\mathrm{c}}} \colon M \to \mathbf{R}$,
$\mathrm{J}_f^{{\mathrm{c}}}(x) \eqdef \log | Df_x |_{E^{\mathrm{c}} (x)}|$ is continuous for every $f\in\cR\cT(M)$.
Recall that $(\mathrm{J}_{f}^{{\mathrm{c}}})_n$ stands for the Birkhoff
$n$-average of $\mathrm{J}_f^{{\mathrm{c}}}$, cf. \eqref{e.averages}.
\begin{theo}\lambdabel{t.l.h(q)}
There is a $C^1$-open and dense subset $\cI(M)$ of $\cR\cT(M)$ such that
for every $f\in \cI(M)$ there are $N\in \mathbb{N}N$ and a neighbourhood $\mathcal \cU_f\subset \cI(M)$ such that
every $g\in \cU_f$ has
a flip-flop family with respect to the map
$(\mathrm{J}_g^{{\mathrm{c}}})_N$ and $g^N$
with sojourns along $M$.
\end{theo}
A crucial point here is that we get a single $N$ such that for every $g$ near $f$ we get a flip-flop family associated with
$g^N$ (we will pay special attention to this fact). A priori, the number $N$ for the flip-flop families constructed in \cite{BBD:16,BDB:} could depend on $f$. Since the flip-flop family for $f^N$ leads (through the criterion in Theorem \ref{thm:mainbis}) to an invariant measure with entropy bounded below by a constant times $\log 2/N$, we need the number $N$ to be locally invariable to get uniform local lower bounds for the entropy of measures we find. This is precisely what we obtain from Theorem~\ref{t.l.h(q)}.
\begin{proof}[Sketch of the proof Theorem~\ref{t.l.h(q)}]
Our hypotheses imply that every
$f\in \cI(M)$ has a pair of saddles $p_f$ and $q_f$ of indices, respectively, $d^\mathrm{uu}$ and $d^\mathrm{uu}+1$. The saddles depend continuously on $f$ and the indices are locally constant. Furthermore, the homoclinic classes of the saddles satisfy $H(p_f,f)=H(q_f,f)=M$ (see \cite[Proposition 7.1]{BDB:}
which just summarises results from \cite{BDPR}).
The discussion below involves the notions of a {\emph{dynamical blender}} and a {\emph{flip-flop configuration.}}
As we do not need their precise definitions and will only use some specific properties of them,
we will just give rough definitions of these concepts an refer to
\cite{BDB:} and \cite{BBD:16} for details. In what follows, the discussion is restricted to our partially hyperbolic setting and to small open subset
of $\cI(M)$ where the index $d^\mathrm{uu}$ is constant.
Recall that a family of discs $\mathfrak{D}$ is \emph{strictly $f$-invariant} if there is
an $\varepsilon$-neigh\-bour\-hood of $\mathfrak{D}$
such that for every disc
$D_0$ in a such a neighbourhood
there is a disc $D_1\in \mathfrak{D}$ with
$D_1\subset f(D_0)$, see \cite[Definition 3.7]{BBD:16}.
A {\emph{dynamical blender}} (in what follows we simply say a {\emph{blender}}) of a diffeomorphism $f$ is a locally maximal (in
an open set $U$) and transitive hyperbolic set
$\Gammamma$ of index $d^\mathrm{uu}+1$
endowed with an strictly $f$-invariant family of discs $\mathfrak{D}_f$ of dimension $d^\mathrm{uu}$
tangent to an invariant expanding cone field ${\mathcal{C}}^{\mathrm{uu}}$ around $E^\mathrm{uu}$. Hence, a blender is $4$-tuple $(\Gammamma_f, U,{\mathcal{C}}^{\mathrm{uu}}, \mathfrak{D}_f)$. In what follows, let us simply denote the blender as $(\Gammamma_f,\mathfrak{D}_f)$.
As the usual hyperbolic sets,
blenders are $C^1$-robust and have continuations.
By \cite[Lemma 3.8]{BBD:16} strictly invariant families are robust: for every $g$ sufficiently close to $f$ the family $\mathfrak{D}_f$ is also strictly invariant for $g$.
As a consequence, if $(\Gammamma_f,\mathfrak{D}_f)$ is a blender of $f$ then $(\Gammamma_g,\mathfrak{D}_f)$ is a blender of $g$ for every $g$ close to $f$,
where $\Gammamma_g$ is the hyperbolic continuation of $\Gammamma_f$. In what follows we will omit the subscripts for simplicity.
We can speak of the index of a blender $(\Gammamma,\mathfrak{D})$ (the dimension of the unstable bundle of $\Gamma$).
Given a saddle of the same index as the blender we say that the blender and the saddle are homoclinically related if their invariant manifolds intersect cyclically and transversely (this is a natural extension of the homoclinic relation of a pair of saddles).
In what follows, we consider blenders which are expanding in
the center direction, that is, with index $d^\mathrm{uu}+1$.
Consider now a saddle $p$ of index $d^\mathrm{uu}$. The saddle $p$ and the blender $(\Gammamma,\mathfrak{D})$ are in
a {\emph{flip-flop configuration}} if $W^\mathrm{u} (p,f)$ contains some disc of the family $\mathfrak{D}$ of the blender
and the unstable manifold of the blender transversely intersects the stable manifold of the saddle
(note that the sums of these manifolds exceeds by one the dimension of the ambient space). By transversality and the openess of the invariant family the flip-flop configurations are also $C^1$-robust.
The results of \cite[Section 6.5.1]{BDB:} are summarised in the following proposition.
\begin{propo}
\lambdabel{p.dandovoltas}
There is an open and dense subset $\cF(M)$ of
$\cR\cT(M)$
such that every diffeomorphisms $f\in \cF(M)$ has a pair of saddles
$p_f$ and $q_f$ of different indices and
a blender $(\Gamma_f,\mathfrak{D}_f)$ such that:
\begin{itemize}
\item
$H(p_f,f)=H(q_f,f)=M$,
\item
$\Gammamma_f$ is
homoclinically related to $p_f$,
\item
$\Gammamma_f$ and $q_f$
are in a flip-flop configuration,
\item there is a metric on $M$ such that
$\mathrm{J}_f^{{\mathrm{c}}}$ is positive in a neighbourhood of $\Gamma_f$ and negative in a neighbourhood
of the orbit of $q_f$.
\end{itemize}
\end{propo}
From now on, we will always consider $M$ with a metric given by Proposition \ref{p.dandovoltas}.
Let us recall another result from \cite{BDB:}.
\begin{theo}[Theorem 6.8 in \cite{BDB:}] \lambdabel{t.p.flipfloptail}
Consider $f\in \operatorname{Diff}^1(M)$ with a
dynamical blender $(\Gamma,\mathfrak{D})$ in a flip-flop configuration with a hyperbolic periodic point $q$.
Let
$\varphi\colon M\to\mathbf{R}$ be a continuous function such that $\varphi|_\Gamma>0$ and $\varphi|_{\cO(q)}<0$.
Then there are $N\geq1$ and a flip-flop family $\mathfrak{F}$
with respect to $\varphi_N$ and
$f^N$ which $f$-sojourns along the homoclinic class
$H(q,f)$.
\end{theo}
We can now apply Theorem~\ref{t.p.flipfloptail} to the flip-flop configuration associated to the blender $(\Gammamma_f,\mathfrak{D}_f)$
and the saddle $q_f$ provided by Proposition~\ref{p.dandovoltas} and the map $\mathrm{J}_f^{{\mathrm{c}}}$.
This provides the flip-flop family associated to the map $\mathrm{J}_f^{{\mathrm{c}}}$. The fact that the sojourns take
place in the whole manifold follows from $H(q_f,f)=M$.
To complete the sketch of the proof of
Theorem~\ref{t.l.h(q)} it remains to get the uniformity of $N$.
To get such a control we need to recall some steps of the construction in \cite{BBD:16}.
Let us explain how to derive the flip-flop family $\mathfrak{F}=\mathfrak{F}^+\sqcup\mathfrak{F}^-$ associated to $f^{N_f}$ and the number $N_f$ from the flip-flop configuration of the saddle $q_f$ and the blender $(\Gammamma_f, \mathfrak{D}_f)$.
The sub-family $\mathfrak{F}^+$ is formed by the discs of $\mathfrak{D}_f$. To define
$\mathfrak{F}^-$ let us assume, for simplicity, that $f(q_f)=q_f$.
We consider an auxiliary
family $\mathfrak{D}_q$ of $C^1$-embedded discs
containing $W^\mathrm{u}_{\deltalta}(q_f,f)$ (for sufficiently small $\deltalta$) in its interior and consisting of small discs
$D$ such that
\begin{enumerate}
\item[a)]
every $D$ intersects transversely $W^\mathrm{s}_\deltalta(q_f,f)$ and is tangent to a small cone field around $E^{\mathrm{uu}}$,
\item[b)]
there is $\lambdambda>1$ such that
$\|Df(v)\| \ge \lambdambda \|v\|$
for every vector $v$ tangent to $D$,
\item[c)]
$f(D)$ contains a disc in $\mathfrak{D}_q$.
\end{enumerate}
For the existence of the family $\mathfrak{D}_q$ and its
precise definition
see \cite[Lemma 4.11]{BBD:16}.
It turns out that for every $g$ nearby $f$ the family $\mathfrak{D}_q$ also satisfies these properties for $g$.
We let $\mathfrak{F}^-=\mathfrak{D}_q$.
Observe now that due to the flip-flop configuration $W^u(q_f,f)$ contains a disc $D'\in \mathfrak{D}_f=\mathfrak{F}^+$.
Hence there is large $k_0$ such that for every disc $D\in \mathfrak{D}_f=\mathfrak{F}^+$ and every $N\ge k_0$ the disc $f^N(D)$ contains a sub-disc close enough
to $D'$ and hence contains a disc in $\mathfrak{D}_f=\mathfrak{F}^+$ (note that this family is necessarily open). Note also that $f^N(D)$ contains a disc of $\mathfrak{D}_q$ by
(c). Observe that the choice of $k_0$ holds for every $g$ nearby $f$.
A similar construction holds for the images of the discs in $\mathfrak{D}=\mathfrak{F}^-$, now we use that $W^s(q_f,f)$ transversely intersects every disc
in $\mathfrak{D}_f=\mathfrak{F}^+$. In this way, we get a uniform $N$ in such a way the family
satisfies condition \eqref{i.flipflop2} in the definition of a flip-flop family (Definition~\ref{i.flipflop33}). In our partially hyperbolic case, condition
\eqref{i.flipflop33} follows because all the discs we consider are tangent to a strong unstable cone field.
It remains to get condition \eqref{i.flipflop1} on the averages of $(\mathrm{J}_f^{{\mathrm{c}}})_N$. For this, some additional shrinking of the discs of the blender is needed. We will follow \cite[Section 4.4]{BBD:16}.
Note that the map $\mathrm{J}_f^{{\mathrm{c}}}$ is positive for the points in the set $\Gammamma_f$ (here we recall that $E^\mathrm{c}$ is expanding
in a neighbourhood $V$ of $\Gammamma_f$ since we consider the metric given by Proposition \ref{p.dandovoltas}).
Consider for each disc $D$ of the family $\mathfrak{D}_f$ a sub-disc $D'$ contained in $V$ such that the family $\mathfrak{D}'_f$ formed by the sets $D'$ is invariant for $f^m$ for some $m$. Again, the same $m$ works for every $g$ sufficiently close to $f$. The precise definition of this new family $\mathfrak{D}'_f$ is in \cite[Definition 4.15]{BBD:16} and the invariance properties are in \cite[Lemmas 4.17 and 4.18]{BBD:16}.
Finally, observe that once we have obtained the flip-flop family $\mathfrak{F}=\mathfrak{F}^+\sqcup\mathfrak{F}^-$ the proof that this family has sojourns is exactly as in \cite[Proposition 5.2]{BDB:}.
This completes our sketch of the proof of Theorem~\ref{t.l.h(q)}.
\end{proof}
\subsection{Proof of Theorem~\ref{t.openanddense}}
The theorem follows immediately from Theorem~\ref{t.flipfloptailqual} and Theorem~\ref{t.l.h(q)}.
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\subsection{Proof of Theorem~\ref{t.openanddense}}
The theorem follows immediately from Theorem~\ref{t.flipfloptailqual} and Theorem~\ref{t.l.h(q)}.
\subsection{Proof of Theorem \ref{t.average}}
Recall that for a periodic point $p_f$ of $f$ we denote by $\mu_{\cO(p_f)}$ the
unique $f$-invariant probability measure supported on the orbit of $p_f$.
Consider now periodic points $p_f$ and $q_f$ of $f$
satisfying $\int \varphi\, d\mu_{\cO(p_f)} <0<
\int \varphi \, d\mu_{\cO(q_f)}$.
To prove Theorem~\ref{t.average} it is enough to consider the case where the saddles $p_f$ and $q_f$ have the same index and
are homoclinically related (which is an open and dense condition in $\mathcal{RT}(M)$).
In \cite[Section 5.3]{BDB:} it is explained how the case where the saddles have different indices is reduced to this ``homoclinically related'' case: after an arbitrarily small perturbation of $f$ one gets $g$ with a
saddle-node $r_g$ with $\int \varphi d\mu_{\cO(r_g)}\ne 0$. Assume that $\int \varphi d\mu_{\cO(r_g)}>0$. In this case
we perturb $g$ to get $h$ such that $r_h$ has the same index as $p_h$ and these points are homoclinically related. Then we are in the
``homoclinically related case''.
We now prove Theorem~\ref{t.average} when the saddles $p_f$ and $q_f$ are homoclinically related. Let us assume for simplicity, that these saddles are fixed points of $f$.
The result follows from the construction in \cite{BDB:},
we will sketch below its main steps.
Recall
that the set $\cD^i(M)$ of
$i$-dimensional (closed) discs $C^1$-embedded in $M$ has a natural topology which is
induced by a metric $\mathfrak{d}$, for details see \cite[Proposition 3.1]{BBD:16}. For small $\varrho>0$ consider the
$\varrho$-neighbourhoods
$\cV^{\mathfrak{d}}_\varrho(p_f)\eqdef \cV^{\mathfrak{d}}_\varrho(W^{\mathrm{u}}_{loc}(p_f,f))$ and $\cV^{\mathfrak{d}}_\varrho(q_f)\eqdef
\cV^{\mathfrak{d}}_\varrho(W^{\mathrm{u}}_{loc}(q_f,f))$
of the local unstable manifolds of $p_f$ and $q_f$ for the distance $\mathfrak{d}$ in $\cD^i(M)$, where $i$ is the dimension of the unstable bundle
of $p_f$ and $q_f$.
We consider the following family $\mathfrak{F}_f=\mathfrak{F}_f^+\sqcup \mathfrak{F}_f^-$ of discs:
\begin{itemize}
\item $\mathfrak{F}_f^-$ is the family of discs in $\cV^\mathfrak{d}_\varrho(p_f)$ contained in $W^{\mathrm{u}}(p_f,f)\cup W^{\mathrm{u}}(q_f,f)$;
\item $\mathfrak{F}_f^+$ is the family of discs in $\cV^\mathfrak{d}_\varrho(q_f)$ contained in $W^{\mathrm{u}}(p_f,f)\cup W^{\mathrm{u}}(q_f,f)$.
\end{itemize}
Note that as $q_f$ and $p_f$ are homoclinically related
these two families are both infinite. Note also that for $\varrho>0$ small enough one has that
$\varphi$ is negative in the discs of $\mathfrak{F}_f^-$ and positive in the discs of $\mathfrak{F}_f^+$.
Note that we can define the families $\mathfrak{F}_g^\pm$ analogously for every $g$ close to $f$, having also that
$\varphi$ is negative in $\mathfrak{F}_f^-$ and positive in $\mathfrak{F}_f^+$.
We have the following result which is an improvement of \cite[Proposition 5.2]{BDB:}. The original result is stated for a single diffeomorphism $f$. Here we have a version valid for a neighbourhood with a uniform control of $n$ in the whole neighbourhood. As in the case in the previous section, this allows us to locally bound the entropy of the measures associated to the flip-flop from below.
\begin{propo}
Consider $f$ and $\varphi$ as above. Then there is $n$ such that
the family $\mathfrak{F}_g$ is a flip-flop family associated to $\varphi$ and $g^n$ and
has $g$-sojourns along the homoclinic class $H(p_g,g)$.
\end{propo}
\begin{proof}
Let us recall the proof of the proposition for $f$ (\cite[Proposition 5.2]{BDB:}).
Since the saddles $p_f$ and $q_f$ are homoclinically related,
there is $n$ such that for every disc $D\in \mathfrak{F}_f^\pm$ the disc $f^n(D)$ contains
discs $D_p\in \cV^\mathfrak{d}_\varrho(p_f)$ and $D_q\in \cV^\mathfrak{d}_\varrho(q_f)$. By construction
$D\in W^{\mathrm{u}}(p_f,f)\cup W^{\mathrm{u}}(q_f,f)$. Observe that for $g$ close enough to $f$ and
every disc $D\in \mathfrak{F}_g^\pm$ the disc $g^n(D)$ also contains
discs $D_p\in \cV^\mathfrak{d}_\varrho(p_g)$ and $D_q\in \cV^\mathfrak{d}_\varrho(q_g)$.
The fact that $\mathfrak{F}_f$ is a flip-flop family is quite
straightforward. The same proof applies to $\mathfrak{F}_g$. For details see \cite[Section 5.2]{BDB:}, where it is also proved that the family has sojourns
in the whole class.
\end{proof}
\end{document}
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\begin{document}
\begin{abstract}
In this paper we consider the class of K3 surfaces defined as hypersurfaces in weighted projective space, that admit a non-symplectic automorphism of non-prime order, excluding the orders 4, 8, and 12. We show that on these surfaces the Berglund-H\"ubsch-Krawitz mirror construction and mirror symmetry for lattice polarized K3 surfaces constructed by Dolgachev agree; that is, both versions of mirror symmetry define the same mirror K3 surface. \end{abstract}
\keywords{K3 surfaces, mirror symmetry, mirror lattices, Berglund-H\"ubsch-Krawitz construction}
\subjclass[2010]{Primary 14J28, 14J33; Secondary 14J17, 11E12, 14J32} \title{BHK mirror symmetry for K3 surfaces with non-symplectic automorphism}
\section*{Introduction}
Since its discovery by physicists nearly 30 years ago, mirror symmetry has been the focus of much interest for both physicists and mathematicians.
Although mirror symmetry has been ``proven'' physically, we have much to learn about the phenomenon mathematically. When we speak of mirror symmetry mathematically, there are many different constructions or rules for determining when a Calabi--Yau manifold is ``mirror'' to another. The constructions are often formulated in terms of families of Calabi--Yau manifolds.
A natural question is whether, in a situation where more than one version can apply, they produce the same mirror (or mirror family). In this article, we consider two versions of mirror symmetry for K3 surfaces, and show that in this case the answer is affirmative, as we might expect.
The first version of mirror symmetry of interest to us is known as BHK mirror symmetry. This was formulated by Berglund--H\"ubsch \cite{berghub}, Berglund--Henningson \cite{berghenn} and Krawitz \cite{krawitz} for Landau--Ginzburg models.
Using the ideas of the Landau--Ginzburg/Calabi--Yau correspondence, BHK mirror symmetry also produces a version of mirror symmetry for certain Calabi--Yau manifolds (see Section~\ref{sec-mirror}).
In the BHK construction, one starts with a quasihomogeneous and invertible polynomial $W$
and a group $G$ of symmetries of $W$ satisfying certain conditions (see Section~\ref{quasi_sec} for more details).
From this data, we obtain the Calabi--Yau (orbifold) defined as the hypersurface $Y_{W,G}=\set{W=0}/G$.
Given an LG pair $(W,G)$, BHK mirror symmetry allows to obtain another LG pair $(W^T,G^T)$
satisfying the same conditions, and therefore another Calabi--Yau (orbifold) $Y_{W^T,G^T}$.
We say that $Y_{W,G}$ and $Y_{W^T,G^T}$ form a BHK mirror pair. In our case, we resolve singularities to obtain K3 surfaces $X_{W,G}$ and $X_{W^T,G^T}$, which we call a BHK mirror pair. When no confusion arise, we will denote these mirror K3 surfaces simply by $X$ and $X^T$, respectively.
Another form of mirror symmetry for K3 surfaces, which we will call LPK3 mirror symmetry, is described by Dolgachev in \cite{dolgachev}.
LPK3 mirror symmetry says that the mirror family of a given K3 surface admitting a polarization by a lattice $M$ is the family of K3 surfaces polarized by the \emph{mirror lattice} $M^\vee$.
We say that the two K3 surfaces are LPK3 mirror when they are lattice polarized and they belong to LPK3 mirror families (see details in Section \ref{sec-LPK3}).
Returning to the question posed earlier, one can ask
whether the BHK mirror symmetry and LPK3 mirror symmetry produce the same mirror.
A similar question was considered by Belcastro in \cite{belcastro}.
She considers a family of K3 surfaces that arise as (the resolution of) hypersurfaces in weighted projective space,
uses the Picard lattice of a general member of the family as polarization, and finds that this particular polarization does not yield very many mirror families.
This polarization fails to yield mirror symmetry for at least two reasons. First, it does not consider the group of symmetries. And secondly---and perhaps more compelling---a result proved by Lyons--Olcken (see \cite{LO}) following Kelly (see \cite{kelly}) shows that the rank of the Picard lattice of $X_{W,G}$ does not depend on $G$ at all.
This fact suggests that we need a finer invariant than the full Picard lattice to exhibit LPK3 mirror symmetry.
We need to find a polarizing lattice that recognizes the role of the group $G$.
The correct polarizing lattice seems to be the invariant lattice
\[
S_X(\sigma)=\{x\in H^2(X,\mathbb Z):\sigma^*x=x \}
\]
of a certain non-symplectic automorphism $\sigma\in\Aut{X}$. This was proven in \cite{ABS} and \cite{CLPS} in the case of K3 surfaces admitting a non--symplectic automorphism prime order.
In what follows, we generalize the results of \cite{ABS} and \cite{CLPS} to K3 surfaces admitting a non--symplectic automorphism $\sigma$ of any finite order, excepting orders 4, 8 and 12.
By polarizing each of the K3 surfaces in question by the invariant lattice $S_X(\sigma)$ of a non-symplectic automorphism $\sigma$ of finite order, we prove that BHK mirror symmetry and LPK3 mirror symmetry agree. This is done as in the previous works, by showing that $S_{X^T}(\sigma^T)$ is the mirror lattice of $S_{X}(\sigma)$.
This situation differs significantly from the case of prime order automorphism in that the invariant lattice is no longer $p$-elementary and there is no longer a (known) relationship between the invariants of $S_X(\sigma)$ and the fixed locus of $\sigma$. Hence, instead of studying the fixed locus in order to recover $S_X(\sigma)$, we determine $S_X(\sigma)$ with other methods.
As for orders 4, 8 and 12, more details are required and the methods are slightly different, so that this will be the object of further work.
The question of whether two versions of mirror symmetry produce the same mirror has been investigated by others as well, but for different constructions of mirror symmetry than we consider here. Partial answers to the question are given by Artebani--Comparin--Guilbot in \cite{good_pairs}, where Batyrev and BHK mirror constructions are both seen as specializations of a more general construction based on the definition of good pairs of polytopes. Rohsiepe also considered Batyrev mirror symmetry in connection with LPK3 mirror symmetry in \cite{Roh}, where he shows a duality for the K3's obtained as hypersurfaces in one of the Fano toric varieties constructed by one of the 4319 3-dimensional reflexive polytopes. As in Belcastro's paper \cite{belcastro}, Rohsiepe used the Picard lattice of a general member of the family of such hypersurfaces to polarize the K3 surfaces. As it turns out, only 14 of the 95 weight systems yield a K3 surface in a Fano ambient space. We do not consider such a restriction in the current paper.
Clarke has also described a framework which he calls an auxilliary Landau--Ginzburg model, which encapsulates several versions of mirror symmetry, including Batyrev--Borisov, BHK, Givental's mirror theorem and Hori--Vafa mirror symmetry (see \cite{clarke}). Kelly also has some results in this direction in \cite{kelly}, where he shows by means of Shioda maps, that certain BHK mirrors are birational. The current article is similar in scope to these articles.
There are also several papers treating non--symplectic automorphisms of K3 surfaces, which are closely related to this paper. These include \cite{order_four} for automorphisms of order four, \cite{order_six} for order six, \cite{Schutt2010} for order $2^p$, \cite{order_eight} for order eight, and \cite{order_sixteen} for order sixteen. In general, it seems difficult to find the invariant lattice of a non--symplectic automorphism on a K3 surface. The current article gives some new methods for computing the invariant lattice, which we hope will yield more general results.
As complementary results, in doing this classification we discovered the existence of one of the cases that couldn't be discovered in the order 16 classification in \cite{order_sixteen} namely a K3 surface admitting a purely non--symplectic automorphism of order sixteen, which has as fixed locus a curve of genus zero, and 10 isolated fixed points. This is number 58 in Table~\ref{tab-16}. Dillies has also found such an example in \cite{dillies16}.
Additionally, our computations unearthed a different result from Dillies in \cite{order_six}. If we look at Table~\ref{tab-6}, we find the invariant lattice for number 29 and one of rows of 5d has an invariant lattice of order 12. These K3 surfaces admit an automorphism of order three, namely $\sigma_6^2$, with invariants $(g,n,k)=(0,8,5)$, but the automorphism $\sigma_6$ fixes one rational curve and 8 isolated points. This is missing from Table~1 in \cite{order_six}. Furthermore, the same can be said for the K3 surfaces in same table which have $v\oplus 4\omega_{2,1}^{1}$ as the invariant lattice, namely one of 8b, 8d, 33a, and 33b. These K3 surfaces admit a non--symplectic automorphism of order three with invariants $(g,n,k)=(0,7,4)$, but $\sigma_6$ fixes one rational curve and seven isolated points. This is also missing from the Table in \cite{order_six}.
The paper is organized as follows.
In Section \ref{sec-background} we recall some definitions and results on K3 surfaces and lattices, while Section \ref{sec-mirror} is dedicated to the introduction of mirror symmetry, both LPK3 and BHK.
The main result of the paper is Theorem \ref{t:main_thm}. Section \ref{sec-method} is dedicated to the explanation of the methods used in the proof.
In Section \ref{sec-ex} we report some meaningful examples, and Section \ref{sec:tables} contains the tables proving the main theorem.
We would like to thank Michela Artebani, Alice Garbagnati, Alessandra Sarti and Matthias Sch\"utt
for many useful discussions and helpful insights. We would also thank Antonio Laface for the help on magma code \cite{magma}.
The first author has been partially supported by Proyecto Fondecyt Postdoctorado N. 3150015 and Proyecto Anillo ACT 1415 PIA Conicyt.
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\section{Background}
\label{sec-background}
In this section we recall some facts about K3 surfaces and lattices.
For notations and theorems, we follow \cite{surfaces, nikulin}
\subsection{K3 Surfaces}
A \emph{K3 surface} is a compact complex surface $X$ with trivial canonical bundle and $\dim H^1(X,\mathcal O_X)=0$. All K3 surfaces considered here will be projective and minimal.
It is well-known that all K3 surfaces are diffeomorphic and K\"ahler.
Given a K3 surface $X$, $H^2(X,\mathbb{Z})$ is free of rank 22,
the Hodge numbers of $X$ are $h^{2,0}(X)=h^{0,2}(X)=1$, $h^{1,1}(X)=20$ and $h^{1,0}(X)=h^{0,1}(X)=0$,
and the Euler characteristic is $24$.
The Picard group of $X$ coincides with the N\'eron--Severi group,
and both are torsion free.
From the facts above, we see that $H^{2,0}(X)$ is one--dimensional.
In fact, it is generated by a nowhere--vanishing two--form $\omega_X$,
which satisfies $\langle \omega_X,\omega_X \rangle=0$ and $\langle \omega_X,\overline{\omega}_X \rangle>0$.
Given an automorphism $\sigma$ of the K3 surfaces $X$,
we get an induced Hodge isometry $\sigma^*$, which preserves $H^{2,0}(X)$,
i.e. $\sigma^*\omega_X=\lambda_\sigma \omega_X$ for some $\lambda_\sigma\in \mathbb{C}^*$.
We call $\sigma$ \emph{symplectic} if $\lambda_\sigma=1$
and \emph{non-symplectic} otherwise.
If $\sigma$ is an automorphism with nonprime order $m$,
we say $\sigma$ is \emph{purely non-symplectic} if $\lambda_\sigma=\xi_m$
with $\xi_m$ a primitive $m$-th root of unity.
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\subsection{Lattice theory}\label{s:lattice}
A \emph{lattice} is a free abelian group $L$ of finite rank
together with a non-degenerate symmetric bilinear form $B\colon L \times L \to \mathbb{Z}$.
A lattice $L$ is \emph{even} if $B(x,x)\in 2\mathbb{Z}$ for each $x\in L$.
The \emph{signature} of $L$ is the signature $(t_+,t_-)$ of $B$.
A lattice $L$ is \emph{hyperbolic} if its signature is $(1,\rk(L)-1)$.
A sublattice $L\subset L'$ is called \emph{primitive} if $L'/L$ is free.
On the other hand, a lattice $L'$ is an \emph{overlattice} of finite index of $L$
if $L\subset L'$ and $L'/L$ is a finite abelian group.
We will refer to it simply as an \emph{overlattice}.
Given a finite abelian group $A$,
a \emph{finite quadratic form} is a map $q:A\to \mathbb{Q}/2\mathbb{Z}$ such that
for all $n\in \mathbb{Z}$ and $a,a'\in A$
$q(na)=n^2q(a)$
and
$q(a+a')-q(a)-q(a')\equiv 2b(a,a') \imod{2\mathbb{Z}}$
where $b:A\times A\to \mathbb{Q}/\mathbb{Z}$ is a finite symmetric bilinear form.
We define orthogonality on subgroups of $A$ via $b$.
Given a lattice $L$, the corresponding bilinear form $B$ induces an embedding
$L \hookrightarrow L^\ast$, where $L^\ast:= \Hom(L,\mathbb{Z})$.
The \emph{discriminant group} $A_{L}:= L^\ast/L$ is a finite abelian group.
In fact, if we write $B$ as a symmetric matrix in terms of a minimal set of generators of $L$,
then the order of $A_{L}$ is equal to $|\det(B)|$.
The bilinear form $B$ can be extended to $L^*\times L^*$ taking values in $\mathbb{Q}$.
If $L$ is even, this induces a finite quadratic form $q_L: A_L\to \mathbb{Q}/2\mathbb{Z}$.
The minimal number of generators of $A_L$ is called the \emph{length} of $L$.
If $A_L$ is trivial, $L$ is called \emph{unimodular}.
For a prime number $p$, $L$ is called \emph{$p$-elementary}
if $A_L \simeq (\mathbb{Z}/p\mathbb{Z})^a$ for some $a\in\mathbb N_0$;
in this case, $a$ is the length of $A_L$.
Two lattices $L$ and $K$ are said to be \emph{orthogonal},
if there exists an even unimodular lattice $S$ such that $L\subset S$ and $L^^{\prime}erp_S\cong K$.
Orthogonality will be a key ingredient in the definition of mirror symmetry for K3 surfaces. The following fact will also be useful.
\begin{prop}[cf. {\cite[Corollary 1.6.2]{nikulin}}]\label{p:orth}
Two lattices $L$ and $K$ are orthogonal if and only if $q_L\cong -q_K$.
\end{prop}
We recall the definition of several lattices that we will encounter later.
The lattice $U$ is the hyperbolic lattice of rank 2 whose bilinear form is given by the matrix $\left( \begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array} \right)$.
The lattices $A_n, D_m, E_6,E_7, E_8, n\geq 1, m\geq 4$ are the even negative definite lattices associated to the respective Dynkin diagrams.
For $n\geq 1$, the lattice $A_n$ has rank $n$ and its discriminant group is $\mathbb{Z}/({n+1})\mathbb{Z}$. If $p$ is prime, $A_{p-1}$ is $p$-elementary (with $a=1$).
For $m\geq 4$, the lattice $D_m$ has rank $m$ and its discriminant group is $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$ for $m$ even, and $\mathbb{Z}/4\mathbb{Z}$ for $m$ odd.
Finally, $E_6,E_7,E_8$ have ranks 6, 7, and 8 and discriminant groups of order 3, 2, and 1, respectively.
For $p\equiv 1 ^{\prime}mod 4$ the lattice $H_p$ is the hyperbolic even lattice of rank 2, whose bilinear form is given by the matrix
\[
H_p=\left( \begin{array}{cc}
\frac{(p+1)}2& 1\\
1&2\end{array} \right).
\]
The discriminant group of $H_p$ is $\mathbb{Z}/p\mathbb{Z}$.
There are two non--isomorphic hyperbolic lattices of rank 2 with discriminant group $\mathbb{Z}/9\mathbb{Z}$ defined by the matrices
\[ L_9=\left( \begin{array}{cc}
-2& 1\\
1&4\end{array} \right),\qquad
M_9=\left( \begin{array}{cc}
-4&5\\
5&-4\end{array} \right).\]
Following \cite{belcastro} we recall that $T_{p,q,r}$ with $p,q,r\in \mathbb{Z}$
is the lattice determined by a graph which has the form of a T,
and $p,q,r$ are the respective lengths of the three legs.
The rank of $T_{p,q,r}$ is $p+q+r-2$ and the discriminant group has order $pqr-pq-qr-pr$.
Given a lattice $L$, we denote by $L(n)$, the lattice with the same rank as $L$, but whose values under the bilinear form $B$ are multiplied by $n$.
Many even lattices are uniquely determined by their rank and the discriminant quadratic form.
To make this statement precise, we introduce the following finite quadratic forms.
The notation follows \cite{belcastro} and the results are proven in \cite{nikulin}.
We define three classes of finite quadratic forms forms, $w_{p,k}^\varepsilonilon$, $u_k$, $v_k$ as follows:
\begin{enumerate}
\item For $p\neq 2$ prime, $k\geq 1$ an integer, and $\varepsilonilon\in \set{^{\prime}m 1}$, let $a$ be the smallest even integer that has $\varepsilonilon$ as quadratic residue modulo $p$. Then we define $w_{p,k}^\varepsilonilon:\mathbb{Z}/{p^k}\mathbb{Z}\to \mathbb{Q}/2\mathbb{Z}$ via $w_{p,k}^\varepsilonilon(1)= ap^{-k}$.
\item For $p=2$, $k\geq 1$ and $\varepsilonilon\in \set{^{\prime}m 1, ^{\prime}m 5}$, we define $w_{2,k}^\varepsilonilon: \mathbb{Z}_{2^k}\to \mathbb{Q}/2\mathbb{Z}$ on the generator via $w_{2,k}^\varepsilonilon (1)=\varepsilonilon\cdot 2^{-k}$.
\item For $k\geq 1$ an integer, we define the forms $u_k$ and $v_k$ on $\mathbb{Z}/{2^k}\mathbb{Z}\times \mathbb{Z}/{2^k}\mathbb{Z}$ via the matrices:
\[
u_k=\left(\begin{matrix}
0 & 2^{-k} \\
2^{-k} & 0
\end{matrix}\right) \quad
v_k=2^{-k}\left(\begin{matrix}
2 & 1 \\
1 & 2
\end{matrix}\right)
\]
\end{enumerate}
For example, if we consider the lattice $L=A_2$, then $A_L\cong \mathbb{Z}/3\mathbb{Z}$ and $q_L$ has value $\tfrac{4}{3}$ on the generator. Thus $q_L\cong \omega_{3,1}^1$.
\begin{thm}[cf. {\cite[Thm. 1.8.1]{nikulin}}]\label{t:relations}
The forms $w_{p,k}^\varepsilonilon$, $u_k$, $v_k$ generate the semi--group of finite quadratic forms.
\end{thm}
In other words every finite quadratic form can be written (not uniquely)
as a direct sum of the generators $w_{p,k}^\varepsilonilon$, $u_k$, $v_k$.
Relations can be found in \cite[Thm. 1.8.2]{nikulin}.
For a finite quadratic form $q$, and a prime number $p$,
we denote $q$ restricted to the $p$--component $(A_q)_p$ of $A$ by $q_p$.
The following results describe the close link between discriminant quadratic forms and even lattices.
\begin{thm}[cf. {\cite[Thm. 1.13.2]{nikulin}}]\label{t:lattice_unique}
An even lattice $S$ with invariants $(t_+,t_-,q)$ is unique if, simultaneously,
\begin{enumerate}
\item $t_+\geq 1, t_-\geq 1, t_++t_-\geq 3$;
\item for each $p\neq 2$, either $\rk S \geq 2+l((A_q)_p)$ or $
q_p\cong w_{p,k}^\varepsilonilon\oplus w_{p.k}^{\varepsilonilon'}\oplus q_p'$;
\item for $p= 2$, either $\rk S \geq 2+l((A_q)_2)$ or one of the following holds
\[
q_2\cong u_k\oplus q_2',\quad
q_2\cong v_k\oplus q_p',\quad
q_2\cong w_{2,k}^\varepsilonilon\oplus w_{2.k}^{\varepsilonilon'}\oplus q_2'.
\]
\end{enumerate}
\end{thm}
\begin{cor}[cf. {\cite[Corollary 1.13.3]{nikulin}}]\label{c:latunique}
An even lattice $S$ with invariants $(t_+,t_-,q)$ exists and is unique
if $t_+ -t_-\equiv \sign q \imod 8$, $t_+ +t_-\geq 2+ l(A_q)$, and $t_+, t_- \geq 1$.
\end{cor}
\begin{cor}[cf. {\cite[Corollary 1.13.4]{nikulin}}]\label{c:U+T}
Let $S$ be an even lattice of signature $(t_+,t_-)$. If $t_+\geq 1$, $t_-\geq 1$ and $t_+ +t_-\geq 3+l(A_S)$, then $S\cong U
\oplus T$ for some lattice $T$.
\end{cor}
In Table \ref{tab-forms}, we list the discriminant form associated to each of the lattices a
ppearing in our calculations (see Sections \ref{sec-ex},\ref{sec:tables}).
A complete description can be found in \cite[Appendix A]{belcastro}.
\begin{table}[h!]\centering
\begin{tabular}{ l c c || l c c|| l c c}
$L$&$\sign L$ &$q_L$& $L$&$\sign L$ &$q_L$& $L$&$\sign L$ &$q_L$\\
\hline
$U$ &(1,1) & trivial &$D_6$ &(0,6) & $(w_{2,1}^1)^2$ & $T_{4,4,4}$ &(1,9) & $v_2$\\
$U(2)$ &(1,1) & $u$ &$D_9$ &(0,9) & $w_{2,2}^{-1}$ & $T_{3,4,4}$ &(1,8) & $w_{2,3}^5$\\
$A_1$ &(0,1) & $w^{-1}_{2,1}$ &$E_6$ &(0,6) & $w_{3,1}^{-1}$ & $T_{2,5,6}$ &(1,10) & $w_{2,3}^{-5}$\\
$A_2$ &(0,2) & $w_{3,1}^{1}$ &$E_7$ &(0,7) & $w_{2,1}^{1}$ & $<2>$ &(1,0) & $w_{2,1}^1$\\
$A_3$ &(0,3) & $w_{2,2}^5$ &$E_8$ &(0,8) & trivial & $<4>$ &(1,0) & $w_{2,2}^1$\\
$A_1(2)$ &(0,1) & $w_{2,2}^{-1}$ &$H_5$ &(1,1) & $w_{5,1}^{-1}$ & $<8>$ &(1,0) & $w_{2,3}^1$\\
$D_4$ &(0,4) & $v$ &$L_9$ &(1,1) & $w_{3,2}^1$ & $<-8>$ &(0,1) & $w_{2,3}^{-1}$\\
$D_5$ &(0,5) & $w_{2,2}^{-5}$ &$M_9$ &(1,1) & $w_{3,2}^{-1}$ & & &\\
\end{tabular}
\caption{Lattices and forms} \label{tab-forms}
\end{table}
Let $L'$ be an overlattice $L'$ of the lattice $L$.
We call $H_{L'}:=L'/L$.
By the chain of embeddings $L\subset L'\subset (L')^*\subset L^*$
one has $H_{L'}\subset A_L$ and $A_{L'}=((L')^*/L)/H_{L'}$.
\begin{prop}[cf. {\cite[Prop. 1.4.1]{nikulin}}]\label{p:overl}
The correspondence $L'\leftrightarrow H_{L'}$ is a 1:1 correspondence
between overlattices of finite index of $L$ and $q_L$-isotopic subgroups of $A_L$,
i.e. subgroups on which the form $q_L$ is 0.
Moreover, $H_{L'}^^{\prime}erp=(L')^*/L$ and $q_{L'}=({q_L}_{|H_{L'}^^{\prime}erp})/H_{L'}$.
\end{prop}
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\subsection{K3 lattices} \label{sec-K3lattice}
Let $X$ be a K3 surface.
It is well--known that $H^2(X,\mathbb{Z})$ is an even unimodular lattice of signature $(3,19)$.
As such, it is isometric to the \emph{K3-lattice} $L_{\text{K3}} = U^3\oplus (E_8)^2$.
We let
\[
S_X = H^2(X,\mathbb{Z}) \cap H^{1,1}(X,\mathbb{C})
\]
denote the {\em Picard} lattice of $X$ in $H^2(X,\mathbb{Z})$ and $T_X = S_X^^{\prime}erp$ denote the {\em transcendental lattice}.
Let $\sigma$ be a non-symplectic automorphism of $X$.
We let $S_X(\sigma)\subseteq H^2(X,\mathbb{Z})$ denote the $\sigma^\ast$-invariant sublattice of $H^2(X,\mathbb{Z})$:
\[S_X(\sigma)=\{x\in H^2(X,\mathbb{Z}): \sigma^*x=x\}.\]
One can check that it is a primitive sublattice of $H^2(X,\mathbb{Z})$. In fact, $S_X(\sigma)$ is a primitive sublattice of $S_X$ and in general $S_X(\sigma)\subsetneq S_X$.
We let $T_X(\sigma) = S_X(\sigma)^^{\prime}erp$ denote its orthogonal complement. The signature of $S_X(\sigma)$ is $(1,t)$ for some $t\leq 19$, i.e. $S_X(\sigma)$ is hyperbolic.
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\section{Mirror symmetry}
\label{sec-mirror}
\subsection{Mirror symmetry for K3 surfaces}
\label{sec-LPK3}
Mirror symmetry for a Calabi--Yau manifold $X$ and its mirror $X^\vee$ can be thought of as an exchanging of the K\"ahler structure on $X$ for the complex structure of $X^\vee$. Thus a first prediction of mirror symmetry is the rotation of the Hodge diamond:
\begin{equation*}
H^{p,q}(X,\mathbb{C})\cong H^{q,N-p}(X,\mathbb{C})
\end{equation*}
where $N$ is the dimension of $X$.
For K3 surfaces, however, the Hodge diamond
is symmetric under the rotation mentioned above. So we need to consider a refinement of this idea. This is accomplished by the notion of lattice polarization. Roughly, we choose a primitive lattice $M\hookrightarrow S_X$, which plays the role of the K\"ahler deformations, and the mirror lattice $M^\vee$, which we now define, plays the role of the complex deformations.
We will refer to this formulation of mirror symmetry simply as LPK3 mirror symmetry.
Following \cite{dolgachev}, let $X$ be a K3 surface and suppose that $M$ is a lattice of signature $(1,t)$. If $j\colon M\hookrightarrow S_X$ is a primitive embedding into the Picard lattice of $X$, the pair $(X,j)$ is called an \emph{$M$-polarized K3 surface}. There is a moduli space of $M$--polarized K3 surfaces with dimension $19-t$.
We will not be concerned about the embedding. As in \cite{CLPS}, we will call the pair $(X,M)$ an \emph{$M$-polarizable} K3 surface if such an embedding $j$ exists. Note that for an $M$-polarizable K3 surface $(X,M)$, the lattice $M$ naturally embeds primitively into $L_{\text{K3}}$.
\begin{defn}\label{mirror_defn}
Let $M$ be a primitive sublattice of $L_{\text{K3}}$ of signature $(1,t)$ with $t\leq 18$ such that $M^^{\prime}erp_{L_{K3}}\cong U\oplus \mirror{M}$.
We define $\mirror{M}$ to be (up to isometry) the \emph{mirror lattice} of $M$.\footnote{As in \cite{CLPS}, our definition in this restricted setting is slightly coarser than the one used by Dolgachev in \cite{dolgachev}, since we do not keep track of the embedding $U\hookrightarrow M^^{\prime}erp$ and instead only consider $M^\vee$ up to isometry.}
\end{defn}
By Theorem \ref{t:lattice_unique}
this definition is independent of the embedding $M$ into $L_{\text{K3}}$. Furthermore under some conditions, (see e.g. Corollary~\ref{c:U+T} and Theorem~\ref{t:lattice_unique}) this definition is also independent of the embedding $U$ into $M^^{\prime}erp$. One can check that these conditions are satisfied for the lattices we consider here.
Note that $\mirror{M}$ also embeds primitively into $L_{\text{K3}}$ and has signature $(1,18-t)$. Furthermore, $q_M\cong -q_{\mirror{M}}$. One easily checks that $(\mirror{M})^^{\prime}erp_{L_{\text{K3}}}\cong U\oplus M$.
Given $(X,M)$ an $M$-polarizable K3 surface and $(X',M')$ an $M'$-polarizable K3 surface, with $M$ and $M'$ primitive sublattices of $S_X$ and $S_{X'}$, resp.,
we say that $(X,M)$ and $(X',M')$ are \emph{LPK3 mirrors} if $M' = \mirror{M}$ (or equivalently $M=\mirror{(M')}$).
Notice that if $M$ has rank $t+1$, then the dimension of the moduli space of $M^\vee$ polarized K3 surfaces is $19-(18-t)$ which agrees with the rank of $M$. Returning to the question of K\"ahler deformations and complex deformations, we see that this definition of mirror symmetry matches the idea behind rotation of the Hodge diamond, as mentioned earlier.
\subsection{Quasihomogeneous polynomials and diagonal symmetries}\label{quasi_sec}
We recall a few facts and definitions (cf. \cite{CLPS} for details).
A \emph{quasihomogeneous} map of degree $d$ with integer weights $w_1, w_2, \dots, w_n$
is $W:\mathbb{C}^n\to \mathbb{C}$ such that for every $\lambda \in \mathbb{C}$,
\[
W(\lambda^{w_1}x_1, \lambda^{w_2}x_2, \dots, \lambda^{w_n}x_n) = \lambda^dW(x_1,x_2, \dots, x_n).
\]
One can assume
$\gcd(w_1, w_2, \dots, w_n)=1$ and say $W$ has the \emph{weight system} $(w_1, w_2, \ldots, w_n; d)$.
Given a quasihomogeneous polynomial $W:\mathbb{C}^n \rightarrow \mathbb{C}$ with a critical point at the origin,
we say it is \emph{non-degenerate} if the origin is the only critical point of $W$ and
the fractional weights $\frac{w_1}{d}, \ldots, \frac{w_n}{d}$ of $W$ are uniquely determined by $W$.
A non-degenerate quasihomogeneous polynomial $W$ (also called \emph{potential} in the literature) is \emph{invertible} if it has the same number of monomials as variables.
If $W$ is invertible we can rescale variables so that $W = \sum_{i=1}^n ^{\prime}rod_{j=1}^n x_j^{a_{ij}}$.
This polynomial can be represented by the square matrix $\A{W} = (a_{ij})$,
which we will call the \emph{exponent matrix} of the polynomial.
Since $W$ is invertible, the matrix $\A{W}$ is an invertible matrix.
The \emph{group $G_W$ of diagonal symmetries} of an invertible polynomial $W$ is
\begin{equation*}
\Gmax{W} = \{(c_1, c_2, \ldots, c_n) \in (\mathbb{C}^*)^n: W(c_1x_1, c_2x_2, \ldots, c_nx_n) = W(x_1, x_2, \ldots, x_n)\}.
\end{equation*}
Observe that, given $\gamma=(c_1, c_2, \ldots, c_n)\in \Gmax{W}$,
the $c_i$'s are roots of unity.
Thus one can consider $\Gmax{W}$ as a subgroup of $(\mathbb{Q} / \mathbb{Z})^n$, using addivite notation and identifying
$(c_1, c_2, \ldots, c_n) = (e^{2 ^{\prime}i i g_1}, e^{2 ^{\prime}i i g_2}, \ldots, e^{2 ^{\prime}i i g_n})$ with $(g_1, g_2, \ldots, g_n)\in(\mathbb{Q} / \mathbb{Z})^n$.
Observe that the order of $\Gmax{W}$ is $|\Gmax{W}| = \det(\A{W})$.
Since $W$ is quasihomogeneous, the \emph{exponential grading operator} $\J{W} = \left(\frac{w_1}{d}, \frac{w_2}{d}, \ldots, \frac{w_n}{d}\right)$ is contained in $\Gmax{W}$.
We denote by $J_W$ the cyclic group of order $d$ generated by $\J{W}$: $J_W=\inn{\J{W}}$.
Moreover, each $\gamma=(g_1,\dots,g_n)$ define a diagonal matrix and thus
$G_W$ is embedded in $\GL_n(\mathbb{C})$.
We define $$\SLgp{W}:=\Gmax{W}\cap \SLn{n},$$
i.e. $\gamma=(g_1,\dots,g_n)\in \SLgp{W}$ if and only if $\sum_i g_i\in \mathbb{Z}$.
The group $\SLgp{W}$ is called the {\em symplectic group} since, by \cite[Proposition 1]{ABS},
an automorphism $\sigma\in \Gmax{W}$ is symplectic if and only if $\det \sigma =1$, that is, if and only if $\sigma\in \SLgp{W}$.
\subsection{K3 surfaces from $(W,G)$}\label{K3_sec}
Reid (in an unpublished work) and Yonemura \cite{yonemura}
have indipendently compiled a list of the 95 normalized weight systems $(w_1,w_2,w_3,w_4;d)$
(``the 95 families'') such that $\mathbb{P}(w_1,w_2,w_3,w_4)$
admits a quasismooth hypersurface of degree $d$ whose minimal resolution is a K3 surface.
We consider one of these weight systems $(w_1,w_2,w_3,w_4;d)$
and an invertible quasihomogeneous polynomial of the form
\begin{equation}\label{eq-W}
W=x_1^m+f(x_2,x_3,x_4).
\end{equation}
Moreover, let $G$ be a group of symmetries such that $J_W\subseteq G \subseteq \SLgp{W}$
and let $\omegaidetilde{G}=G/J_W$.
The polynomial $W$ defines a hypersurface $Y_{W,G}\subset \mathbb{P}(w_1,w_2,w_3,w_4)/\omegaidetilde{G}$
and one shows that the minimal resolution $X_{W,G}$ of $Y_{W,G}$ is a K3 surface (see \cite{ABS,CLPS}).
The group $\Gmax{W}$ acts on $Y_{W,G}$ via automorphisms, which extend to automorphisms on the K3 surface $X_{W,G}$.
The given form of $W$ ensures that the K3 surface $X_{W,G}$
admits a purely non-symplectic automorphism of order $m$:
\[
\sigma_m:^{\prime}p{x_1}{x_2}{x_3}{x_4}\mapsto ^{\prime}p{\zeta_mx_1}{x_2}{x_3}{x_4}
\]
where $\zeta_m$ is a primitive $m$-th root of unity.
With additive notation, it is $\sigma_m=\left(\tfrac 1m,0,0,0\right)$.
\subsection{BHK mirror symmetry}\label{BHK_sec}
Now we can describe the second relevant formulation of mirror symmetry coming from mirror symmetry for Landau--Ginzburg models and which we call BHK (from Berglund-H\"ubsch-Krawitz) mirror symmetry.
This particular formulation of mirror symmetry was developed initially by Berglund--H\"ubsch in \cite{berghub}, and later refined by Berglund--Henningson in \cite{berghenn} and Krawitz in \cite{krawitz}. Because of the LG/CY correspondence and a theorem from Chiodo--Ruan \cite{BHCR}, this mirror symmetry of LG models can be translated into mirror symmetry for Calabi--Yau varieties (or orbifolds).
We consider $(W,G)$ with $W$ invertible and $W=\sum_{i=1}^n ^{\prime}rod_{j=1}^n x_j^{a_{ij}}$
and define another pair $(W^T,G^T)$, called the BHK mirror.
We first define the polynomial $W^T$ as $$W^T = \sum_{i=1}^n ^{\prime}rod_{j=1}^n x_j^{a_{ji}},$$
i.e. the matrix of exponents of $W^T$ is $A_W^T$.
By the classification of invertible polynomials (cf. \cite[Theorem 1]{KrSk}),
$W^T$ is invertible.
Next, using additive notation, one defines the dual group $G^T$ of $G$ as
\begin{equation}\label{dualG_def}
G^T = \set{\; g \in \Gmax{W^T} \; | \;\;g\A{W} h^T \in \mathbb{Z} \text{ for all } h \in G \; }.
\end{equation}
The following useful properties of the dual group can be found in \cite[Proposition 3]{ABS}:
\begin{prop}[cf. {\cite[Proposition 3]{ABS}}]
Given $G$ and $G^T$ as before, one has:
\begin{enumerate}
\item $(G^T)^T = G$.
\item If $G_1\subset G_2$, then $G_2^T\subset G_1^T$ and $G_2/G_1\cong G_1^T/G_2^T$.
\item $(\Gmax{W})^T = \{0\}$, $(\{0\})^T = \Gmax{W^T}$.
\item $(J_W)^T=\SLgp{W^T}$. In particular, if $J_W\subset G$, then $G^T\subset \SLgp{W}$.
\end{enumerate}
\end{prop}
Given the pair $(W,G)$ with $W$ invertible with respect to one of the 95 weight systems,
we associated to it the K3 surface $X_{W,G}$.
One can check that in this case the weight system of $W^T$ also belongs to the 95.
By the previous result, $J_{W^T}\subseteq G^T\subseteq \SL_{W^T}$, so that $X_{W^T,G^T}$ is again a K3 surface.
We call $X_{W^T,G^T}$ the \emph{BHK mirror of $X_{W,G}$}.
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\subsection{K3 surfaces from $(W,G)$}\label{K3_sec}
Reid (in an unpublished work) and Yonemura \cite{yonemura}
have indipendently compiled a list of the 95 normalized weight systems $(w_1,w_2,w_3,w_4;d)$
(``the 95 families'') such that $\mathbb{P}(w_1,w_2,w_3,w_4)$
admits a quasismooth hypersurface of degree $d$ whose minimal resolution is a K3 surface.
We consider one of these weight systems $(w_1,w_2,w_3,w_4;d)$
and an invertible quasihomogeneous polynomial of the form
\begin{equation}\label{eq-W}
W=x_1^m+f(x_2,x_3,x_4).
\end{equation}
Moreover, let $G$ be a group of symmetries such that $J_W\subseteq G \subseteq \SLgp{W}$
and let $\omegaidetilde{G}=G/J_W$.
The polynomial $W$ defines a hypersurface $Y_{W,G}\subset \mathbb{P}(w_1,w_2,w_3,w_4)/\omegaidetilde{G}$
and one shows that the minimal resolution $X_{W,G}$ of $Y_{W,G}$ is a K3 surface (see \cite{ABS,CLPS}).
The group $\Gmax{W}$ acts on $Y_{W,G}$ via automorphisms, which extend to automorphisms on the K3 surface $X_{W,G}$.
The given form of $W$ ensures that the K3 surface $X_{W,G}$
admits a purely non-symplectic automorphism of order $m$:
\[
\sigma_m:^{\prime}p{x_1}{x_2}{x_3}{x_4}\mapsto ^{\prime}p{\zeta_mx_1}{x_2}{x_3}{x_4}
\]
where $\zeta_m$ is a primitive $m$-th root of unity.
With additive notation, it is $\sigma_m=\left(\tfrac 1m,0,0,0\right)$.
\subsection{BHK mirror symmetry}\label{BHK_sec}
Now we can describe the second relevant formulation of mirror symmetry coming from mirror symmetry for Landau--Ginzburg models and which we call BHK (from Berglund-H\"ubsch-Krawitz) mirror symmetry.
This particular formulation of mirror symmetry was developed initially by Berglund--H\"ubsch in \cite{berghub}, and later refined by Berglund--Henningson in \cite{berghenn} and Krawitz in \cite{krawitz}. Because of the LG/CY correspondence and a theorem from Chiodo--Ruan \cite{BHCR}, this mirror symmetry of LG models can be translated into mirror symmetry for Calabi--Yau varieties (or orbifolds).
We consider $(W,G)$ with $W$ invertible and $W=\sum_{i=1}^n ^{\prime}rod_{j=1}^n x_j^{a_{ij}}$
and define another pair $(W^T,G^T)$, called the BHK mirror.
We first define the polynomial $W^T$ as $$W^T = \sum_{i=1}^n ^{\prime}rod_{j=1}^n x_j^{a_{ji}},$$
i.e. the matrix of exponents of $W^T$ is $A_W^T$.
By the classification of invertible polynomials (cf. \cite[Theorem 1]{KrSk}),
$W^T$ is invertible.
Next, using additive notation, one defines the dual group $G^T$ of $G$ as
\begin{equation}\label{dualG_def}
G^T = \set{\; g \in \Gmax{W^T} \; | \;\;g\A{W} h^T \in \mathbb{Z} \text{ for all } h \in G \; }.
\end{equation}
The following useful properties of the dual group can be found in \cite[Proposition 3]{ABS}:
\begin{prop}[cf. {\cite[Proposition 3]{ABS}}]
Given $G$ and $G^T$ as before, one has:
\begin{enumerate}
\item $(G^T)^T = G$.
\item If $G_1\subset G_2$, then $G_2^T\subset G_1^T$ and $G_2/G_1\cong G_1^T/G_2^T$.
\item $(\Gmax{W})^T = \{0\}$, $(\{0\})^T = \Gmax{W^T}$.
\item $(J_W)^T=\SLgp{W^T}$. In particular, if $J_W\subset G$, then $G^T\subset \SLgp{W}$.
\end{enumerate}
\end{prop}
Given the pair $(W,G)$ with $W$ invertible with respect to one of the 95 weight systems,
we associated to it the K3 surface $X_{W,G}$.
One can check that in this case the weight system of $W^T$ also belongs to the 95.
By the previous result, $J_{W^T}\subseteq G^T\subseteq \SL_{W^T}$, so that $X_{W^T,G^T}$ is again a K3 surface.
We call $X_{W^T,G^T}$ the \emph{BHK mirror of $X_{W,G}$}.
\subsection{Main theorem}\label{main_sec}
We have described two kinds of mirror symmetry for K3 surfaces: LPK3 mirror symetry and BHK one.
Since mirror symmetry describes a single physical phenomenon,
we expect the two constructions to be compatible in situations where both apply.
We will now state our main theorem, which shows that BHK and LPK3 mirror symmetry agree for the K3 surfaces $X_{W,G}$, when $W$ is of the form \eqref{eq-W}.
When no confusion arises, we will denote the mirror K3 surfaces $X_{W,G}$ and $X_{W^T,G^T}$ simply by $X$ and $X^T$.
Consider the data $(W,G,\sigma_m)$,
where \begin{itemize}
\item $W$ is an invertible polynomial of the form \eqref{eq-W}
whose weight system belong to the 95 families of Reid and Yonemura,
\item $\sigma_m=(\frac 1m,0,0,0)$ is the non-symplectic automorphism of order $m$,
\item $G$ is a group of diagonal symmetries of $W$ such that $J_W\subseteq G\subseteq \SL_W$.\end{itemize}
By section \ref{sec-K3lattice}, the invariant lattice $S_X(\sigma_m)$ is a primitive sublattice of $S_X$
and $(X_{W,G}, S_X(\sigma_m))$ is a $S_X(\sigma_m)$--polarizable K3 surface.
Let $r$ be the rank of $S_X(\sigma_m)$.
The BHK mirror is given by $(W^T,G^T,\sigma_m^T)$, where $\sigma_m^T$ is the non-symplectic automorphism of order $m$ on $X_{W^T,G^T}$.
Notice that $\sigma_m$ and $\sigma_m^T$ have the same form, namely $(\tfrac{1}{m},0,0,0)$, but they act on different surfaces.
\begin{thm}\label{t:main_thm}
Suppose $m\neq 4,8,12$. If $W$ is a polynomial of the form \eqref{eq-W}, quasihomogeneous with respect to one of the 95 weight systems for K3 surfaces as in Section~\ref{K3_sec} and $G$ is a group of diagonal symmetries satisfying $J_W\subseteq G\subset\SL_W$, then $\left(X_{W^T,G^T}, S_{X^T}(\sigma_p^T\right))$ is an LPK3 mirror of $\left(X_{W,G}, S_X(\sigma_p)\right)$.
\end{thm}
The theorem is proved by showing that $$S_X(\sigma_m)^\vee\cong S_{X^T}(\sigma_m^T).$$
As we have seen in section \ref{s:lattice}, this amounts to checking that the invariants $(r,q_{S_X(\sigma_m)})$ for $X_{W,G}$ and $(r^T,q_{S_{X^T}(\sigma_m^T)})$ for $X_{W^T,G^T}$ satisfy $r=20-r^T$ and $q_{S_X(\sigma_m)}\cong -q_{S_{X^T}(\sigma_m^T)}$. Thus the heart of the proof is determining $q_{S_X(\sigma_m)}$ (or equivalently in our case $S_X(\sigma_m)$).
In the following section, we will describe how this is done.
It involves computing the invariant lattice and its overlattices.
We list the results in tables in Section \ref{sec:tables}.
Unfortunately, our method does not work for $m=4,8,12$ due to the presence of many overlattices, so that we cannot exactly pinpoint the invariant lattice.
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\section{Methods}
\label{sec-method}
In the setting of Theorem \ref{t:main_thm}, one has to show that $S_X(\sigma_m)^\vee\cong S_{X^T}(\sigma_m^T)$.
Whenever $m=p$ a prime number, Theorem \ref{t:main_thm} was proved using a similar method in \cite{ABS} for $m=2$ and \cite{CLPS} for other primes.
There is not a general method of proof in either article; instead the theorem is checked in every case.
In \cite{ABS} and \cite{CLPS} there are several tools introduced in order to facilitate computation of the invariant lattice.
The proof we give here follows roughly the same idea, however the methods used in the previous articles for computing $S_X(\sigma_m)$ are no longer valid, when $m$ is not prime.
In order to illustrate the differences, we highlight briefly the method used in case of $p$ prime.
Then we will describe the proof of the theorem, in case $m$ is not prime.
\subsection{Method for $m=p$ prime}\label{s:methodsprime}
As mentioned, the argument given in \cite{ABS} and \cite{CLPS} essentially boils down to determining the invariant lattice $S_X(\sigma_p)$ for $X_{W,G}$.
The method for determining this lattice relies on the following powerful theorems.
\begin{thm}[\cite{other_primes}] Given a $K3$ surface with a non-symplectic automorphism $\sigma$ of order $p$, a prime, the invariant lattice $S_X(\sigma)$ is $p$--elementary, i.e. $A_{S_X(\sigma)}\cong (\mathbb{Z}/p\mathbb{Z})^a$.
\end{thm}
\begin{thm}[\cite{RS,nikulin}] For a prime $p\neq 2$, a hyperbolic, $p$--elementary lattice $L$ with rank $r\geq 2$ is completely determined by the invariants $(r,a)$, where $a$ is the length. An indefinite 2--elementary lattice is determined by the invariants $(r,a,\delta)$, where $\delta\in{0,1}$ and $\delta=0$ if the discriminant quadratic form takes values 0 or 1 only and $\delta=1$ otherwise.
\end{thm}
By Proposition \ref{p:orth}, the orthogonal complement in $L_{K3}$ of a $p$--elementary lattice with invariants $(r,a)$ is a $p$--elementary lattice with invariants $(22-r,a)$. For both a given 2--elementary lattice and its orthogonal complement, the third invariant $\delta$ agrees.
In the setting of Theorem \ref{t:main_thm}, Corollary \ref{c:U+T} shows that we also have $S_X(\sigma_p)^^{\prime}erp\cong U\oplus M$ where $M$ is a hyperbolic $p$--elementary with invariants $(20-r,a)$. Thus for $p\neq 2$ it is enough to verify that $(r,a)$ for $X_{W,G}$ and $(r^T,a_T)$ for $X_{W^T,G^T}$ satisfy $r=20-r^T$ and $a=a_T$. For $p=2$ the third invariant $\delta$ must also be compared, which the authors checked in \cite{ABS}.
In order to compare $(r,a)$, we first look at the topology of the fixed point locus.
\begin{thm}[cf. \cite{nikulin2, other_primes}]\label{summary_invariants}
Let $X$ be a K3 surface with a non-symplectic automorphism $\sigma$ of prime order $p\neq 2$. Then the fixed locus $X^\sigma$ is nonempty, and consists of either isolated points or a disjoint union of smooth curves and isolated points of the following form:
\begin{equation}
X^\sigma=C\cup R_1\cup\ldots\cup R_k\cup \{p_1,\ldots,p_n\}\label{e:fixedlocus}.
\end{equation}
Here $C$ is a curve of genus $g\geq 0$, $R_i$ are rational curves and $p_i$ are isolated points.
If $p=2$, then the fixed locus is either empty, the disjoint union of two elliptic curves, or is of the form \eqref{e:fixedlocus} with $n=0$.
\end{thm}
In \cite{ABS} and \cite{CLPS}, $\sigma_p$ always fixes a curve. Furthermore, the case of two elliptic curves does not appear in this setting described there. Therefore, the fixed locus is determined by the invariants $(g,k,n)$.
In \cite{other_primes}, the authors give formulas to calculate $(r,a)$ given $(g,k,n)$ for each prime $p$ (see \cite[Theorem 0.1]{other_primes}).
Thus, in order to prove Theorem \ref{t:main_thm} for $p$ prime, one first computes the invariants $(g,k,n)$,
and from them computes the invariants $(r,a)$ (if $p=2$, additional computation are required to obtain $\delta$).
Then one compares the invariants for BHK mirrors as described above.
\input{methodnotprime}
\subsection{Proof of main Theorem}
We now provide the details of the proof of Theorem \ref{t:main_thm}.
For each K3 surface $X_{W,G}$ we have the invariants $(r,q_{S_X(\sigma_m)})$ for the invariant lattice $S_X(\sigma_m)$, as discussed in the previous section. We know the invariant lattice has signature $(1,r-1)$, and so the orthogonal complement has signature $(2,20-r)$. We check that the conditions of Corollary \ref{c:U+T} are fulfilled so that $S_X(\sigma_m)^^{\prime}erp_{L_{K3}}\cong U\oplus M$ for some lattice $M$\footnote{In one case, the conditions are not fulfilled, namely $m=6$, $r=1$. However, in this case, we know the lattice $U\oplus\langle4\rangle$ has the given invariants, and by Corollary~\ref{c:latunique} this is the only such lattice}. This lattice $M$ is hyperbolic with signature $(1,19-r)$ and has discriminant quadratic form $-q_{S_X(\sigma_m)}$. One can see from the tables that the conditions of Theorem~\ref{t:lattice_unique} are satisfied. Hence, there is exactly one lattice with these invariants. To prove the theorem, therefore, we need simply to check that $(20-r, -q_{S_X(\sigma_m)})$ are the invariants for the invariant lattice $S_{X^T}(\sigma_m^T)$ for $X_{W^T,G^T}$. This can be checked by consulting the tables in Section \ref{sec:tables}.
This concludes the proof.
Tables contain all possible invertible polynomial of the from \eqref{eq-W} with non--symplectic automorphism of order $m$, and for each polynomial, we list the orders of the possible groups $G/J_W$ satisfying $J_W\subset G\subset \SL_W$.
In most cases $\SL_W/J_W$ is cyclic, so the properties of $G^T$ make it clear what the dual group is. However for two examples, one can see by the multiplicity of subgroups with the same order that the group $\SL_W/J_W$ is not cyclic. These two examples are $x^3+y^3+z^6+w^6$ (number 3d) and $x^2+y^4+z^6+w^{12}$ (number 8d) in Table \ref{tab-6}. In these cases, we will clear up any ambiguities in the following sections.
From now on we will make a change of notation from $(x_1,x_2,x_3,x_4)$ to $(x,y,z,w)$ for the variables of $W$, so that the variables are arranged with the weights in nonincreasing order. In other words, it is possible that $x_1$ corresponds to any of $x,y,z$, or $w$. This convention is also used in Tables of Section \ref{sec:tables}.
\begin{rem}
It is possible that a given K3 surface admits a purely non--symplectic automorphism of different orders. It turns out that it doesn't matter which automorphism one uses to exhibit LPK3 mirror symmetry, the notion still agrees with BHK mirror symmetry, in the sense of Theorem~\ref{t:main_thm}, as long as the defining polynomial is of the proper form \eqref{eq-W}. We expect that the theorem still holds for K3 surfaces that don't take the form of \eqref{eq-W}, but that is a topic for further investigation.
\end{rem}
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\section{Examples}
\label{sec-ex}
In this section we will first give examples to illustrate each of the methods that was used to determine $S_X(\sigma_m)$. Then we will describe the subgroups in the two cases where $\SL_W/J_W$ is not cyclic.
\begin{ex}\label{ex-method1}Method I:
This first example will illustrate Method I for determining $S_X(\sigma_m)$. Let us consider the K3 surface with equation
\[
W=x^2+y^3+z^9+yw^{12}=0
\]
in the weighted projective space $\mathbb{P}(9,6,2,1)$ with degree 18. This is number 12b in Table~\ref{tab-9}. There are two non--symplectic automorphisms of interest $\sigma_2:(x,y,z,w)\mapsto(-x,y,z,w)$ and $\sigma_9:(x,y,z,w)\mapsto(x,y,\mu_9z,w)$. The invariant lattice $S_X(\sigma_2)$ was dealt with in \cite{ABS}, so we focus on $\sigma_9$.
Here $|G_W|=36\cdot 18$, $|J_W|=18$ and the weight system for the BHK mirror is $(18,11,4,3;36)$ so that $[G_{W}:\SL_W]=36$. Thus $|\SL_W/J_W|=1$.
Looking at the action of $\CC^{\ast}$ on the weighted projective space $\mathbb{P}(9,6,2,1)$, we find the following isotropy:
\begin{align*}
\mu_3 &: \text{fixes }z=w=0, x^2+y^3=0\\
\mu_2 &: \text{fixes }x=w=0, y^3+z^9=0.
\end{align*}
The first row provides a single point with $\Zfin{3}$ isotropy ($A_2$ singularity), and the second provides three points each with $\Zfin{2}$ isotropy (3 $A_1$'s).
Their resolution gives the configuration of curves on $X_{W,G}$ depicted in Figure \ref{fig:res}. In this depiction, we have not indicated the three intersection points between $C_x$ and $C_z$.
\begin{figure}
\caption{Resolution of singularities on $X_W$}
\label{fig:res}
\end{figure}
The set $\mathcal{E}$ consists of these five exceptional curves. Denote by $E_1$, $E_2$ and $E_3$ the three $A_1$ fibers, and $E_4$ and $E_5$ the two curves in the $A_2$ fiber. Looking at the form of $W$, we see that the curves $C_x=\{x=0\}$, $C_z=\{z=0\}$ and $C_w=\{w=0\}$ are smooth. The curve $C_y=\{y=0\}$ is not smooth. Thus the set $\mathcal{K}$ consists of these three smooth curves. The curve $C_x$ has genus 7, $C_z$ has genus 1, and $C_w$
has genus 0.
There are two important representatives of the coset $\sigma_9J_W$ in $G_W$ which will help us compute the fixed locus for $\sigma_9$, namely $(0,0,\tfrac 19,0)$ and $(0,\tfrac 23,0,\tfrac 49 )$. These representatives show us that the curve $C_z=\{z=0\}$ is fixed and the point defined by $\{w=y=0, x^2+z^9=0\}$ is also fixed.
Because $C_z$ is fixed, $E_4$ and $E_5$ are invariant (though not fixed pointwisely).
The point of intersection of $C_w$ and the $A_2$ exceptional fiber and this other point are the only fixed points on $C_w$. Thus the three $A_1$ singularities are permuted by the action. By Lemma \ref{l:rank}, since there are three orbits, $S_X(\sigma_9)$ has rank 4.
We now compute the lattice $L_\mathcal{B}$, generated by $\mathcal{B}=\set{E_1+E_2+E_3, E_4,E_5, C_x,C_z,C_w}$. Since there are six generators, two of them are redundant, for example $C_x$ and $C_z$.
Consider the lattice $L$ generated by $E_1+E_2+E_3$, $E_4$, $E_5$, and $C_w$. This lattice has bilinear form
\[
\left(\begin{matrix}
-6 &0 &0 &3\\
0 &-2 &1 &0\\
0 &1 &-2 &1\\
3 &0 &1 &-2\\
\end{matrix}\right)
\] which has discriminant form $\omega_{3,1}^1$.
By Proposition \ref{p:overl}, there are no non--trivial even overlattices of this lattice, hence $L=L_\mathcal{B}=S_X(\sigma_9)$. Thus we have the invariants $(r,q_{S_X(\sigma_9)})=(4,w_{3,1}^{1})$. In fact, $S_X(\sigma_9)\cong U\oplus A_2$.
\end{ex}
\begin{rem}
This method also yields some other interesting facts regarding the Picard lattice of these surfaces.
In \cite{belcastro}, Belcastro computes the Picard lattice for a generic hypersurface with these weights and degree as $U\oplus D_4$. However, if we look at the non--symplectic automorphism $\sigma_9^3$, we
can compute the invariants $g=1,n=4,k=1$, and therefore $r=10, a=4$ for the invariant lattice, giving us the invariant lattice $S_X(\sigma_9^3)=U\oplus A_2 \oplus E_6$. This shows us in particular, that the Picard lattice of this
surface is bigger than the Picard lattice for a generic quasihomogeneous polynomial with these weights.
\end{rem}
\begin{ex}\label{ex-method2}Method II:
In order to illustrate Method II, we repeat the computations for the BHK mirror of the previous example:
\[
W^T=x^2+y^3w+z^9+w^{12}
\]
with weight system $(18,11,4,3;36)$. This is number 43a in Table~\ref{tab-9}. Here again $|\SL_W/J_W|=1$. As before, we also have an involution, but we consider only $\sigma_9^T$.
Looking at the action of $\CC^{\ast}$ on $\mathbb{C}^4$ and resolving the singularities we have
an $A_{10}$ given by resolving the point $(0,1,0,0)$,
$2A_2$ coming from the two points with $y=z=0$ fixed by $\mu_3$ and
an $A_1$ coming from the point with $y=w=0$ fixed by $\mu_2$.
This time $\mathcal{E}$ has 15 curves and $\mathcal{K}=\set{C_x,C_y,C_z}$ as in Figure~\ref{fig:res9}. Again we do not depict the three points of intersection between $C_x$ and $C_y$.
\begin{figure}
\caption{Resolution of singularities on $X_{W^T}
\label{fig:res9}
\end{figure}
Two relevant representatives of $\sigma_9^T$ in $G_{W^T}/J_{W^T}$ are $(0,0,\tfrac 19,0)$ and $(0,\tfrac 49,0, \tfrac 23)$. From these we see that the curve $C_z$ is fixed. It has genus 0.
Furthermore the exceptional curve from the $A_1$ singularity at $y=w=0$ is fixed pointwisely, as well as one of the curves in the exceptional $A_{10}$. Since $C_z$ is fixed, each of the exceptional curves is left invariant under $\sigma_9^T$. From Lemma \ref{l:rank}, the rank of the invariant lattice $S_{X^T}(\sigma_9^T)$ is $r=16$.
In this case, we compute the invariant lattice for $(\sigma_9^T)^3$, which is a non--symplectic automorphism of order 3.
The curves $C_z$ and $C_y$ are fixed; both have genus zero. Three of the curves on the $A_{10}$ chain are also fixed, as in Lemma~\ref{l:rationaltree}. Furthermore, the remainin intersection points of the chains of exceptional curves are fixed, and an additional point on the $A_1$. Thus the invariants are $(g,n,k)=(1,4,7)$.
Using the results cited in Section \ref{s:methodsprime}, the invariants for the 3--elementary lattice $(\sigma_9^T)^3$ are $(16,1)$. Since $S_{X^T}(\sigma_9^T)$ is a primitive sublattice of this 3--elementary lattice, and both have the same rank, they are equal. Therefore we have invariants $(r,q)=(16, w_{3,1}^{-1})$ and the lattice is $S_X(\sigma_9)=U\oplus E_6\oplus E_8$.
\end{ex}
Comparing the ranks, and noticing that $\omega_{3,1}^1=-\omega_{3,1}^{-1}$, we see the BHK mirror matches the LPK3 mirror symmetry.
\begin{ex} Method III:\label{ex-meth3}
Let $W:=x^2+y^4+yz^4+w^{16}$ with $m=16$ in weight system $(8,4,3,1;16)$. This is number 37b in Table \ref{tab-16}. The order of $\SL_W/J_W$ is 2. This appears to be the same K3 surface investigated in \cite[Example~3.2]{order_sixteen}.
Computing singularities we obtain an $A_2$ at the point $(0:0:1:0)$ and
two $A_3$'s at the two points with $z=w=0$.
Resolving these, we obtain the configuration of curves showed in Figure \ref{fig:resIII}. The curves $C_x$ and $C_z$ intersect in four points, which are not depicted.
\begin{figure}
\caption{Resolution of singularities for $X_W$}
\label{fig:resIII}
\end{figure}
The genus of the curve $C_w$ is 0, the genus of $C_z$ is 1, the genus of $C_x$ is 6. However, $C_y$ consists of two components, each a copy of $\mathbb{P}^1$.
The automorphism $\sigma_{16}=(0,0,0,\tfrac{1}{16})$ fixes $C_w$, and therefore leaves all of the exceptional curves invariant. Thus we have $|\mathcal{E}/\sigma_{16}|=8$, and $r=9$. Furthermore, $\mathcal{K}$ consists of the curves $C_w$, $C_z$ and the two curves that make up $C_y$.
Using an explicit form of the intersection matrix, one can check that the lattice $L_\mathcal{B}$ is actually generated by the exceptional curves and $C_w$.
one sees that it is a lattice of type $T_{3,4,4}$. The discriminant group of $T_{3,4,4}$ is $\mathbb Z/8\mathbb Z$ and the corresponding form $q$ is $w^{5}_{2,3}$. This form has one overlattice. However, Belcastro has computed the Picard Lattice for a general member of the family of K3 surfaces with this weight system as $T_{3,4,4}$. Thus $L_\mathcal{B}\cong T_{3,4,4}$ embeds primitively into $S_X(\sigma_{16})$ and so they are equal, e.g. $S_X(\sigma_{16})=L_\mathcal{B}$ with invariants $(r,q)=(9,w^5_{2,3})$.
\end{ex}
\begin{rem}
There is another case with with the same invariant lattice in the same weight system, namely number 37a. The reasoning is similar to what we have just outlined.
\end{rem}
Finally, we will describe both of the cases requiring what we have called Method IV. These two cases are similar in that we use the Picard lattice to help determine $S_X(\sigma_{m})$. We will need the following proposition.
\begin{prop}[{\cite[Prop. 1.15.1]{nikulin}}]\label{p:primembedding}
The primitive embeddings of a lattice $L$ into an even lattice with invariants $(m_+,m_-,q)$ are determined by the sets $(H_L, H_q,\gamma;K,\gamma_K)$, where $H_L\subset A_L$ and $H_q\subset A_q$ are subgroups, $\gamma:q_S|_{H_S}\to q|_{H_q}$ is an isomorphism of subgroups preserving the quadratic forms to these subgroups, $K$ is an even lattice with invariants $(m_+-t_+, m_--t_-, -\delta)$, where $\delta\cong q_S\oplus (-q)|_{\Gamma_\gamma^^{\prime}erp/\Gamma_\gamma}$, $\Gamma_\gamma$ being the pushout of $\gamma$ in $A_S\oplus A_q$, and, finally, $\gamma_K:q_K\to (-\delta)$ is an isomorphism of quadratic forms.
\end{prop}
From this proposition, we can determine all primitive embeddings of one even lattice into another. We will use this in the next example.
\begin{ex}Method IV:\label{ex-meth41}
We now consider the BHK dual to the previous example. This is the first entry for 37b in Table~\ref{tab-16}. As mentioned in the introduction, this provides an example to the case in \cite{order_sixteen}, where no example could be found. In this case, we have
\[
W^T=W=x^2+y^4+yz^4+w^{16},
\]
and $G^T=\SL_W$, and we know $|\SL_W/J_W|=2$. In fact the group is generated by $(\tfrac{1}{2},0, \tfrac{1}{2},0)$, so we see that the points with $x=z=0$ are fixed. Another representative in the same coset is $(\tfrac{1}{2},0, 0, \tfrac{1}{2})$. Thus we see that the intersection points on the $A_2$ chain from the previous example are fixed, as well as other point with $x=w=0$. The two $A_3$ chains are permuted by the action. Thus on $X_{W^T,G^T}$ we get the configuration of curves of Figure \ref{fig:res16}.
\begin{figure}
\caption{Resolution of singularities on $X_{W^T,G^T}
\label{fig:res16}
\end{figure}
Using the Riemann--Hurwitz Theorem, we can compute the genus of the coordinate curves. The curve $C_x$ is covered by a curve of genus 6 with 6 fixed points, so it has genus 2. Similarly, we see that the genus of $C_w$ is 0 and the genus of $C_z$ is 0.
The two components of the curve $\set{y=0}$ from the previous example are permuted, to give us $C_y$ of genus 0.
The non--symplectic automorphism $\sigma_{16}=(0,0,0,\tfrac{1}{16})$, fixes $C_w$, and therefore the chains of exceptional curves intersecting $C_w$ are invariant. It is not difficult to see also that the four exceptional curves intersecting $C_x$ and $C_z$ are permuted. Thus $r=11$.
One can check that $C_z$, $C_y$ and $C_x$ are superfluous, giving us a lattice $L_\mathcal{B}$
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The automorphism $\sigma_{16}=(0,0,0,\tfrac{1}{16})$ fixes $C_w$, and therefore leaves all of the exceptional curves invariant. Thus we have $|\mathcal{E}/\sigma_{16}|=8$, and $r=9$. Furthermore, $\mathcal{K}$ consists of the curves $C_w$, $C_z$ and the two curves that make up $C_y$.
Using an explicit form of the intersection matrix, one can check that the lattice $L_\mathcal{B}$ is actually generated by the exceptional curves and $C_w$.
one sees that it is a lattice of type $T_{3,4,4}$. The discriminant group of $T_{3,4,4}$ is $\mathbb Z/8\mathbb Z$ and the corresponding form $q$ is $w^{5}_{2,3}$. This form has one overlattice. However, Belcastro has computed the Picard Lattice for a general member of the family of K3 surfaces with this weight system as $T_{3,4,4}$. Thus $L_\mathcal{B}\cong T_{3,4,4}$ embeds primitively into $S_X(\sigma_{16})$ and so they are equal, e.g. $S_X(\sigma_{16})=L_\mathcal{B}$ with invariants $(r,q)=(9,w^5_{2,3})$.
\end{ex}
\begin{rem}
There is another case with with the same invariant lattice in the same weight system, namely number 37a. The reasoning is similar to what we have just outlined.
\end{rem}
Finally, we will describe both of the cases requiring what we have called Method IV. These two cases are similar in that we use the Picard lattice to help determine $S_X(\sigma_{m})$. We will need the following proposition.
\begin{prop}[{\cite[Prop. 1.15.1]{nikulin}}]\label{p:primembedding}
The primitive embeddings of a lattice $L$ into an even lattice with invariants $(m_+,m_-,q)$ are determined by the sets $(H_L, H_q,\gamma;K,\gamma_K)$, where $H_L\subset A_L$ and $H_q\subset A_q$ are subgroups, $\gamma:q_S|_{H_S}\to q|_{H_q}$ is an isomorphism of subgroups preserving the quadratic forms to these subgroups, $K$ is an even lattice with invariants $(m_+-t_+, m_--t_-, -\delta)$, where $\delta\cong q_S\oplus (-q)|_{\Gamma_\gamma^^{\prime}erp/\Gamma_\gamma}$, $\Gamma_\gamma$ being the pushout of $\gamma$ in $A_S\oplus A_q$, and, finally, $\gamma_K:q_K\to (-\delta)$ is an isomorphism of quadratic forms.
\end{prop}
From this proposition, we can determine all primitive embeddings of one even lattice into another. We will use this in the next example.
\begin{ex}Method IV:\label{ex-meth41}
We now consider the BHK dual to the previous example. This is the first entry for 37b in Table~\ref{tab-16}. As mentioned in the introduction, this provides an example to the case in \cite{order_sixteen}, where no example could be found. In this case, we have
\[
W^T=W=x^2+y^4+yz^4+w^{16},
\]
and $G^T=\SL_W$, and we know $|\SL_W/J_W|=2$. In fact the group is generated by $(\tfrac{1}{2},0, \tfrac{1}{2},0)$, so we see that the points with $x=z=0$ are fixed. Another representative in the same coset is $(\tfrac{1}{2},0, 0, \tfrac{1}{2})$. Thus we see that the intersection points on the $A_2$ chain from the previous example are fixed, as well as other point with $x=w=0$. The two $A_3$ chains are permuted by the action. Thus on $X_{W^T,G^T}$ we get the configuration of curves of Figure \ref{fig:res16}.
\begin{figure}
\caption{Resolution of singularities on $X_{W^T,G^T}
\label{fig:res16}
\end{figure}
Using the Riemann--Hurwitz Theorem, we can compute the genus of the coordinate curves. The curve $C_x$ is covered by a curve of genus 6 with 6 fixed points, so it has genus 2. Similarly, we see that the genus of $C_w$ is 0 and the genus of $C_z$ is 0.
The two components of the curve $\set{y=0}$ from the previous example are permuted, to give us $C_y$ of genus 0.
The non--symplectic automorphism $\sigma_{16}=(0,0,0,\tfrac{1}{16})$, fixes $C_w$, and therefore the chains of exceptional curves intersecting $C_w$ are invariant. It is not difficult to see also that the four exceptional curves intersecting $C_x$ and $C_z$ are permuted. Thus $r=11$.
One can check that $C_z$, $C_y$ and $C_x$ are superfluous, giving us a lattice $L_\mathcal{B}$
with discriminant form $w^{-5}_{2,3}$.
There is one isotropic subgroup $H$ and hence one overlattice of $L_\mathcal{B}$. By Proposition~\ref{p:overl} this overlattice has discriminant form $\omega^{-1}_{2,1}$.
Since $S_X(\sigma_{16})$ is an overlattice of $L_\mathcal{B}$, the two possibilities for $S_X(\sigma_{16})$ are $U\oplus E_8\oplus A_1$ or $T_{2,5,6}$. Using Proposition~\ref{p:primembedding} we will show that $U\oplus E_8\oplus A_1$ does not embed primitively into $S_{X_{W^T,G^T}}$, so that $S_X(\sigma_{16})=T_{2,5,6}$.
In \cite{order_sixteen} Al Tabbaa-Sarti-Taki have computed the Picard Lattice for K3 surfaces with non--symplectic automorphisms of order 16, and found that in our case, the Picard lattice is $U(2)\oplus D_4\oplus E_8$. This lattice is 2--elementary with $u\oplus v$ as discriminant quadratic form. In particular, this quadratic form takes values 0 or 1 (i.e. $\delta=0$).
On the other hand, $\omega_{2,1}^{-1}$ has value $\tfrac{3}{2}$ on the generator for $\mathbb{Z}/2\mathbb{Z}$. By Proposition~\ref{p:primembedding}, a primitive embedding of $U\oplus E_8\oplus A_1$ into the Picard lattice $U(2)\oplus D_4\oplus E_8$ must therefore correspond to the trivial subgroup. The existence of such a primitive embedding depends on the existence of an even lattice with invariants $(0,3,u\oplus v\oplus \omega_{2,1}^1)$. The length of this discriminant quadratic form is 5, whereas the rank of the desired lattice is 3, and so no such lattice exists (see \cite[Theorem 1.10.1]{nikulin}).
We conclude that the invariant lattice is $S_X(\sigma_{16})\cong T_{2,5,6}$, which has invariants $(11, \omega_{2,3}^{-5})$.
\end{ex}
\begin{rem}
The other case with $m=16$, $r=11$ is number 58 in Table~\ref{tab-16}. The method for computing the invariant lattice in this case is very similar to what we have just computed. Alternatively, that case can also be computed with Method III.
\end{rem}
\begin{ex}Method IV:\label{ex-meth42}
The other case that requires Method IV is $m=9$, $r=12$. This occurs for two of the K3 surfaces, namely 18a and 18b, both instances using the group $\SL_W$. Both of these cases are similar, so we describe only the first.
Using methods similar to those described in the previous examples, we get the configuration of curves depicted in Figure~\ref{fig:resoIV}.
\begin{figure}
\caption{Resolution of curves on $X_{W,G}
\label{fig:resoIV}
\end{figure}
For the discussion that follows, we denote by $E_1$ the exceptional curve in the $A_5$ chain, which intersects $C_x$.
The automorphism $\sigma_9$ permutes the three $A_2$ chains (yielding one orbit for each curve in the chain for a total of 2 orbits), but leaves the other nine exceptional curves, as well as the coordinate curves, invariant. This gives us $r=12$, and we can compute the lattice $L_\mathcal{B}\cong M_9\oplus A_2\oplus E_8$ with discriminant quadratic form $\omega_{3,1}^1\oplus\omega_{3,2}^1$. There is one isotropic subgroup of this lattice, corresponding to the overlattice $U\oplus A_2\oplus E_8$. So we must find some way to show that $L_\mathcal{B}$ is primitively embedded in the Picard lattice, for then it is the invariant lattice.
We can determine the Picard lattice $S_{X_{W,G}}$. We first notice that $\sigma_9^3$ has order 3. Furthermore, its fixed locus has invariants $(g,n,k)=(0,3,7)$. Therefore its invariant lattice $S_X(\sigma_9^3)$ is a 3-elementary lattice with invariants $(16,3)$, i.e. it is the lattice $U\oplus E_8\oplus 3A_2$, with discriminant quadratic form $3\omega_{3,1}^{1}$. Since the transcendental lattice $S_{X_{W,G}}^^{\prime}erp$ has order divisible by $^{\prime}hi(9)=6$, $S_X(\sigma_9^3)$ is the Picard lattice.
In fact, we
will determine a basis for $S_{X_{W,G}}$. Consider the set $\mathcal{E}_1$ consisting of all of the exceptional curves (15 of them), and $\mathcal{K}$ consisting of all irreducible components of (the strict transforms of) the coordinate curves. Let $\mathcal{B}_1=\mathcal{E}_1\cup\mathcal{K}$. This set generates $S_{X_{W,G}}$. One can check by direct computation (\cite{magma}) that $C_x$ and $E_1$ are redundant.
Now we consider the set $\mathcal{B}$, generating the lattice $L_\mathcal{B}$. Again, we compute that $C_x$ and $E_1$ are redundant, so we get $L_\mathcal{B}$ generated by the two orbits from the $A_2$ chains, the remaining exceptional curves, and $C_w$ and $C_z$. Two of the generators for $L_\mathcal{B}$ are just sums of generators of $S_{X_{W,G}}$. Thus a change of basis shows that $S_X/L_\mathcal{B}$ is a free group of rank 4, and so $L_\mathcal{B}$ is primitively embedded.
\end{ex}
The other example is similar. Instead of three $A_2$ chains, there are three $A_1$'s. One can check that the set $\set{y=0}$ is composed of three curves, each of genus zero. Each of these curves intersects one of the $A_1$ curves. These are permuted by the action of $\sigma_9$. Up to a relabelling, we obtain the same configuration of curves, and the same Picard lattice.
\label{ex-noncyclic}
The only cases where $\SL_W/J_W$ is not cyclic are $x^3+y^3+z^6+w^6$ (number 3d) and $x^2+y^4+z^6+w^{12}$ (number 8d) in Table~\ref{tab-6}.
We analyze them separately.
\begin{ex}
The first polynomial we consider is $W=x^3+y^3+z^6+w^6$ in $\mathbb{P}(2,2,1,1)$. This is number 3d in Table~\ref{tab-6}.
In this case the order of $\SL_W/J_W$ is 9 and it results that $\SL_W/J_W=\mathbb{Z}/3\mathbb{Z}\oplus\mathbb{Z}/3\mathbb{Z}$ since there are no elements of order 9.
The group $J_W$ is generated by $j_W=(\frac12,\frac13,\frac16,\frac16)$ and two generators for $\SL_W/J_W$ are $g_1=(\frac13,\frac23,0,0)$ and $g_2=(\frac13,\frac13,\frac13,0)$.
Also we name $g_3=(\frac13,0,\frac23,0)$ and $g_4=(0,\frac13,0,\frac23)$.
There are four subgroups of $\SL_W/J_W$ of order 3,
namely $G_i=<g_i,\J{W}>, i=1,2,3,4$.
We can also observe that $G_1^T=G_2$, $G_3^T=G_3$ and $G_4^T=G_4$.
Now we consider the non-symplectic automorphism $\sigma_{6}=(0,0,0,\tfrac{1}{6})$. One may notice, there is another automorphism of order 6, namely $(0,0,\tfrac{1}{6},0)$, but due to symmetry (i.e. exchanging $z$ and $w$), we must only consider one of them.
Using the same methods as before for each group, we can compute the invariant lattice for the corresponding K3 surface. In each case, the lattice $L_\mathcal{B}$ has no overlattices, so we can use Method I. When $G=G_3$, the invariant lattice has $(r,q)=(10,4w_{3,1}^1)$, which is self-dual. The same is true for $G=G_4$.
When $G=G_1$ we get an invariant lattice with rank 16 and the discriminant form is $v\oplus w_{3,1}^1$. This is the dual of the invariant lattice we get with the choice $G=G_2$ and so it proves the theorem for this case.
\end{ex}
\begin{ex}
Finally, we examine the polynomial $W=x^2+y^4+z^6+w^{12}$ in $\mathbb{P}(6,3,2,1)$. This is number 8d in Table~\ref{tab-6}. There are non--symplectic automorphisms of order 2, 4 and 12, but we again focus on the non-symplectic automorphism of order 6: $\sigma_6=(0,0,\tfrac{1}{6},0)$.
The order of $\SL_W/J_W$ is 4 and since there are no elements of order 4, we conclude that $\SL_W/J_W=\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$.
The elements
\[g_1=\left(\frac12,0,\frac12,0\right),\ g_2=\left(0,\frac12,\frac12,0\right),\ g_3=\left(\frac12,\frac12,0,0\right)\]
each have order 2 and represent different cosets in $\SL_W/J_W$; let $G_i:=<g_i,\J{W}>,i=1,2,3$.
Observe that $G_1^T=G_1$ while $G_2^T=G_3$.
When $G=G_1$, with Method I we compute the invariant lattice and obtain $(r,q)=(10,v\oplus 4w_{2,1}^{-1})$.
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In fact, we
will determine a basis for $S_{X_{W,G}}$. Consider the set $\mathcal{E}_1$ consisting of all of the exceptional curves (15 of them), and $\mathcal{K}$ consisting of all irreducible components of (the strict transforms of) the coordinate curves. Let $\mathcal{B}_1=\mathcal{E}_1\cup\mathcal{K}$. This set generates $S_{X_{W,G}}$. One can check by direct computation (\cite{magma}) that $C_x$ and $E_1$ are redundant.
Now we consider the set $\mathcal{B}$, generating the lattice $L_\mathcal{B}$. Again, we compute that $C_x$ and $E_1$ are redundant, so we get $L_\mathcal{B}$ generated by the two orbits from the $A_2$ chains, the remaining exceptional curves, and $C_w$ and $C_z$. Two of the generators for $L_\mathcal{B}$ are just sums of generators of $S_{X_{W,G}}$. Thus a change of basis shows that $S_X/L_\mathcal{B}$ is a free group of rank 4, and so $L_\mathcal{B}$ is primitively embedded.
\end{ex}
The other example is similar. Instead of three $A_2$ chains, there are three $A_1$'s. One can check that the set $\set{y=0}$ is composed of three curves, each of genus zero. Each of these curves intersects one of the $A_1$ curves. These are permuted by the action of $\sigma_9$. Up to a relabelling, we obtain the same configuration of curves, and the same Picard lattice.
\label{ex-noncyclic}
The only cases where $\SL_W/J_W$ is not cyclic are $x^3+y^3+z^6+w^6$ (number 3d) and $x^2+y^4+z^6+w^{12}$ (number 8d) in Table~\ref{tab-6}.
We analyze them separately.
\begin{ex}
The first polynomial we consider is $W=x^3+y^3+z^6+w^6$ in $\mathbb{P}(2,2,1,1)$. This is number 3d in Table~\ref{tab-6}.
In this case the order of $\SL_W/J_W$ is 9 and it results that $\SL_W/J_W=\mathbb{Z}/3\mathbb{Z}\oplus\mathbb{Z}/3\mathbb{Z}$ since there are no elements of order 9.
The group $J_W$ is generated by $j_W=(\frac12,\frac13,\frac16,\frac16)$ and two generators for $\SL_W/J_W$ are $g_1=(\frac13,\frac23,0,0)$ and $g_2=(\frac13,\frac13,\frac13,0)$.
Also we name $g_3=(\frac13,0,\frac23,0)$ and $g_4=(0,\frac13,0,\frac23)$.
There are four subgroups of $\SL_W/J_W$ of order 3,
namely $G_i=<g_i,\J{W}>, i=1,2,3,4$.
We can also observe that $G_1^T=G_2$, $G_3^T=G_3$ and $G_4^T=G_4$.
Now we consider the non-symplectic automorphism $\sigma_{6}=(0,0,0,\tfrac{1}{6})$. One may notice, there is another automorphism of order 6, namely $(0,0,\tfrac{1}{6},0)$, but due to symmetry (i.e. exchanging $z$ and $w$), we must only consider one of them.
Using the same methods as before for each group, we can compute the invariant lattice for the corresponding K3 surface. In each case, the lattice $L_\mathcal{B}$ has no overlattices, so we can use Method I. When $G=G_3$, the invariant lattice has $(r,q)=(10,4w_{3,1}^1)$, which is self-dual. The same is true for $G=G_4$.
When $G=G_1$ we get an invariant lattice with rank 16 and the discriminant form is $v\oplus w_{3,1}^1$. This is the dual of the invariant lattice we get with the choice $G=G_2$ and so it proves the theorem for this case.
\end{ex}
\begin{ex}
Finally, we examine the polynomial $W=x^2+y^4+z^6+w^{12}$ in $\mathbb{P}(6,3,2,1)$. This is number 8d in Table~\ref{tab-6}. There are non--symplectic automorphisms of order 2, 4 and 12, but we again focus on the non-symplectic automorphism of order 6: $\sigma_6=(0,0,\tfrac{1}{6},0)$.
The order of $\SL_W/J_W$ is 4 and since there are no elements of order 4, we conclude that $\SL_W/J_W=\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$.
The elements
\[g_1=\left(\frac12,0,\frac12,0\right),\ g_2=\left(0,\frac12,\frac12,0\right),\ g_3=\left(\frac12,\frac12,0,0\right)\]
each have order 2 and represent different cosets in $\SL_W/J_W$; let $G_i:=<g_i,\J{W}>,i=1,2,3$.
Observe that $G_1^T=G_1$ while $G_2^T=G_3$.
When $G=G_1$, with Method I we compute the invariant lattice and obtain $(r,q)=(10,v\oplus 4w_{2,1}^{-1})$.
As for $G_2$, we use again Method I and obtain $(r,q)=(14,2w_{2,1}^{-1}\oplus w_{3,1}^1)$, while for $G_3$ we get $(r,q)=(6,2w_{2,1}^{1}\oplus w_{3,1}^{-1})$.
Observing they are mirror of each other, we can conclude that the Theorem is proved in this case.
\end{ex}
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\section{Tables}
\label{sec:tables}
\footnotesize{
In each table, we have arranged the surfaces by weight system. Each weight system is listed by the number assigned to it by Yonemura in \cite{yonemura}. In each weight system, we have listed all possible invertible polynomials of the form \eqref{eq-W} with non--symplectic automorphism of order $m$, and for each polynomial, we list the orders of the possible groups $G/J_W$ satisfying $J_W\subseteq G\subseteq \SL_W$. The invariants $(r,q_{S_X(\sigma_m)})$ are then given, as well as the number of the BHK mirror dual. Finally, we have also indicated which method was used to determine $q_{S_X(\sigma_m)}$.
When consulting the tables, it will be helpful to know that $\omega_{5,1}^\varepsilonilon=-\omega_{5,1}^\varepsilonilon$, and that $4\omega_{3,1}^{-1}=4\omega_{3,1}^1$, $4\omega_{2,1}^{-1}=4\omega_{2,1}^1$. The first fact follows simply by definition. The latter two follow from \cite[Theorem 1.8.2]{nikulin}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
14& (21,14,6,1;42)& $x^2+y^3+z^7+w^{42}$& 1&1&$(10,<0>)$&14&I\\
\caption{Table for $m=42$} \label{tab-42} \end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
38& (15,8,6,1;30) &$x^2+y^3z+z^5+w^{30}$& 1&1&$(11,w_{2,1}^{-1})$ &50&I \\
50& (15,10,4,1;30) &$x^2+y^3+yz^5+w^{30}$& 1&1&$(9,w_{2,1}^{1})$ &38&I \\
\caption{Table for $m=30$} \label{tab-30} \end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
13a &(12,8,3,1;24) &$x^2+y^3+xz^4+w^{24}$ &1 &1 &$(8,w_{3,1}^{-1})$ &20&I\\
13b &(12,8,3,1;24) &$x^2+y^3+z^8+w^{24}$ &2 &2 &$(12, w_{3,1}^{1})$ &13b&I\\
& & & &1 &$(8,w_{3,1}^{-1})$ &13b&I\\
20 &(9,8,6,1;24) &$x^2z+y^3+z^4+w^{24}$ &1 &1 &$(12, w_{3,1}^{1})$ &13a&I\\
\caption{Table for $m=24$} \label{tab-24}
\end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
78 &(11,6,4,1;22) &$x^2+y^3z+yz^4+w^{22}$ &1 &1 &$(10,w_{2,1}^{-1}\oplus w_{2,1}^{1})$ &78&II\\
\caption{Table for $m=22$} \label{tab-22} \end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
9a &(10,5,4,1;20) &$x^2+xy^2+z^5+w^{20}$ &1 &1 &$(10,w_{5,1}^{-1})$ &9a&I\\
9b &(10,5,4,1;20) &$x^2+y^4+z^5+w^{20}$ &2 &2 &$(10,w_{5,1}^{-1})$ &9b&I\\
& & & &1 &$(10,w_{5,1}^{-1})$ &9b&I\\
\caption{Table for $m=20$} \label{tab-20} \end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
12a &(9,6,2,1;18) &$x^2+y^3+yz^6+w^{18}$ &2 &2 &$(11,w_{2,1}^1\oplus w_{3,1}^1)$ &39a&I\\
& & & &1 &$(6,v)$ &39a&I\\
12b &(9,6,2,1;18) &$x^2+y^3+z^9+w^{18}$ &3 &3 &$(14,v)$ &12b&I\\
& & & &1 &$(6,v)$ &12b&I\\
39a &(9,5,3,1;18) &$x^2+y^3z+z^6+w^{18}$ &2 &2 &$(14,v)$ &12a&I\\
& & & &1 &$(9,w_{2,1}^{-1}\oplus w_{3,1}^{-1})$ &12a&I\\
39b &(9,5,3,1;18) &$x^2+y^3z+xz^3+w^{18}$ &1 &1 &$(9,w_{2,1}^{-1}\oplus w_{3,1}^{-1})$ &60&I\\
60 &(7,6,4,1;18) &$x^2z+y^3+yz^3+w^{18}$ &1 &1 &$(11,w_{2,1}^1\oplus w_{3,1}^1)$ &39b&I\\
\caption{Table for $m=18$} \label{tab-18} \end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
37a &(8,4,3,1;16) &$x^2+xy^2+yz^4+w^{16}$ &1 &1 &$(9,w_{2,3}^{5})$ &58&III\\
37b &(8,4,3,1;16) &$x^2+y^4+yz^4+w^{16}$ &2 &2 &$(11,w_{2,3}^{-5})$ &37b&IV\\
& & & &1 &$(9,w_{2,3}^{5})$ &37b&III\\
58 &(6,5,4,1;16) &$x^2z+xy^2+z^4+w^{16}$ &1 &1 &$(11,w_{2,3}^{-5})$ &37a&IV\\
\caption{Table for $m=16$} \label{tab-16} \end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
11a &(15,10,3,2;30) &$x^2+y^3+xz^5+w^{15}$ &1 &1 &$(10,w_{3,1}^{-1}\oplus w_{3,1}^{1})$ &22a&II\\
11b &(15,10,3,2;30) &$x^2+y^3+z^{10}+w^{15}$ &1 &1 &$(10,w_{3,1}^{-1}\oplus w_{3,1}^{1})$ &11b&II\\
22a &(6,5,3,1;15) &$x^2z+y^3+z^5+w^{15}$ &1 &1 &$(10,w_{3,1}^{-1}\oplus w_{3,1}^{1})$ &11a&II\\
22b &(6,5,3,1;15) &$x^2z+y^3+xz^3+w^{15}$ &1 &1 &$(10,w_{3,1}^{-1}\oplus w_{3,1}^{1})$ &22b&II\\
\caption{Table for $m=15$} \label{tab-15}
\end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
40a &(7,4,2,1;14) &$x^2+y^3z+z^{7}+w^{14}$ &1 &1 &$(7,v\oplus w_{2,1}^{-1})$ &47 &III\\
40b &(7,4,2,1;14) &$x^2+y^3z+yz^{5}+w^{14}$ &2 &2 &$(13,v\oplus w_{2,1}^{1})$ &40b&II \\
& & & &1 &$(7,v\oplus w_{2,1}^{-1})$ &40b &III\\
47 &(21,14,4,3;42) &$x^2+y^3+yz^{7}+w^{14}$ &1 &1 &$(13,v\oplus w_{2,1}^{1})$ &40a&II \\
\caption{Table for $m=14$} \label{tab-14} \end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
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\caption{Table for $m=18$} \label{tab-18} \end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
37a &(8,4,3,1;16) &$x^2+xy^2+yz^4+w^{16}$ &1 &1 &$(9,w_{2,3}^{5})$ &58&III\\
37b &(8,4,3,1;16) &$x^2+y^4+yz^4+w^{16}$ &2 &2 &$(11,w_{2,3}^{-5})$ &37b&IV\\
& & & &1 &$(9,w_{2,3}^{5})$ &37b&III\\
58 &(6,5,4,1;16) &$x^2z+xy^2+z^4+w^{16}$ &1 &1 &$(11,w_{2,3}^{-5})$ &37a&IV\\
\caption{Table for $m=16$} \label{tab-16} \end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
11a &(15,10,3,2;30) &$x^2+y^3+xz^5+w^{15}$ &1 &1 &$(10,w_{3,1}^{-1}\oplus w_{3,1}^{1})$ &22a&II\\
11b &(15,10,3,2;30) &$x^2+y^3+z^{10}+w^{15}$ &1 &1 &$(10,w_{3,1}^{-1}\oplus w_{3,1}^{1})$ &11b&II\\
22a &(6,5,3,1;15) &$x^2z+y^3+z^5+w^{15}$ &1 &1 &$(10,w_{3,1}^{-1}\oplus w_{3,1}^{1})$ &11a&II\\
22b &(6,5,3,1;15) &$x^2z+y^3+xz^3+w^{15}$ &1 &1 &$(10,w_{3,1}^{-1}\oplus w_{3,1}^{1})$ &22b&II\\
\caption{Table for $m=15$} \label{tab-15}
\end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
40a &(7,4,2,1;14) &$x^2+y^3z+z^{7}+w^{14}$ &1 &1 &$(7,v\oplus w_{2,1}^{-1})$ &47 &III\\
40b &(7,4,2,1;14) &$x^2+y^3z+yz^{5}+w^{14}$ &2 &2 &$(13,v\oplus w_{2,1}^{1})$ &40b&II \\
& & & &1 &$(7,v\oplus w_{2,1}^{-1})$ &40b &III\\
47 &(21,14,4,3;42) &$x^2+y^3+yz^{7}+w^{14}$ &1 &1 &$(13,v\oplus w_{2,1}^{1})$ &40a&II \\
\caption{Table for $m=14$} \label{tab-14} \end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
6a &(5,2,2,1;10) &$x^2+y^4z+z^5+w^{10}$ &2 &2 &$(8,w_{5,1}^{-1}\oplus 2w_{2,1}^1)$ &36a&I \\
& & & &1 &$(6,u\oplus v)$ &36a&II \\
6b &(5,2,2,1;10) &$x^2+y^5+z^5+w^{10}$ &5 &5 &$(14,u\oplus v)$ &6b &II\\
& & & &1 &$(6,u\oplus v)$ &6b &II\\
6c &(5,2,2,1;10) &$x^2+y^4z+yz^4+w^{10}$ &3 &3 &$(14,u\oplus v)$ &6c &II\\
& & & &1 &$(6,u\oplus v)$ &6c &II\\
11a &(15,10,3,2;30) &$x^2+y^3+z^{10}+yw^{10}$ &2 &2 &$(17,w_{2,1}^{1})$ &42a&I \\
& & & &1 &$(10,v\oplus v)$ &42a&II \\
11b &(15,10,3,2;30)&$x^2+y^3+z^{10}+w^{15}$ &1 &1 &$(10,v\oplus v)$ &11b&II \\
36a &(10,5,3,2;20) &$x^2+y^4+yz^5+w^{10}$ &2 &2 &$(14,u\oplus v)$ &6a &II\\
& & & &1 &$(12,w_{5,1}^{-1}\oplus 2w_{2,1}^{-1})$ &6a&I \\
36b &(10,5,3,2;20) &$x^2+xy^2+yz^5+w^{10}$ &1 &1 &$(12,w_{5,1}^{-1}\oplus 2w_{2,1}^{-1})$ &63&I \\
42a &(5,3,1,1;10) &$x^2+y^3w+z^{10}+w^{10}$ &2 &2 &$(10,v\oplus v)$ &11a&II \\
& & & &1 &$(3,w_{2,1}^{-1})$ &11a&I \\
42b &(5,3,1,1;10) &$x^2+y^3z+xz^5+w^{10}$ &1 &1 &$(3,w_{2,1}^{-1})$ &68&I \\
42c &(5,3,1,1;10) &$x^2+y^3z+yz^7+w^{10}$ &4 &4 &$(17,w_{2,1}^{1})$ &42c&I \\
& & & &2 &$(10,v\oplus v)$ &42c&II\\
& & & &1 &$(3,w_{2,1}^{-1})$ &42c&I \\
63 &(4,3,2,1;10) &$x^2z+y^2x+z^5+w^{10}$ &1 &1 &$(8,w_{5,1}^{-1}\oplus 2w_{2,1}^1)$ &36b&I \\
68 &(13,10,4,3;30) &$x^2z+y^3+yz^5+w^{10}$ &1 &1 &$(17,w_{2,1}^{1})$ &42b&I \\
\caption{Table for $m=10$} \label{tab-10} \end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
12a &(9,6,2,1;18) &$x^2+y^3+z^9+xw^9$ &3 &3 &$(16,w_{3,1}^{-1})$ &25a&I \\
& & & &1 &$(4,w_{3,1}^1)$ &25a&I \\
12b &(9,6,2,1;18) &$x^2+y^3+z^9+yw^{12}$ &1 &1 &$(4,w_{3,1}^1)$ &43a&I \\
12c &(9,6,2,1;18) &$x^2+y^3+z^9+w^{18}$ &3 &3 &$(16,w_{3,1}^{-1})$ &12c&I \\
& & & &1 &$(4,w_{3,1}^1)$ &12c&I \\
18a &(3,3,2,1;9) &$x^3+y^3+xz^3+w^9$ &3 &3 &$(12,w_{3,1}^1\oplus w_{3,2}^1)$ &18a&IV \\
& & & &1 &$(8,w_{3,1}^{-1}\oplus w_{3,2}^{-1})$ &18a&III \\
18b &(3,3,2,1;9) &$x^3+xy^2+yz^3+w^9$ &2 &2 &$(12,w_{3,1}^1\oplus w_{3,2}^1)$ &18b&IV \\
& & & &1 &$(8,w_{3,1}^{-1}\oplus w_{3,2}^{-1})$ &18b&III \\
25a &(4,3,1,1;9) &$x^2w+y^3+z^9+w^9$ &3 &3 &$(16,w_{3,1}^{-1})$ &12a&I \\
& & & &1 &$(4,w_{3,1}^1)$ &12a&I \\
25b &(4,3,1,1;9) &$x^2w+y^3+z^9+yw^6$ &1 &1 &$(4,w_{3,1}^1)$ &43b&I \\
25c &(4,3,1,1;9) &$x^2w+y^3+z^9+xw^5$ &3 &3 &$(16,w_{3,1}^{-1})$ &25a&I \\
& & & &1 &$(4,w_{3,1}^1)$ &25a&I \\
43a &(18,11,4,3;36) &$x^2+y^3w+z^9+w^{12}$ &1 &1 &$(16,w_{3,1}^{-1})$ &12b&I \\
43b &(18,11,4,3;36) &$x^2+y^3w+z^9+xw^6$ &1 &1 &$(16,w_{3,1}^{-1})$ &25b&I \\
\caption{Table for $m=9$} \label{tab-9} \end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
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\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
12a &(9,6,2,1;18) &$x^2+y^3+z^9+xw^9$ &3 &3 &$(16,w_{3,1}^{-1})$ &25a&I \\
& & & &1 &$(4,w_{3,1}^1)$ &25a&I \\
12b &(9,6,2,1;18) &$x^2+y^3+z^9+yw^{12}$ &1 &1 &$(4,w_{3,1}^1)$ &43a&I \\
12c &(9,6,2,1;18) &$x^2+y^3+z^9+w^{18}$ &3 &3 &$(16,w_{3,1}^{-1})$ &12c&I \\
& & & &1 &$(4,w_{3,1}^1)$ &12c&I \\
18a &(3,3,2,1;9) &$x^3+y^3+xz^3+w^9$ &3 &3 &$(12,w_{3,1}^1\oplus w_{3,2}^1)$ &18a&IV \\
& & & &1 &$(8,w_{3,1}^{-1}\oplus w_{3,2}^{-1})$ &18a&III \\
18b &(3,3,2,1;9) &$x^3+xy^2+yz^3+w^9$ &2 &2 &$(12,w_{3,1}^1\oplus w_{3,2}^1)$ &18b&IV \\
& & & &1 &$(8,w_{3,1}^{-1}\oplus w_{3,2}^{-1})$ &18b&III \\
25a &(4,3,1,1;9) &$x^2w+y^3+z^9+w^9$ &3 &3 &$(16,w_{3,1}^{-1})$ &12a&I \\
& & & &1 &$(4,w_{3,1}^1)$ &12a&I \\
25b &(4,3,1,1;9) &$x^2w+y^3+z^9+yw^6$ &1 &1 &$(4,w_{3,1}^1)$ &43b&I \\
25c &(4,3,1,1;9) &$x^2w+y^3+z^9+xw^5$ &3 &3 &$(16,w_{3,1}^{-1})$ &25a&I \\
& & & &1 &$(4,w_{3,1}^1)$ &25a&I \\
43a &(18,11,4,3;36) &$x^2+y^3w+z^9+w^{12}$ &1 &1 &$(16,w_{3,1}^{-1})$ &12b&I \\
43b &(18,11,4,3;36) &$x^2+y^3w+z^9+xw^6$ &1 &1 &$(16,w_{3,1}^{-1})$ &25b&I \\
\caption{Table for $m=9$} \label{tab-9} \end{longtable}
\begin{longtable}{p{12pt}|c|c|c|c|c|c|c}
\multicolumn{7}{c}{}\\
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\
\hline
\midrule
\endfirsthead
No. &Weights&Polynomial&SL/J&G/J&$(r,q)$&BHK dual&Method\\\hline \midrule
\endhead
2a &(4,3,3,2;12) &$x^3+y^3z+z^4+w^6$ &3 &3 &$(16,v\oplus w_{3,1}^1)$ &3a&I \\
& & & &1 &$(10,4w_{3,1}^1)$ &3a&II \\
2b &(4,3,3,2;12) &$x^3+y^3z+yz^3+w^6$ &1 &1 &$(10,4w_{3,1}^1)$ &2b &II\\
2c &(4,3,3,2;12) &$x^3+y^4+z^4+w^6$ &2 &2 &$(10,4w_{3,1}^1)$ &2c&II \\
& & & &1 &$(10,4w_{3,1}^1)$ &2c&II \\
3a &(2,2,1,1;6) &$x^3+y^3+yz^4+w^6$ &3 &3 &$(10,4w_{3,1}^1)$ &2a &II\\
& & & &1 &$(4,v\oplus w_{3,1}^{-1})$ &2a &I\\
3b &(2,2,1,1;6) &$x^2y+y^3+z^6+w^6$ &6 &6 &$(19,w_{2,1}^{-1})$ &5a &I\\
& & & &3 &$(16,v\oplus w_{3,1}^1)$ &5a&I \\
& & & &2 &$(9,3w_{2,1}^{-1}\oplus 2w_{3,1}^{-1})$ &5a&I \\
& & & &1 &$(4,v\oplus w_{3,1}^{-1})$ &5a &I\\
3c &(2,2,1,1;6) &$x^3+xy^2+yz^4+w^6$ &1 &1 &$(4,v\oplus w_{3,1}^{-1})$ &57 &I\\
3d &(2,2,1,1;6) &$x^3+y^3+z^6+w^6$ &9 &9 &$(16,v\oplus w_{3,1}^1)$ &3d&I \\
& & & &3 &$(16,v\oplus w_{3,1}^1)$ &3d &I\\
& & & &3 &$(10,4w_{3,1}^1)$ &3d&II \\
& & & &3 &$(10,4w_{3,1}^1)$ &3d&II \\
& & & &3 &$(4,v\oplus w_{3,1}^{-1})$ &3d &I\\
& & & &1 &$(4,v\oplus w_{3,1}^{-1})$ &3d &I\\
3e &(2,2,1,1;6) &$x^2y+xy^2+z^6+w^6$ &3 &3 &$(16,v\oplus w_{3,1}^1)$ &3e &I\\
& & & &1 &$(4,v\oplus w_{3,1}^{-1})$ &3e &I\\
5a &(3,1,1,1;6) &$x^2+xy^3+z^6+w^6$ &6 &6 &$(16,v\oplus w_{3,1}^1)$ &3b&I \\
& & & &3 &$(11,3w_{2,1}^1\oplus 2w_{3,1}^{1})$ &3b&I \\
& & & &2 &$(4,v\oplus w_{3,1}^{-1})$ &3b &I\\
& & & &1 &$(1,w_{2,1}^1)$ &3b&I \\
5b &(3,1,1,1;6) &$x^2+y^5w+z^6+w^6$ &2 &2 &$(8,6w_{2,1}^{-1})$ &29 &II\\
& & & &1 &$(1,w_{2,1}^1)$ &29&I \\
5c &(3,1,1,1;6) &$x^2+xy^3+yz^5+w^6$ &1 &1 &$(1,w_{2,1}^1)$ &56 &I\\
5d &(3,1,1,1;6) &$x^2+y^6+z^5w+zw^5$ &8 &8 &$(19,w_{2,1}^{-1})$ &5d &I\\
& & & &4 &$(12,6w_{2,1}^1)$ &5d&II \\
& & & &2 &$(8,6w_{2,1}^{-1})$ &5d &II\\
& & & &1 &$(1,w_{2,1}^1)$ &5d&I \\
5e &(3,1,1,1;6) &$x^2+y^6+z^6+w^6$ &12 &12 &$(19,w_{2,1}^{-1})$ &5e &I\\
& & & &6 &$(12,6w_{2,1}^1)$ &5e &II\\
& & & &4 &$(9,3w_{2,1}^{-1}\oplus 2w_{3,1}^{-1})$ &5e&I \\
& & & &3 &$(11,3w_{2,1}^1\oplus 2w_{3,1}^{1})$ &5e &I\\
& & & &2 &$(8,6w_{2,1}^{-1})$ &5e&II \\
& & & &1 &$(1,w_{2,1}^1)$ &5e &I\\
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8a &(6,3,2,1;12) &$x^2+y^4+z^6+xw^6$ &2 &2 &$(14,2w_{2,1}^{-1}\oplus w_{3,1}^1)$ &23 &I\\
& & & &1 &$(6,2w_{2,1}^1\oplus w_{3,1}^{-1})$ &23 &I\\
8b &(6,3,2,1;12) &$x^2+y^4+z^6+yw^{9}$ &2 &2 &$(10,v\oplus 4w_{2,1}^{-1})$ &33a&I \\
& & & &1 &$(6,2w_{2,1}^1\oplus w_{3,1}^{-1})$ &33a &I\\
8c &(6,3,2,1;12) &$x^2+xy^2+z^6+yw^{9}$ &1 &1 &$(6,2w_{2,1}^1\oplus w_{3,1}^{-1})$ &70 &I\\
8d &(6,3,2,1;12) &$x^2+y^4+z^6+w^{12}$ &4 &4 &$(14,2w_{2,1}^{-1}\oplus w_{3,1}^1)$ &8d&I \\
& & & &2 &$(14,2w_{2,1}^{-1}\oplus w_{3,1}^1)$ &8d &I\\
& & & &2 &$(6,2w_{2,1}^1\oplus w_{3,1}^{-1})$ &8d&I \\
& & & &2 &$(10,v\oplus 4w_{2,1}^{-1})$ &8d&II \\
& & & &1 &$(6,2w_{2,1}^1\oplus w_{3,1}^{-1})$ &8d &I\\
8e &(6,3,2,1;12) &$x^2+xy^2+z^6+w^{12}$ &2 &2 &$(14,2w_{2,1}^{-1}\oplus w_{3,1}^1)$ &8e&I \\
& & & &1 &$(6,2w_{2,1}^1\oplus w_{3,1}^{-1})$ &8e &I\\
23 &(5,3,2,2;12) &$x^2w+y^4+z^6+w^6$ &2 &2 &$(14,2w_{2,1}^{-1}\oplus w_{3,1}^1)$ &8a &I\\
& & & &1 &$(6,2w_{2,1}^1\oplus w_{3,1}^{-1})$ &8a &I\\
29 &(15,6,5,4;30) &$x^2+y^5+z^6+yw^6$ &2 &2 &$(19,w_{2,1}^{-1})$ &5b&I \\
& & & &1 &$(12,6w_{2,1}^1)$ &5b&II \\
33a &(9,4,3,2;18) &$x^2+y^4w+z^6+w^{9}$ &2 &2 &$(14,2w_{2,1}^{-1}\oplus w_{3,1}^1)$ &8b &I\\
& & & &1 &$(10,v\oplus 4w_{2,1}^{-1})$ &8b &II\\
33b &(9,4,3,2;18) &$x^2+y^4w+z^6+yw^{7}$&1 &1 &$(10,v\oplus 4w_{2,1}^{-1})$ &33b&II \\
56 &(11,8,6,5;30) &$x^2y+y^3z+z^5+w^6$ &1 &1 &$(19,w_{2,1}^{-1})$ &5c &I\\
57 &(9,6,5,4;24) &$x^2y+y^4+xz^3+w^6$ &1 &1 &$(16,v\oplus w_{3,1}^1)$ &3c &I\\
70 &(8,5,3,2;18) &$x^2w+xy^2+z^6+w^{9}$&1 &1 &$(14,2w_{2,1}^{-1}\oplus w_{3,1}^1)$ &8c &I\\
\caption{Table for $m=6$} \label{tab-6} \end{longtable}
}
\normalsize
\appendix
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\section{Computer code for computing lattices}
In order to compute the lattices using the configuration of curves, we used the following Magma code, developed by Antonio Laface and added here with his permission.
This first function takes an even bilinear form $B$, and outputs generators of the discriminant group and the values of $q_B$ on these generators.
\begin{alltt}
disc:=function(M)
S,A,B:=SmithForm(M);
l:=[[S[i,i],i]: i in [1..NumberOfColumns(S)]| S[i,i] notin {0,1}];
sA:=Matrix(Rationals(),ColumnSubmatrixRange(B,l[1][2],l[#l][2]));
for i in [1..#l] do
MultiplyColumn(\(\sim\)sA,1/l[i][1],i);
end for;
Q:=Transpose(sA)*Matrix(Rationals(),M)*sA;
for i,j in [1..NumberOfColumns(Q)] do
if i ne j then
Q[i,j]:=Q[i,j]-Floor(Q[i,j]);
else
Q[i,j]:=Q[i,j]-Floor(Q[i,j])+ (Floor(Q[i,j]) mod 2);
end if;
end for;
return [l[i][1]: i in [1..#l]], Q;
end function;
\end{alltt}
The next function determines whether a given even bilinear form has overlattices. Input is a matrix $M$ and a number $n$. The output is the subgroup of $A_L$ that takes values equal to $n$ modulo $2\mathbb{Z}$. For isotropic subgroups of the discriminant group, use $n=0$.
\begin{verbatim}
isot:=function(M,n)
v,U:=disc(M);
Q:=Rationals();
A:=AbelianGroup(v);
return [Eltseq(a) : a in A |
mod2(Matrix(Q,1,#v,Eltseq(a))*U*Matrix(Q,#v,1,Eltseq(a)))[1,1] eq n];
end function;
\end{verbatim}
The function {\verb mod2 } is as follows:
\begin{verbatim}
mod2:=function(Q);
for i,j in [1..Nrows(Q)] do
if i ne j then Q[i,j]:=Q[i,j]-Floor(Q[i,j]);
else Q[i,j]:=Q[i,j]-2*Floor(Q[i,j]/2);
end if;
end for;
return Q;
end function;
\end{verbatim}
Finally, the following function compares two discriminant quadratic forms, and lets us know if they are the same finite quadratic form or not. This is not always easy to check due to the relations in Proposition~\ref{t:relations}.
\begin{verbatim}
dicompare:=function(M,Q)
v,U:=disc(M);
w,D:=disc(Q);
if v ne w then return false; end if;
A:=AbelianGroup(v);
Aut:=AutomorphismGroup(A);
f,G:=PermutationRepresentation(Aut);
h:=Inverse(f);
ll:=[Matrix(Rationals(),[Eltseq(Image(h(g),A.i)) : i in [1..Ngens(A)]]) : g in G];
dd:=[mod2(a*U*Transpose(a)) : a in ll];
return D in dd;
end function;
\end{verbatim}
\end{document}
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\begin{document}
\title{The cohomology of the free loop spaces of $SU(n+1)/T^n$}
\author{Matthew I. Burfitt}
\address{\scriptsize{Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom}}
\email{[email protected]}
\author{Jelena Grbi\'c}
\address{\scriptsize{School of Mathematics, University of Southampton,Southampton SO17 1BJ, United Kingdom}}
\email{[email protected]}
\subjclass[2010]{}
\keywords{}
\thanks{Research supported in part by The Leverhulme Trust Research Project Grant RPG-2012-560.}
\maketitle
\begin{abstract}
We study the cohomology of the free loop space of $SU(n+1)/T^n$, the simplest example of a complete flag manifolds and an important homogeneous space. Through this enhanced analysis we reveal rich new combinatorial structures arising in the cohomology algebra of the free loop spaces. We build new theory to allow for the computation of $H^*(\Lambda(SU(n+1)/T^{n});\mathbb{Z})$, a significantly more complicated structure than other known examples. In addition to our theoretical results, we explicitly implement a novel integral Gr\"obner basis procedure for computation. This procedure is applicable to any Leray-Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra. The power of this procedure is illustrated by the explicit calculation of $H^*( \Lambda(SU(4)/T^3);\mathbb{Z})$. We also provide a python library with examples of all procedures used in the paper.
\end{abstract}
\section{Introduction}
The free loop space $\Lambda X$ of a topological space $X$ is defined to be the mapping space $Map(S^1,X)$, the space of all unpointed maps from the circle to $X$.
This differs from the based loops space $\Omega X=Map_*(S^1,X)$, the space of all pointed maps from the circle to $X$.
The two loop spaces are connected by the evaluation fibration.
The based loop space functor is an important classical object in algebraic topology and has been well studied.
However, the topology of free loop spaces, while required for many applications,
behaves in a much more complex way
and is still only well understood in a handful of examples.
A primary motivation for studying the topology of the free loop space is the important role loops on a manifold play in both mathematics and physics.
Given a Riemannian manifold $(M,g)$, the closed geodesics parametrised by $S^1$ are the critical points of the energy functional
\begin{equation*}\label{eq:Energy}
E\colon \Lambda M\to \mathbb R, \quad E(\gamma):= \frac{1}{2}\int_{S^1} ||\dot{\gamma}(t)||^2 dt.
\end{equation*}
Morse theory applied to the energy functional $E$ gives a description of the loop space $\Lambda M$ by successive attachments of bundles over the critical submanifolds.
Knowledge of the topology of $\Lambda M$ therefore implies existence results for critical points of $E$.
Computations of the cohomology of the free loop space have therefore received much attention over the last several decades.
In the simplest case when $X$ is an $H$-space,
there is a homotopy equivalence
\[
\Lambda X \simeq \Omega X \times X.
\]
Progress past this is restricted to specific examples by applying specialised methods relevant to their particular case.
Broadly speaking there are three main related approaches to studying the cohomology of the free loop space:
the Hochschild cohomology of the normalized singular chains on the base loop space \cite{Goodwillie85, Burghelea86, Dupont03, Ndombo2002, Idrissi00, Menichi00},
the Eilenberg-Moore spectral sequence of the fibre square realising $\Lambda X$ as the pullback back of a pair of diagonal maps
\cite{Smith81, Kuribayashi91, Kuribayashi99, Kuribayashi04}
and the cohomology Leray-Serre spectral sequence of the path-loop (or Wegraum) fibration \cite{McCleary1987, cohololgy_Lprojective, Burfitt2018}.
Computational examples in the literature include
the complex projective space with integral coefficients \cite{Crabb88},
Grassman and Stiefel manifolds with coefficients in a finite field in many cases \cite{Kuribayashi91},
wedge products of same dimensional spheres with integral coefficients \cite{Parhizgar97},
the classifying space of a compact simply connected Lie groups with coefficients in a finite field \cite{Kuribayashi99}
and
simply connected $4$-manifolds with rational coefficients \cite{Onishchenko12}.
The importance of the topology of the free loop space of closed oriented manifolds is further highlighted by the seminal work of Chas and Sullivan \cite{StringTopology} where two new algebraic operations, the loop product and Batalin–Vilkovisky operator where introduced, collectively referred to as string topology operations.
In studying string topology operations the homology of the free loop space has also been considered in a several additional cases.
Integral homology of the free loop space of complex Stiefel manifold \cite{Tamanoi07},
spaces whose cohomology is an exterior algebra with field coefficients \cite{Bohmann2021},
$(n-1)$-connected manifolds up to dimension $3n-2$ with homology coefficients over a field \cite{Berglund15}
and many cases of $(n-1)$-connected $2n$-manifolds with integral coefficients \cite{Beben17}.
Cohen-Jones-Yan \cite{Cohen2004} have also shown that there is a spectral sequence of algebras converging to the homology of the free loop space and the loop product demonstrating its use on spheres and complex proactive spaces.
A straight forward Leray-Serre spectral sequence approach to computing the cohomology of the free loop space was presented by Seeliger \cite{cohololgy_Lprojective} and demonstrated on the free loop space of complex projective spaces.
Greatly extending these ideas the authors \cite{Burfitt2018} previously obtained the integral cohomology of the free loop space of the complete flag manifolds of rank $2$ simple Lie groups.
The other work closely related to this paper is that of McCleary and Ziller~\cite{McCleary1991, McCleary1987}, where it is shown using the Leray-Serre spectral sequence on the path-loop fibration and a classical theorems Gromoll-Meyer~\cite{Gromoll69}, that all homogeneous spaces with the exception of those of rank $1$ have infinitely many geometrically distinct closed geodesics.
This extends earlier work of Ziller~\cite{Ziller1977}, in which Morse theory is applied to obtain the $\mathbb{Z}_2$ Betti number of the free loop space of globally symmetric spaces.
In this paper we explore the cohomology of the free loop space of homogeneous spaces by studying $H^*(\Lambda(SU(n+1)/T^n);\mathbb{Z})$ for $n \geq 2$.
In doing so we uncover surprising combinatorial structure, make use of computational commutative algebra and develop computer aided algorithms.
The power of the theory is illustrated by obtaining the integral cohomology of $\Lambda(SU(4)/T^3)$, a significantly more complex example than obtaining $H^*(\Lambda S^3;\mathbb{Z})$ in the case when $n=2$ or $H^*(\Lambda(SU(3)/T^2);\mathbb{Z})$ in the case the $n=3$ previously considered in \cite{Burfitt2018}.
We apply classical homotopy theoretic arguments to the path-loop
fibration and its pull back along the diagonal map on $SU(n+1)/T^n$ to derive the differentials in the Serre spectral sequence of the free loop space evaluation fibration converging to $H^*(\Lambda(SU(n+1)/T^n);\mathbb{Z})$.
Classically, the elementary symmetric polynomials are used for the basis of symmetric functions in order to write down generators of the quotient ideal in $H^*(SU(n+1)/T^n;\mathbb{Z})$.
However, that choice of the basis elements does not lead to a description of the differentials that can be easily applied to developing further theory.
In this work we choose the basis consisting of complete homogeneous symmetric polynomials.
By doing so we acquire a new unexpectedly sophisticated combinatorial structure on the differentials.
The consequences of the choice of basis is first highlighted by Theorem~\ref{thm:monomial sum} where the ideal generated by complete homogeneous symmetric generator is shown to straightforwardly rearrange to a reduced Gr\"obner basis.
This demonstrates our new approach of using Gr\"obner basis for understating cohomology algebras expressed as a polynomial quotients by analysing them from the perspective of computational commutative algebra and computer aided algorithms.
We explicitly apply Gr\"obner basis to spectral sequences in Proposition~\ref{thm:SpectralGrobner} and lay out an integral Gr\"obner basis procedure for performing computations applicable to any Leray-Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra.
The enhancements to the classical Buchberger algorithm in Section~\ref{sec:SpectralGrobner} combine to provided a powerful procedure that makes the later application in Section~\ref{sec:LSU4/T3} computationally possible.
It is the characterisation in Proposition~\ref{thm:SpectralGrobner} that motivates Theorem~\ref{thm:Ideals}, which provides part of the computation of $H^*(\Lambda(SU(n+1)/T^n);\mathbb{Z})$ for arbitrary $n$.
Theorem~\ref{thm:Ideals} is used along side the direct application of Proposition~\ref{thm:SpectralGrobner} in Section~\ref{sec:LSU4/T3} to obtain an expression for the module structure of $H^*(\Lambda(SU(4)/T^3);\mathbb{Z})$ up to a small uncertainty in torsion type.
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\section{Background}\label{sec:Background}
\subsection{Symmetric polynomials}\label{sec:SymPoly}
A polynomial in $\mathbb{Z}[\gamma_1,\dots,\gamma_n]$ is called symmetric if it is invariant under permutations of the indices of variables $\gamma_1,\dots,\gamma_n$.
The study of symmetric polynomials goes back more than three hundred years, originally used in the study of roots of single variable polynomials.
Today symmetric polynomials have applications in a diverse range of areas of mathematics.
In the paper the relevance of the symmetric polynomials is brought by their presence in the cohomology rings of complete flag manifolds,
in Section~\ref{sec:CohomCompFlag}.
In this section we summaries some basic concepts from the theory of symmetric polynomials that will be essential for our later work.
A compete introduction to the topic can be found in \cite[\S $7$]{ECstanly} or \cite[\S $I$]{Macdonald}.
\subsubsection{Elementary symmetric polynomials}\label{sec:elementary}
Much of the language used to described symmetric polynomials is the language of partitions.
An $n$ \emph{partition} $\lambda$ is a sequence of non-negative integers $(\lambda_1,\dots,\lambda_k)$, for some integer $k\geq 1$, such that
\begin{equation*}
\lambda_1\geq\cdots\geq\lambda_k \;\; \text{and} \;\; \lambda_1+\cdots+\lambda_k=n.
\end{equation*}
By convention we consider partition $(\lambda_1,\dots,\lambda_k)$ and $(\lambda_1,\dots,\lambda_k,0,\dots,0)$ to be equal
and abbreviate an $n$ partition $\lambda$ by $\lambda \vdash n$.
The elementary symmetric polynomials are a special collection of symmetric polynomials that form a basis of the symmetric polynomials, which we make explicit in Theorem~\ref{thm:FunThmSym}.
For each integer $n\geq 1$ and $1\leq l \leq n$, the \emph{elementary symmetric polynomial} $\sigma_l\in\mathbb{Z}[\gamma_1,\dots,\gamma_n]$ in $n$ variables is given by
\begin{equation*}
\sigma_l=\sum_{1\leq i_1<\cdots<i_l\leq n}{\gamma_{i_1}\cdots\gamma_{i_l}}.
\end{equation*}
For a partition $\lambda=(\lambda_1,\dots,\lambda_k)$, denote by $\sigma_\lambda$ the symmetric polynomial $\sigma_{\lambda_1}\cdots\sigma_{\lambda_k}$.
The following theorem is sometimes known as the fundamental theorem of symmetric polynomials.
\begin{theorem}[{\cite[\S $7.4$]{ECstanly}}]\label{thm:FunThmSym}
For each $n\geq 1$, the set of $\sigma_\lambda$, where $\lambda$ ranges over all $n$ partitions, forms an additive basis of all symmetric polynomials.
That is, for $1\leq i \leq n$, the set of $\sigma_i$ are independent and generate
the symmetric polynomials as an algebra.
\end{theorem}
\hspace*{\fill} $\square$
\subsubsection{Complete homogeneous symmetric polynomials}\label{sec:homogeneous}
The complete homogeneous symmetric polynomials are another collection of $n$ symmetric polynomials in $n$ variables for each $n\geq 1$.
In a sense, which is made explicit in \cite[\S $7.6$]{ECstanly},
the complete homogeneous symmetric polynomials can be thought of as dual to the elementary symmetric polynomials.
For each integer $n\geq 1$ and $1\leq l \leq n$, define the \emph{complete homogeneous symmetric polynomials} $h_l\in\mathbb{Z}[\gamma_1,\dots,\gamma_n]$ in $n$ variables by
\begin{equation}\label{defn:CompleteHomogeneous}
h_l=\sum_{1\leq i_1\leq\cdots\leq i_l\leq n}{\gamma_{i_1}\cdots\gamma_{i_l}}.
\end{equation}
For a partition $\lambda=(\lambda_1,\dots,\lambda_k)$, denote by $h_\lambda$ the symmetric polynomial $h_{\lambda_1}\cdots h_{\lambda_k}$.
Setting $\sigma_0=h_0=1$, the following identity derived for infinite variables in \cite[\S $I$,2]{Macdonald}, gives the relationship between the elementary symmetric and complete homogeneous symmetric polynomials.
Evaluating all but $n$ variables to $0$, for any $m \leq n$ gives
\begin{equation}\label{eq:ElementaryHomogeneousRelation}
\sum_{t=0}^{m}(-1)^t \sigma_t h_{n-t} = 0.
\end{equation}
As $\sigma_0=h_0=1$, equation~(\ref{eq:ElementaryHomogeneousRelation}) can be used to inductively derive expressions for either elementally symmetric polynomials in terms of complete homogeneous polynomials or the other way round.
Moreover the complete homogeneous symmetric polynomials
also form a basis of the symmetric polynomials.
\subsection{Gr\"{o}bner bases}\label{sec:Grobner}
Gr\"{o}bner basis provide a powerful tool to perform computations on ideals in commutative algebra.
Their use however extends far beyond such calculations having applications within mathematics, computer science physics and engineering.
We now briefly describe the Gr\"{o}bner basis theory used later in the paper, for further details see \cite{Grobner2} or \cite{Grobner1}.
The following results are stated over a Euclidean or principal ideal domain $R$.
However throughout this paper we consider only the case when $R=\mathbb{Z}$.
Gr\"{o}bner basis theory can be generalised to other rings and stronger results can be recovered over a field.
Given a finite subset $A$ of $R[x_1,\dots,x_n]$, we denote by $\langle A\rangle$ the ideal generated by elements of $A$.
Form now on we assume a total monomial ordering on the polynomial ring $R[x_1,\dots,x_n]$ which respects multiplication.
In the course of this paper we take this order to be the lexicographic ordering.
The \emph{leading term} of a polynomial with respect to an order is the term largest with respect to the order, the \emph{leading monomial} is the leading term multiplied by its coefficient the \emph{leading coefficient}.
For $f,g,p\in R[x_1,\dots,x_n]$, the polynomial $g$ is said to be {\it reduced} from $f$ by $p$, written
\begin{equation*}
f \xrightarrow{p} g
\end{equation*}
if there exists a monomial $m$ in $f$ such that the leading monomial $l_p$ of $p$ divides $m$, say $m= m'l_p$ for some monomial $m'\in R[x_1,\dots,x_n]$ and $g=f-m'p$.
Let $R$ be a principal ideal domain and let $G$ be a finite subset of $R[x_1,\dots,x_n]$.
Then $G$ is a {\it Gr\"{o}bner basis} if any of the following equivalent conditions hold.
\begin{enumerate}
\item
The ideal of leading terms of $\langle G\rangle$ is equal to the ideal generated by the leading terms of $G$.
\item
All elements of $\langle G\rangle$ can be reduced to zero by elements of $G$.
\item
Leading terms of elements in $\langle G\rangle$ are divisible by a leading terms of an elements in $G$.
\end{enumerate}
A set is called {\it decidable} if for any two elements input, there is an algorithm that can determine whether they are equal.
A ring is called {\it computable} if it is decidable as a set and there is an effectively computable algorithm for addition, multiplication and subtraction in the ring for an input of a pair of elements.
A principal ideal domain is called a {\it computable principal ideal domain} if it is a computable ring, there is an algorithm that can effectively compute whether a given pair of elements is divisible and an extended Euclidean algorithm can be effectively computed.
Euclidean domain is a {\it computable Euclidean domain} if it is a computable ring and there is an algorithm that effectively computes division with remainder.
The integers are a computable Euclidean domain.
Moreover, as division with remainder can be applied to construct an extended Euclidean algorithm, so every computable Euclidean domain is also a computable principle ideal domain.
If $R$ is a computable principle ideal domain, then for any ideal in $R[x_1,\dots,x_n]$, there exists a Gr\"{o}bner basis.
In particular, for finite $A\subseteq R[x_1,\dots,x_n]$ there is an algorithm
to obtain a Gr\"{o}bner basis $G$ such that $\langle G\rangle=\langle A\rangle$.
Over a field the most efficient algorithm is known as the Buchberger algorithm and can easily be implemented by a computer and a similar algorithm can used for principal ideal domains.
Over a Euclidean domain computation speed might be improved with the implementing of more advanced algorithms \cite{Lichtblau13, Eder19}.
A basic Gr\"{o}bner basis algorithm can be deduced from the following theorem.
Let $g_1,g_2\in R[x_1,\dots,x_n]$ be non-zero with leading terms $t_1,t_2$ and leading coefficients $c_1,c_2$.
Set $b_1,b_2\in R$ be such that $b_1c_1=b_2c_2=\lcm(c_1,c_2)$ and $s_1,s_2\in R[x_1,\dots,x_n]$ be such that $s_1t_1=s_2t_2=\lcm(t_1,t_2)$.
Then the $S$-polynomial of $g_1$ and $g_2$ is given by
\begin{equation*}
Spol(g_1,g_2) = b_1s_1g_1 - b_2s_2g_2.
\end{equation*}
Set $d_1,d_2\in R$ to be $d_1c_1=d_2c_2=\gcd(c_1,c_2)$.
Then $G$-polynomial of $g_1$ and $g_2$ is given by
\begin{equation*}
Gpol(g_1,g_2) = d_1s_1g_1 + d_2s_2g_2.
\end{equation*}
\begin{theorem}[\cite{Grobner1}]\label{thm:GrobnerAlg}
A finite subset $G$ of $R[x_1,\dots,x_n]$ is a Gr\"{o}bner basis if for any $g_1,g_2\in G$
\begin{enumerate}
\item
$Spol(g_1,g_2)$ reduces to $0$ by $G$ and
\item
$Gpol(g_1,g_2)$ is reducible by an element of $G$ in its leading term.
\end{enumerate}
\end{theorem}
\hspace*{\fill} $\square$
In the computational part of our work it is important that a Gr\"{o}bner basis can be used to compute the intersection of ideals. This procedure is made explicit in the next remark.
\begin{remark}\label{rmk:intersectionGrobner}
Let $A=\{a_1,\dots,a_s\}$ and $B=\{b_1,\dots,b_l\}$ be subsets of $R[x_1,\dots,x_n]$.
Take a Gr\"{o}bner basis $G$ of
\begin{equation*}
\{ya_1,\dots,ya_t,(1-y)b_1,\dots,(1-y)b_l\}
\end{equation*}
in $R[x_1,\dots,x_n,y]$ using a monomial ordering in which monomials containing $y$ are larger than $y$ free monomials.
Then a Gr\"{o}bner basis of $\langle A\rangle\cap\langle B\rangle$ is given by the elements of $G$ that do not contain $y$.
\end{remark}
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\begin{theorem}[\cite{Grobner1}]\label{thm:GrobnerAlg}
A finite subset $G$ of $R[x_1,\dots,x_n]$ is a Gr\"{o}bner basis if for any $g_1,g_2\in G$
\begin{enumerate}
\item
$Spol(g_1,g_2)$ reduces to $0$ by $G$ and
\item
$Gpol(g_1,g_2)$ is reducible by an element of $G$ in its leading term.
\end{enumerate}
\end{theorem}
\hspace*{\fill} $\square$
In the computational part of our work it is important that a Gr\"{o}bner basis can be used to compute the intersection of ideals. This procedure is made explicit in the next remark.
\begin{remark}\label{rmk:intersectionGrobner}
Let $A=\{a_1,\dots,a_s\}$ and $B=\{b_1,\dots,b_l\}$ be subsets of $R[x_1,\dots,x_n]$.
Take a Gr\"{o}bner basis $G$ of
\begin{equation*}
\{ya_1,\dots,ya_t,(1-y)b_1,\dots,(1-y)b_l\}
\end{equation*}
in $R[x_1,\dots,x_n,y]$ using a monomial ordering in which monomials containing $y$ are larger than $y$ free monomials.
Then a Gr\"{o}bner basis of $\langle A\rangle\cap\langle B\rangle$ is given by the elements of $G$ that do not contain $y$.
\end{remark}
A Gr\"{o}bner basis over a principle ideal domain however is not unique.
Importantly, different choices of ordering produce very different Gr\"{o}bner basis even when the basis is given in a reduced form.
From now on let $E$ be a Euclidean domain.
In this case, there may be a choice of remainder upon division resulting in alternative division algorithms and this might change the output of a Gr\"{o}bner basis algorithm.
However, assume the division algorithm is fixed.
To find a Gr\"{o}bner basis which is unique in $E[x_1,\dots,x_n]$ we are required to be more precise about reduction.
For $f,g,p\in E[x_1,\dots,x_n]$, the polynomial $g$ is said to be \emph{E-reduced} from $f$ by $p$ to $g$,
if there exists a monomial $m=at$ in $f$ with
the leading term $l_p$ of $p$ dividing $t$ such that $t=sl_p$ and
\begin{equation*}
g=f-qsp
\end{equation*}
for some non-zero $q\in E$ the quotient of $a$ upon division with unique remainder by $l_p$.
A Gr\"{o}bner basis basis $G$ in $E[x_1,\dots,x_n]$ is said to be \emph{reduced} if all polynomials in $G$ cannot be $E$-reduced by any other polynomial in $G$.
\begin{theorem}[\cite{Kandri-Rody88}]
A reduced Gr\"{o}bner basis $G$ over $\mathbb{Z}[x_1,\dots,x_n]$ for which all leading monomials have positive coefficients is unique.
\end{theorem}
\hspace*{\fill} $\square$
In general, with coefficients in a Euclidean domain the uniqueness of a reduced Gr\"{o}bner bias holds up to multiplication by a units. However, in this paper we only consider Gr\"{o}bner basis of integer polynomials.
The following theorem expands upon part (2) of the equivalent Gr\"{o}bner basis definitions above.
\begin{theorem}[\cite{Grobner1}]\label{thm:GrobnerOver}
Let $G$ be a Gr\"{o}bner basis in $E[x_1,\dots,x_n]$.
Then all elements in $E[x_1,\dots,x_n]$ E-reduce by elements of $G$ to a unique representative in
\begin{equation*}
\frac{E[x_1,\dots,x_n]}{\langle G\rangle}.
\end{equation*}
\end{theorem}
\hspace*{\fill} $\square$
\subsection{Cohomology of complete flag manifolds}\label{sec:CohomCompFlag}
A manifold $M$ is called homogeneous if it can be equipped with a transitive $G$ action for some Lie groups $G$.
In this case, $M \cong G/H$ for some Lie subgroup $H$ of $G$ isomorphic to the orbit of a point in $M$.
A Lie subgroup $T$ of a Lie group $G$ isomorphic to a torus is called maximal
if any Lie subgroup also isomorphic to a torus containing $T$ coincidences with $T$.
The next proposition is straightforward to show, see for example \cite[\S $5.3$]{MT2} Theorem $3.15$.
\begin{proposition}\label{prop:tori}
The conjugate of a torus in $G$ is a torus and all maximal tori are conjugate.
In addition given a maximal torus $T$, for all $x\in G$ there exists an element $g\in G$ such that $g^{-1}xg\in T$.
Hence the union of all maximal tori is $G$.
\end{proposition}
\hspace*{\fill} $\square$
It is therefore unambiguous to refer to the maximal torus $T$ of $G$ and consider the quotient $G/T$, which is isomorphic regardless of the choice of $T$.
The homogeneous space $G/T$ is called the {\it complete flag manifold} of $G$.
The rank of Lie group $G$ is the dimension of a maximal torus $T$.
The ranks of classical simple Lie groups can be deduced by considering the standard maximal tori,
see for example~\cite{MT}.
Borel \cite{Borel} studied in detail the cohomology of homogeneous spaces,
in particular deducing the rational cohomology of $G/T$.
Following later work of Bott, Samelson, Toda, Watanabe and others the integral cohomology of complete flag manifolds of all simple Lie groups were deduced.
The integral cohomology of the complete flag manifolds of the special unitary groups is as follows.
\begin{theorem}[\cite{Borel}, \cite{AplicationsOfMorse}]\label{thm:H*SU/T}
For each integer $n\geq 0$, the cohomology of the complete flag manifold of the simple Lie group $SU(n+1)$ is given by
\begin{equation*}
H^*(SU(n+1)/T^n;\mathbb{Z})=\frac{\mathbb{Z}[\gamma_1,\dots,\gamma_{n+1}]}{\langle\sigma_1,\dots,\sigma_{n+1}\rangle}
\end{equation*}
where $|\gamma_i|=2$.
\end{theorem}
\hspace*{\fill} $\square$
\subsection{Based loop space cohomology of $SU(n)$}\label{sec:LoopLie}
The Hopf algebras of the based loop space of Lie groups were studied by Bott in \cite{bott1958}.
More recently, Grbi{\'c} and Terzi{\'c} \cite{homology_Lflags} showed that the integral homology of the based loop space of a complete flag manifold is torsion free
and found the integral Pontrjagin homology algebras of the complete flag manifolds of compact connected simple Lie groups
$SU(n)$, $Sp(n)$, $SO(n)$, $G_2$, $F_4$ and $E_6$.
Recall that the integral divided polynomial algebra on variables $x_1,\dots,x_n$ is given by
\begin{equation*}
\Gamma_{\mathbb{Z}}[x_1,\dots,x_n]=\frac{\mathbb{Z}[(x_i)_1,(x_i)_2,\dots]}{\langle(x_i)_k-k!x_i^k\rangle}
\end{equation*}
where $1\leq i \leq n$, $k\geq 1$ and $x_i=(x_i)_1$.
The next theorem is obtained
using a Leray-Serre spectral sequence argument applied to the path space fibrations
\begin{equation*}
\Omega SU(n) \to PSU(n) \to SU(n).
\end{equation*}
\begin{theorem}\label{thm:LoopSU(n)}
For each $n\geq 1$, the cohomology of the based loop space of the classical simple Lie group $SU(n)$ is given by
\begin{equation*}
H^*(\Omega(SU(n));\mathbb{Z})=\Gamma_{\mathbb{Z}}[x_2,x_4,\dots,x_{2n-2}]
\end{equation*}
where $|x_i|=i$ for $i=2,4,\dots,2n-2$.
\end{theorem}
\hspace*{\fill} $\square$
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\section{Combintorial coefficients}\label{sec:Comintorial}
Before studying the the cohomology of the free loop space of $SU(n+1)/T^n$
we first analyse some of the combinatorial structures that appear in the cohomology algebras.
\subsection{Binomial coefficients}\label{subsec:Binomial}
The binomial coefficients $\binom{n}{k}$ are defined to be the number of size $k$ subsets of a size $n$ set
and they satisfy the recurrence relation
$\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}$.
It is easily shown by induction on $n$ that for $0\leq k \leq n$, $\binom{n}{k}=\frac{n!}{k!(n-k)!}$ and is zero otherwise.
The binomial coefficients also satisfy the well known formulas
\begin{equation}\label{eq:binom}
\sum_{k=0}^n{\binom{n}{k}}=2^n \;\; \text{and} \;\; \sum^{n}_{k=0}{(-1)^k \binom{n}{k}}=0.
\end{equation}
\subsection{Multinomial coefficients}
Throughout this paper for integers $k\geq 2$ and $n,a_1,\dots,a_k \geq 0$, we set
\begin{equation}\label{eq:Multinomial}
\binom{n}{a_1,\dots,a_k}=\frac{n!}{a_1!\cdots a_k!}.
\end{equation}
The following expansion in term of binomial coefficients is easily verified,
\begin{equation*}\label{eq:MultinomilExpansionOriginal}
\binom{n}{a_1,\dots,a_k}=\binom{n}{a_1}\binom{n-a_1}{a_2}\cdots\binom{n-a_1-\cdots- a_{k-1}}{a_k}.
\end{equation*}
In particular
\begin{equation}\label{eq:MultinomilExpansion}
\binom{n}{a_1,\dots,a_k}=\binom{n}{a_2,\dots,a_{k-1}}\binom{n-a_1-\cdots-a_{k-1}}{a_k}.
\end{equation}
When $a_1+\cdots+a_k=n$, the expressions in equation~(\ref{eq:Multinomial}) are called the the \emph{multinomial coefficients}.
In this case the combinatorial interpretation of the coefficients is as the number of way to partition a size $n$ set into subsets of sizes $a_1,\dots,a_k$.
\subsection{Multiset coefficients}
The number of size $k$ multisets that can be formed from elements of a size $n$ set is denoted by $\multiset{n}{k}$ and these numbers are called the {\it multiset coefficients}.
It is well know that $\multiset{n}{k}=\binom{n+k-1}{k}$,
hence $\multiset{n}{k}=\multiset{n-1}{k}+\multiset{n}{k-1}$.
To the best of our knowledge the identity in the next lemma has not been shown before.
\begin{lemma}\label{lem:combino}
For integers $n,m\geq 1$,
\begin{equation*}
\sum^n_{k=0}{(-1)^k\binom{n}{k}\multiset{n}{m-k}}=0.
\end{equation*}
\end{lemma}
\begin{proof}
We prove the statement by induction on $n$.
When $n=1$,
\begin{equation*}
\sum^n_{k=0}{(-1)^k\binom{n}{k}\multiset{n}{m-k}}
=\binom{1}{0}\multiset{1}{m}-\binom{1}{1}\multiset{1}{m-1}
=\binom{m}{m}-\binom{m-1}{m-1}=0.
\end{equation*}
Suppose the lemma holds for $n=t-1\geq1$, then
\begin{align*}
&\sum^t_{k=0}{(-1)^k\binom{t}{k}\multiset{t}{m-k}}
=\sum^t_{k=0}{(-1)^k\bigg(\binom{t-1}{k}+\binom{t-1}{k-1}\bigg)\multiset{t}{m-k}} \\
&=\sum^t_{k=0}{(-1)^k\bigg(\binom{t-1}{k-1}\multiset{t}{m-k}+\binom{t-1}{k}\multiset{t-1}{m-k}+\binom{t-1}{k}\multiset{t}{m-k-1}\bigg)}=0
\end{align*}
as the middle term sum $\sum^{t-1}_{k=0}{\binom{t-1}{k}\multiset{t-1}{m-k}}=0$ by assumption and all other terms cancel except for $\binom{t-1}{-1}\multiset{t}{m}$, $\binom{t-1}{t}\multiset{t-1}{m-t}$ and $\binom{t-1}{t}\multiset{t}{m-t-1}$
all of which are zero.
\end{proof}
\subsection{Stirling numbers of the second kind}\label{subsec:Stirling}
Along with binomial, multinomial and multiset coefficients, Sterling numbers of the second kind appear as part of the so called \emph{12-fold way}, a class of enumerative problems concerned with the combinations of balls in boxes problems.
The \emph{Stirling numbers of the second kind} $\stirling{n}{m}$, denote the number of ways to partition an $n$ element set into $m$ non-empty subsets.
The Stirling numbers of the second kind satisfy a recurrence relations
\begin{equation}\label{eq:StirlingRecurence}
\stirling{n}{m} = \stirling{n-1}{m-1} + m \stirling{n-1}{m}
\end{equation}
for integers $n\geq m \geq 1$.
From the combinatoral definitions, we obtain the relationship between Stirling numbers of the second kind and multinomial coefficients, given by
\begin{equation}\label{eq:StirlingExpansion}
m!\stirling{n}{m}=
\sum_{\substack{a_1,\dots,a_m \geq 1 \\ a_1+\cdots+a_m=n }}
\binom{n}{a_1,\dots,a_m}.
\end{equation}
To the best of our knowledge the identity in the next lemma has not been shown before.
\begin{lemma}\label{lem:StirlingIdentity}
For each $n \geq 1$,
\begin{equation*}
\sum^n_{m=1}{(-1)^m m! \stirling{n}{m}} = (-1)^n.
\end{equation*}
\end{lemma}
\begin{proof}
The formula is easily seen to hold for the case $n=1$.
Using the recurrence relation in equation~(\ref{eq:StirlingRecurence}) and induction on $n$,
\begin{align*}
\sum^n_{m=1}{(-1)^m m! \stirling{n}{m}}
=& \sum^n_{m=1}{(-1)^m m! \left( \stirling{n-1}{m-1} + m \stirling{n-1}{m} \right) } \\
=&\sum^n_{m=1}{(-1)^m m! \stirling{n-1}{m-1}} + \sum^n_{m=1}{(-1)^m m!m \stirling{n-1}{m}} \\
=&\sum^{n-1}_{m=1}{(-1)^{m+1} (m+1)! \stirling{n-1}{m}} + \sum^{n-1}_{m=1}{(-1)^m m!m \stirling{n-1}{m}} \\
=&\sum^{n-1}_{m=1}{(-1)^{m+1} m!(m+1) \stirling{n-1}{m}} + \sum^{n-1}_{m=1}{(-1)^m m!m \stirling{n-1}{m}} \\
=&\sum^n_{m=1}{(-1)^{m+1}\left( m!(m+1)-m!m \right) \stirling{n-1}{m}} \\
=&\sum^n_{m=1}{(-1)^{m+1} m! \stirling{n-1}{m}}
=(-1)^{n}.
\end{align*}
\end{proof}
By an expansion using the recurrence relation in equation~(\ref{eq:StirlingRecurence}) and Lemma~\ref{lem:StirlingIdentity},
the well known relation $\sum^n_{m=1}{(-1)^m (m-1)! \stirling{n}{m}} = 0$ can be easily derived.
\section{Alternative forms of the symmetric ideal}\label{sec:IdeaForms}
Replacing $\sigma_i$ with the complete homogeneous symmetric polynomials $h_i$ as generators of the symmetric ideal, leads to a simplification of the generator expressions, practical for working with $H^*(SU(n+1)/T^n)$ as demonstrated in the next section.
For each integer $n\geq 1$ and all integers $1 \leq k' \leq k \leq n$,
define $\Phi(k,k')$ to be the sum of all monomials in $\mathbb{Z}[x_1,\dots,x_n]$ of degree $k$ in variables $x_1,\dots,x_{n-k'+1}$.
\begin{theorem}\label{thm:monomial sum}
In the ring $\frac{\mathbb{Z}[x_1,\dots,x_n]}{\langle h_1,\dots,h_n\rangle}$
for each $1 \leq k' \leq k \leq n$, $\Phi(k,k')=0$.
In addition
\begin{equation}\label{eq:OrigIdeal}
\langle h_1,\dots,h_n\rangle=\langle\Phi(1,1),\dots,\Phi(n,n)\rangle.
\end{equation}
Moreover these new ideal generators form a reduced Gr\"obner bases for the ideal of $n$ variable symmetric polynomials with respect to the lexicographic term order on variables $x_1<\cdots<x_n$.
\end{theorem}
\begin{proof}
Note that by definition, $h_k=\Phi(k,1)$.
We prove by induction on $k$ that for each $1 \leq k' \leq k \leq n$, $\Phi(k,k')\in \langle h_1,\dots,h_n\rangle$.
When $k=1$, by definition
$h_1=\Phi(1,1)$.
Assume the statement is true for all $k<m\leq n$.
By induction, $\Phi(m-1,m')\in [h_1,\dots,h_n]$ for all $1 \leq m'\leq m-1$.
Note that $\Phi(m-1,m')x_{n-m'+1}$ is the sum of all monomials of degree $m$ in variables $x_1,\dots,x_{n-m'+1}$ divisible by $x_{n-m'+1}$.
Hence, for each $1\leq m'\leq m-1$
\begin{equation*}
h_m-\Phi(m-1,1)x_n-\cdots-\Phi(m-1,m'-1)x_{n-m'+2}=\Phi(m,m').
\end{equation*}
At each stage of the proof the next $\Phi(k,k)$ is obtained as a sum of $h_k$ and polynomials obtained from $h_1,\dots,h_{k-1}$.
Hence $\langle\Phi(1,1),\dots,\Phi(n,n)\rangle$ and $\langle h_1,\dots,h_n \rangle$ are equal.
\end{proof}
For integers $0\leq a\leq b$, denote by $h_a^b$ the complete homogeneous polynomial in variables $x_1,\dots,x_b$ of degree $a$.
Then equation~(\ref{eq:OrigIdeal}) can be written as
\begin{equation}\label{eq:SipleRedusingHomogenious}
\langle h_1^n,\dots,h_n^n\rangle=\langle h_1^n,h_2^{n-1},\dots,h_n^1\rangle.
\end{equation}
A useful intermediate form of Theorem~\ref{thm:monomial sum} is separately set out in the next proposition.
\begin{proposition}\label{prop:Homogeneous-1}
For each $n\geq 1$,
\begin{equation*}
\langle h_1^n,\dots,h_n^n\rangle=\langle h_1^n,h_2^{n-1}\dots,h_n^{n-1}\rangle.
\end{equation*}
\end{proposition}
\begin{proof}
For each $1\leq i\leq n-1$
\begin{equation*}
h^n_{i+1}-x_n h^n_i=h^{n-1}_{i+1}.
\end{equation*}
We can rearrange the ideal to achieve the desired result by performing the above elimination in sequence on the ideal for $i={n-1}$ to $i=1$.
\end{proof}
\begin{remark}\label{remk:SymQotForms}
By Theorem~\ref{thm:monomial sum} and Proposition~\ref{prop:Homogeneous-1} eliminating the last variable in $\mathbb{Z}[x_1,\dots,x_n]$,
by rewriting $h_1$ as $x_n=-x_1-\cdots-x_{n-1}$ gives
\begin{equation*}
\frac{\mathbb{Z}[x_1,\dots,x_n]}{\langle h_1^n,\dots,h_n^n\rangle}
\cong\frac{\mathbb{Z}[x_1,\dots,x_{n-1}]}{\langle h_2^{n-1},\dots,h_n^{n-1}\rangle}
\cong\frac{\mathbb{Z}[x_1,\dots,x_{n-1}]}{\langle h_2^{n-1},\dots,h_n^{1}\rangle}.
\end{equation*}
\end{remark}
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\section{Determining spectral sequence differentials}\label{sec:FreeLoopSU(n+1)/Tn}
The aim of this section is to determine the differentials in a spectral sequence converging to the free loop cohomology $H^{*}(\Lambda(SU(n+1)/T^n);\mathbb{Z})$ for $n \geq 1$.
The case when $n=0$ is trivial as $SU(1)$ is a point.
The approach of the argument is similar to that of \cite{cohololgy_Lprojective},
in which the cohomology of the free loop spaces of spheres and complex projective space are calculated using spectral sequence techniques.
However the details in the case of the complete flag manifold of the special unitary group are considerably more complex, requiring a sophisticated combinatorial argument arising form the structure of the complete homogeneous symmetric function not present for simpler spaces.
\subsection{Differentials in the spectral sequence of the diagonal fibration}\label{sec:evalSS}
For any space $X$, the map $eval \colon Map(I,X) \to X\times X$ is given by $\alpha \mapsto (\alpha(0),\alpha(1))$.
It can be shown directly that $eval$ is a fibration with fiber $\Omega X$.
In this section we compute the differentials in the cohomology Serre spectral sequence of this fibration for the case $X=SU(n+1)/T^n$.
The aim is to compute $H^{*}(\Lambda(SU(n+1)/T^n);\mathbb{Z})$.
The map $eval \colon \Lambda X \to X$ given by evaluation at the base point of a free loop is also a fibration with fiber $\Omega X$.
This is studied in Section~\ref{sec:diff} by considering a map of fibrations from the evaluation fibration for $SU(n+1)/T^n$ to the diagonal fibration
and hence the induced map on spectral sequences. For the rest of this section we consider the fibration
\begin{equation}\label{eq:evalfib}
\Omega (SU(n+1)/T^n) \to Map(I,SU(n+1)/T^n) \xrightarrow{eval} SU(n+1)/T^n\times SU(n+1)/T^n.
\end{equation}
By extending the fibration $T^n \to SU(n+1) \to SU(n+1)/T^n$, we obtain the homotopy fibration sequence
\begin{equation}\label{eq:SU/Tfib}
\Omega(SU(n+1)) \to \Omega(SU(n+1)/T^n) \to T^n \to SU(n+1).
\end{equation}
It is well known see \cite{CohomologyOmega(G/U)}, that the furthest right map above that is, the inclusion of the maximal torus into $SU(n+1)$ is null-homotopic.
Hence there is a homotopy section $T^n \to \Omega(SU(n+1)/T^n)$.
Therefore, as \eqref{eq:SU/Tfib}
is a principle fibration, there is a space decomposition $\Omega(SU(n+1)/T^n) \simeq \Omega(SU(n+1)) \times T^n$.
Using the K\"{u}nneth formula, we obtain the algebra isomorphisum
\begin{equation}\label{eq:BaseLoopFlag}
H^*(\Omega(SU(n+1)/T^n);\mathbb{Z}) \cong H^*(\Omega(SU(n+1);\mathbb{Z}) \otimes H^*(T^n;\mathbb{Z})
\cong \Gamma_{\mathbb{Z}}[x'_2,x'_4,\dots,x'_{2n}] \otimes \Lambda_{\mathbb{Z}}(y'_1,\dots,y'_n)
\end{equation}
where $\Gamma_{\mathbb{Z}}[x'_2,x'_4,\dots,x'_{2n}]$ is the integral divided polynomial algebra on $x'_2,\dots,x'_{2n}$
with $|x'_i|=i$ for each $i=2,\dots,2n$ and
$\Lambda(y'_1,\dots,y'_n)$ is an exterior algebra generated by $y'_1,\dots,y'_n$
with $|y'_j|=1$ for each $j=1,\dots,n$.
Since
\begin{equation*}
Map(I,SU(n+1)/T^n)\simeq SU(n+1)/T^n
\end{equation*}
by Theorem~\ref{thm:H*SU/T}
all cohomology algebras of spaces in fibration (\ref{eq:evalfib}) are known.
By studying the long exact sequence of homotopy groups associated to the fibration $T^n \to SU(n+1) \to SU(n+1)/T^n$,
we obtain that $SU(n+1)/T^n$ and hence $SU(n+1)/T^n \times SU(n+1)/T^n$ are simply connected.
Therefore the cohomology Serre spectral sequence of fibration (\ref{eq:evalfib}), denoted by $\{\bar{E}_r,\bar{d}^r\}$,
converges to $H^*(SU(n+1)/T^n;\mathbb{Z})$ with $\bar{E}_2$-page
\begin{equation*}
\bar{E}^{p,q}_2=H^p(SU(n+1)/T^n\times SU(n+1)/T^n;H^q(\Omega(SU(n+1)/T^n);\mathbb{Z})).
\end{equation*}
In the following arguments we use the notation
\begin{center}
$H^*(Map(I,SU(n+1)/T^n);\mathbb{Z}) \cong \frac{\mathbb{Z}[\lambda_1,\dots,\lambda_{n+1}]}{\langle\sigma^{\lambda}_1,\dots,\sigma^{\lambda}_{n+1}\rangle}$
\end{center}
and
\begin{center}
$H^*(SU(n+1)/T^n \times SU(n+1)/T^n;\mathbb{Z}) \cong
\frac{\mathbb{Z}[\alpha_1,\dots,\alpha_{n+1}]}{\langle\sigma^{\alpha}_1,\dots,\sigma^{\alpha}_{n+1}\rangle} \otimes
\frac{\mathbb{Z}[\beta_1,\dots,\beta_{n+1}]}{\langle\sigma^{\beta}_1,\dots,\sigma^{\beta}_{n+1}\rangle}
$
\end{center}
where $|\alpha_i|=|\beta_i|=|\lambda_i|=2$ for each $i=1,\dots,n+1$ and $\sigma^{\lambda}_i,\sigma^{\alpha}_i$
and $\sigma^{\beta}_i$ are the elementary symmetric polynomials in $\lambda_i,\alpha_i$ and $\beta_i$, respectively.
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,
nodes in empty cells,nodes={minimum width=5ex,
minimum height=5ex,outer sep=-5pt},
column sep=1ex,row sep=1ex]{
& \vdots & \vdots & & & & & & \\
& 2n\; & \langle x'_{2n} \rangle& & & & & & \\
& \vdots & \vdots & & & & & & \\
& 6 & \langle x'_6 \rangle & & & &\dots & & \\
H^{*}(\Omega(SU(n+1)/T^n;\mathbb{Z})) & 4 & \langle x_4\rangle & & & & & & \\
& 2 & \langle x'_2\rangle & &\dots& & & & \\
& & & & & & & & \\
& 1 & \langle y'_i\rangle & \;\;\; \lcdot \;\;\; &\lcdot&\;\;\lcdot\;\;& \cdots& \lcdot & \cdots \\
& 0 & & \langle\alpha_i,\beta_i\rangle &\lcdot&\lcdot & \cdots& \lcdot & \cdots \\
&\quad\strut & 0 & 2 & 4 & 6 & \cdots& 2n & \cdots \strut \\};
\draw[-stealth] (m-2-3.south east) -- (m-8-8.north west);
\draw[-stealth] (m-5-3.south east) -- (m-8-5.north);
\draw[-stealth] (m-4-3.south east) -- (m-8-6.north);
\draw[-stealth] (m-6-3.south) -- (m-8-4.north);
\draw[-stealth] (m-8-3.south east) -- (m-9-4.north west);
\draw[-stealth] (m-8-4.south east) -- (m-9-5.north west);
\draw[-stealth] (m-8-5.south east) -- (m-9-6.north west);
\draw[-stealth] (m-8-7.south east) -- (m-9-8.north west);
\draw[thick] (m-1-2.east) -- (m-10-2.east) ;
\draw[thick] (m-10-2.north) -- (m-10-9.north) ;
\end{tikzpicture}
\label{fig:evalSS}
\end{center}
\begin{center}
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; H^{*}(SU(n+1)/T^n\times SU(n+1)/T^n;\mathbb{Z})$
\end{center}
\begin{center}
\captionof{figure}{Generators in the integral cohomology Leray-Serre spectral sequence $\{\bar{E}_r,\bar{d}^r\}$ converging to $H^{*}(Map(I,SU(n+1));\mathbb{Z})$.}
\end{center}
In the remainder of this section we describe explicitly the images of differentials shown in Figure~\ref{fig:evalSS}
and show that all other differential not generated by these differentials using the Leibniz rule are zero.
It will often be useful to use the alternative basis
\begin{equation}\label{eq:ChangeBasis}
v_i=\alpha_i-\beta_i \text{ and } u_i=\beta_i
\end{equation}
for $H^{*}(SU(n+1)/T^n\times SU(n+1)/T^n;\mathbb{Z})$, where $i=1,\dots,n+1$.
The following lemma determines completely the $\bar{d}^2$ differential on $\bar{E}_2^{*,1}$.
\begin{lemma}\label{lem:E^2_{*,1}d^2}
With the notation above, in the cohomology Leray-Serre spectral sequence of fibration (\ref{eq:evalfib}),
there is a choice of basis $y'_1,\dots,y'_n$ such that
\begin{center}
$\bar{d}^2(y'_i)=v_i$
\end{center}
for each $i=1,\dots,n$.
\end{lemma}
\begin{proof}
There is a homotopy commutative diagram
\begin{equation*}\label{fig:evalcd}
\xymatrix{
{SU(n+1)/T^n} \ar[r]^(.37){\Delta} & {SU(n+1)/T^n\times SU(n+1)/T^n} \ar@{=}[d] \\
{Map(I,SU(n+1)/T^n)} \ar[r]_(.42){eval} \ar[u]_{p_0} & {SU(n+1)/T^n\times SU(n+1)/T^n} }
\end{equation*}
where $p_0$, given by $\psi \mapsto \psi(0)$, is a homotopy equivalence and $\Delta$ is the diagonal map.
As the cup product is induced by the diagonal map, $eval^*$ has the same image as the cup product.
For dimensional reasons, $\bar{d}^2$ is the only possible non-zero differential ending at any $\bar{E}_*^{2,0}$ and no non-zero differential have domain in any $\bar{E}_*^{2,0}$.
Therefore in order for the spectral sequence to converge to $H^{*}(Map(I,SU(n+1)/T^n))$,
the image of $\bar{d}^2 \colon \bar{E}_2^{0,1} \to \bar{E}_2^{2,0}$ must be the kernel of the cup product on $H^*(SU(n+1)/T^n\times SU(n+1)/T^n;\mathbb{Z})$,
which is generated by $v_1,\dots,v_n$.
\end{proof}
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\begin{remark}\label{rmk:unique}
The only remaining differentials on generators left to determine are those with domain in
$\langle x'_2,x'_4\dots,x'_{2n} \rangle$ on some page $\bar{E}_r$ for $r \geq 2$.
For dimensional reasons, the elements $x'_2,x'_4,\dots,x'_{2n}$ cannot be in the image of any differential.
By Lemma~\ref{lem:E^2_{*,1}d^2}, the generators $u_1,\dots,u_n$ must survive to the $\bar{E}_{\infty}$-page,
so generators $x'_2,x'_4,\dots,x'_{2n}$ cannot.
This is due to dimensional reasons combined with the fact that the spectral sequence must converge to $H^*(SU(n+1)/T^n)$.
Now assume inductively that for each $i=1,\dots,n$ and $1\leq j<i$, $\bar{d}^{2j}$ is constructed.
For dimensional reasons and due to all lower rows except $\bar{E}_r^{*,2}$ and $\bar{E}_r^{*,1}$ being annihilated by differentials already determined at lower values of
$1\leq j<i$, the only possible non-zero differential beginning at $x'_{2i}$
is $\bar{d}^{2i}:\bar{E}_{2i}^{0,2i}\to \bar{E}_{2i}^{2i,1}$.
The image of each of the differentials $\bar{d}^{2i}$ will therefore be a unique class in $\bar{E}_{2i}^{2i,1}$
in the kernel of $\bar{d}^2$ not already contained in the image of any $\bar{d}^r$ for $r<2i$.
We note also that the elements $(x_{2i})_m$ for each $m \geq 2$ are also generators on the $\bar{E}_2$-page of the spectral sequence.
The differentials in the spectral sequence are also completely determined on all $(x_{2i})_m$ by their image on $x_{2i}$ in the following way.
Using the relations $x_{2i}^m-m!(x_{2i})_m$ and the Leibniz rule, it follows that $\bar{d}^2(x_{2i}^m) = m\bar{d}^2(x_{2i})x_{2i}^{m-1}$ and hence again using the relations
\begin{equation*}
\bar{d}^2((x_{i2})_m) = \bar{d}^2(x_{2i})(x_{2i})_{m-1}.
\end{equation*}
Due to the fact that $\bar{d}^{2i}(x_{2i})$ must be non-trivial, there are no torsion elements on the spectral sequence pages $E_i^{*,*}$ and $(x_{2i})_m$ cannot be in the image of any differential,
we obtain that
$\bar{d}^r((x_{2i})_m) = \bar{d}^r(x_{2i})(x_{2i})_{m-1} = 0$ for $2\leq r<2i$.
Therefore we can apply the same augments used to derive the $E_2$-page equation above on the $E_{2i}$-page we have that
\begin{equation}\label{eq:DifOnDivPoly}
\bar{d}^{2i}((x_{2i})_m) = \bar{d}^{2i}(x_{2i})(x_{2i})_{m-1}.
\end{equation}
\end{remark}
We have $\bar{d}^2(u_i)=\bar{d}^2(v_i)=0$ and by Lemma~\ref{lem:E^2_{*,1}d^2} we may assume that $\bar{d}^2(y'_i)=v_i$ for each $i=1,\dots,n$.
All non-zero generators $\gamma \in \bar{E}_2^{*,1}$ can be expressed in the form
\begin{center}
$\gamma = y'_k u_{i_1} \cdots u_{i_s} v_{j_1} \cdots v_{j_t}$
\end{center}
for some $1\leq k \leq n$, $1\leq i_1<\cdots<i_s\leq n$ and $1\leq j_1<\cdots<j_t\leq n$.
Therefore, $\bar{d}^2(\gamma)$ is zero only if it is contained in
$\langle \sigma^{\alpha}_1,\dots,\sigma^{\alpha}_{n+1}, \sigma^{\beta}_1,\dots,\sigma^{\beta}_{n+1}\rangle$.
Hence it is important to understand the structure of the symmetric polynomials
$\sigma^{\alpha}_1,\dots,\sigma^{\alpha}_{n+1}, \sigma^{\beta}_1,\dots,\sigma^{\beta}_{n+1}$.
From Remark~\ref{remk:SymQotForms}, we see that
polynomials $\sigma^{\alpha}_1$ and $\sigma^{\beta}_1$ can be thought of as expressions for $\alpha_{n+1}$ and $\beta_{n+1}$ in terms of the other generators of the ideal and generators
$\sigma^{\alpha}_2,\dots,\sigma^{\alpha}_{n+1}, \sigma^{\beta}_2,\dots,\sigma^{\beta}_{n+1}$
can be replaced with homogeneous symmetric polynomials in $\alpha_1,\dots,\alpha_{n}$
and $\beta_1,\dots,\beta_{n}$.
It turns out that this rearranged basis is more convenient for deducing the differentials in the spectral sequence.
Using the next two lemmas, we determine the correspondence between each $\sigma_l^\alpha$ and $\sigma_l^\beta$ and generators of the kernel of $\bar{d}^2$, which provides us with the image of the remaining differentials.
For each $n \geq 1$, let $A=(a_1,\dots,a_n),B=(b_1,\dots,b_n)\in \mathbb{Z}^n_{\geq 0}$,
such that not all of $a_1,\dots,a_n$ are zero.
Denote by $L(a_1,\dots,a_n)$ the number of non-zero entries in $(a_1,\dots,a_n)$.
Define the quotient of the permutation group on $n$ elements $\zeta_{(b_1,\dots,b_l)}^{(a_1,\dots,a_l)} \coloneqq S_n/\sim$ by
\begin{equation*}
\pi \sim \rho \iff (a_{\pi(1)},\dots,a_{\pi(n)})=(a_{\rho(1)},\dots,a_{\rho(n)})
\text{ and } (b_{\pi(1)},\dots,b_{\pi(n)})=(b_{\rho(1)},\dots,b_{\rho(n)}).
\end{equation*}
Let $1\leq x(a_1,\dots,a_n) \leq n$ be the minimal integers such that
\begin{equation}\label{eq:x}
a_{x(a_1,\dots,a_n)} \geq 1.
\end{equation}
Notice that $x(a_{\pi(1)},\dots,a_{\pi(n)})=x(a_{\rho(1)},\dots,a_{\rho(n)})$ if $\pi \sim \rho$.
Define element $s_{(b_1,\dots,b_n)}^{(a_1,\dots,a_n)}$ of $E_2^{2l-1,1}$ by
\begin{equation*}
s_{(b_1,\dots,b_n)}^{(a_1,\dots,a_n)} \coloneqq
\sum_{\substack{\pi \in \zeta_{(b_1,\dots,b_n)}^{(a_1,\dots,a_n)}}}
{y'_{x(a_{\pi(1)},\dots,a_{\pi(n)})}v_{1}^{a_{\pi(1)}}\cdots v_{x(a_{\pi(1)},\dots,a_{\pi(n)})}^{a_{x(a_{\pi(1)},\dots,a_{\pi(n)})}-1}\cdots v_{n}^{a_{\pi(n)}} u_{1}^{b_{\pi(1)}}\cdots u_{n}^{b_{\pi(n)}}}.
\end{equation*}
\begin{lemma}\label{lem:SmallS}
With $s_{(b_1,\dots,b_n)}^{(a_1,\dots,a_n)}$ as given above,
\begin{align*}
\bar{d}^{2}&(s_{(b_1,\dots,b_n)}^{(a_1,\dots,a_n)})=
\\
&\sum_{\substack {\pi \in \zeta_{(b_1,\dots,b_l)}^{(a_1,\dots,a_l)} \\
0\leq t_j \leq a_j, \; 1\leq j \leq n}}
\Bigg{(}\prod_{1\leq k \leq n}{(-1)^{a_k-t_k}\binom{a_k}{t_k}}\Bigg{)}
\alpha_{1}^{t_{\pi(1)}}\cdots \alpha_{n}^{t_{\pi(n)}}
\beta_{1}^{b_{\pi(1)}+a_{\pi(1)}-t_{\pi(1)}}\cdots\beta_{n}^{b_{\pi(n)}+a_{\pi(n)}-t_{\pi(n)}}
.
\end{align*}
\end{lemma}
\begin{proof}
Applying Lemma~\ref{lem:E^2_{*,1}d^2}, the Leibniz rule and the change of basis (\ref{eq:ChangeBasis}),
\begin{align*}
\bar{d}^{2}&(s_{(b_1,\dots,b_n)}^{(a_1,\dots,a_n)})=
\sum_{\substack{\pi \in \zeta_{(b_1,\dots,b_l)}^{(a_1,\dots,a_n)}}}
(\alpha_{1}-\beta_{1})^{a_{\pi(1)}} \cdots (\alpha_{n}-\beta_{n})^{a_{\pi(n)}}
\beta_{1}^{b_{\pi(1)}}\cdots \beta_{n}^{b_{\pi(n)}}.
\end{align*}
Using the binomial expansion on the terms $(\alpha_{i_j}-\beta_{i_j})^{a_j}$ for each $1\leq j \leq n$, we obtain
\begin{align*}
\bar{d}^{2}(s_{(b_1,\dots,b_n)}^{(a_1,\dots,a_n)})= & \\
\sum_{\pi \in \zeta_{(b_1,\dots,b_n)}^{(a_1,\dots,a_n)}} &
\Bigg{(}
\bigg{(}\sum_{0\leq t\leq a_{\pi(1)}}{(-1)^{a_{\pi(1)}-t}\binom{a_{\pi(1)}}{t}\alpha_{1}^t\beta_{1}^{a_{\pi(1)}-t}}
\bigg{)}
\cdots \\ & \cdots
\bigg{(}\sum_{0\leq t\leq a_{\pi(n)}}{(-1)^{a_{\pi(n)}-t}\binom{a_{\pi(n)}}{t}\alpha_{n}^t\beta_{n}^{a_{\pi(n)}-t}}
\bigg{)}
\beta_{1}^{b_{\pi(1)}}\cdots\beta_{n}^{b_{\pi(n)}}
\Bigg{)}.
\end{align*}
The expression in the statement of the lemma now follows by collecting the terms
of the same product of $\alpha_{i_j}$'s and $\beta_{i_j}$'s.
\end{proof}
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Using the notation above, define
\begin{align}\label{eq:BigS}
& S^{(c_1,\dots,c_l)} \coloneqq \nonumber \\
& \sum_{\substack{ 0\leq t_j \leq c_j, \\ 1\leq j \leq n, \\ \text{some } t_j > 0 }}
s_{(c_1-t_1,\dots,c_l-t_n)}^{(t_1,\dots,t_n)}
\prod_{1 \leq i \leq n}
\sum_{\substack{(a_1,\dots,a_{c_i}) \in \mathbb{Z}_{\geq 0} \\ |(a_1,\dots,a_{c_i})|=c_i-t_i, \\ a_{k-1} = 0 \implies a_{k} = 0, \\ 2 \leq k \leq c_i}}
(-1)^{t_i+L(a_1,\dots,a_{c_i})}
\binom{c_i}{a_1,\dots,a_{c_i}}.
\end{align}
The second differential when applied to $S^{(c_1,\dots,c_n)}$ gives an expression in terms
of only products of $\alpha_{i_j}$'s or only $\beta_{i_j}$'s of type $(c_1,\dots,c_n)$.
\begin{lemma}\label{lem:d2BigS}
With $S^{(c_1,\dots,c_n)}$ given as above,
\begin{equation*}
\bar{d}^2(S^{(c_1,\dots,c_n)}) =
\sum_{\pi \in \zeta_{(c_1,\dots,c_n)}^{(c_1,\dots,c_n)}}
{\alpha_{1}^{c_{\pi(1)}}\cdots \alpha_{n}^{c_{\pi(n)}}}
+(-1)^{c_1+\cdots+c_n+1}\sum_{\pi \in \zeta_{(b_1,\dots,b_n)}^{(a_1,\dots,a_n)}}
{\beta_{1}^{c_{\pi(1)}}\cdots \beta_{n}^{c_{\pi(n)}}}.
\end{equation*}
\end{lemma}
\begin{proof}
It follows from Lemma~\ref{lem:SmallS} that the only monomials in $\bar{d}^2(S^{(c_1,\dots,c_l)})$ are integer multiples of
\begin{equation}\label{eq:SumTerms}
\alpha_{1}^{t_{\pi(1)}}\cdots \alpha_{n}^{t_{\pi(n)}}
\beta_{1}^{c_{\pi(1)}-t_{\pi(1)}}\cdots \beta_{n}^{c_{\pi(n)}-t_{\pi(n)}}
\end{equation}
for some $(t_1,\dots,t_n)\leq (c_1,\dots,c_n)$
and $\pi \in \zeta_{(c_1-t_1,\dots,c_n-t_n)}^{(t_1,\dots,t_n)}$.
When $(t_1,\dots,t_n)=(c_1,\dots,c_n)$, then the monomials in (\ref{eq:SumTerms}) are
\begin{equation*}
\alpha_{1}^{c_{\pi(1)}}\cdots \alpha_{n}^{c_{\pi(n)}}
\end{equation*}
for each $\pi \in \zeta_{(0,\dots,0)}^{(c_1,\dots,c_n)}$.
By Lemma~\ref{lem:SmallS}, these monomial terms only appear in the image of the differential $\bar{d}^2$
of $s^{(c_1,\dots,c_n)}_{(0,\dots,0)}$ and appear with multiplicity one.
Hence the statement
of the lemma
holds when restricted to such monomials.
By induction on $l=|(c_1,\dots,c_n)-(t_1,\dots,t_n)|$ we now prove that the statement holds for all other terms containing monomials in~(\ref{eq:SumTerms}).
Take $0\leq (t'_1,\dots,t'_n) < (c_1,\dots,c_n)$
such that $|(c_1,\dots,c_n)-(t'_1,\dots,t'_n)|=l\geq 1$.
Since the terms of monomials in (\ref{eq:SumTerms}) for $(t'_1,\dots,t'_n)$
only occur in the image of the $\bar{d}^2$ differential of $s^{(t_1,\dots,t_n)}_{(c_1-t_1,\dots,c_n-t_n)}$
for $ 0 < (t_1,\dots,t_n) \leq (c_1,\dots,c_n)$ such that $|(c_1,\dots,c_n)-(t_1,\dots,t_n)|\leq l$,
the multiplicity of $s^{(t_1,\dots,t_n)}_{(c_1-t_1,\dots,c_n-t_n)}$ in (\ref{eq:BigS}) for
$|(c_1,\dots,c_n)-(t_1,\dots,t_n)| > l$ does not effect the multiplicity of the terms
corresponding to monomials with $(t'_1,\dots,t'_n)$ in (\ref{eq:SumTerms}).
Hence, we need only show that the sum of terms of monomials in (\ref{eq:SumTerms}) for
$(t'_1,\dots,t'_n)\neq(0,\dots,0)$ arising from
$|(c_1,\dots,c_n)-(t'_1,\dots,t'_n)| > l$ is the negative of the constant
\begin{equation}\label{eq:CancelCoef}
K_{(t'_1,\dots,t'_n)} =
\prod_{1 \leq i \leq n}
\sum_{\substack{(a_1,\dots,a_{c_i}) \in \mathbb{Z}_{\geq 0} \\ |(a_1,\dots,a_{c_i})|=c_i-t'_i, \\ a_{k-1} = 0 \implies a_{k} = 0, \\ 2 \leq k \leq c_i}}
(-1)^{t'_i+L(a_1,\dots,a_{c_i})}\binom{c_i}{a_1,\dots,a_{c_i}}.
\end{equation}
The only remaining case is the terms of monomials from (\ref{eq:SumTerms}) when $(t'_1,\dots,t'_n)=(0,\dots,0)$,
which requires the sum to be $(-1)^{c_1+\cdots+c_n+1}$.
Returning first to the main cases, the sum we are interested in is the same for any
$\pi \in \zeta_{(c_1-t'_1,\dots,c_n-t'_n)}^{(t'_1,\dots,t'_n)}$.
So without loss of generality, we assume that $\pi$ is the identity.
In this case using the formula from Lemma~\ref{lem:SmallS}, we consider
\begin{equation*}
\sum_{\substack{(t'_1,\dots,t'_n) < (t_1,\dots,t_n) \leq (c_1,\dots,c_n)}}
{K_{t_1,\dots,t_n}}\prod_{1\leq j \leq n}{(-1)^{t_j-t'_j}\binom{t_j}{t_j-t'_j}}
\end{equation*}
where we have additionally used the fact that $\binom{t_j}{t'_j}=\binom{t_j}{t_j-t'_j}$,
which using (\ref{eq:CancelCoef}) can be inductively rewritten as
\begin{align}\label{eq:InductionExpression}
\sum_{(t'_1,\dots,t'_n) < (t_1,\dots,t_n) \leq (c_1,\dots,c_n)}
\prod_{1\leq j \leq n}{(-1)^{t_j-t'_j}\binom{t_j}{t_j-t'_j}}
\\ \cdot
\prod_{1 \leq i \leq n}
\sum_{\substack{(a_1,\dots,a_{c_i}) \in \mathbb{Z}_{\geq 0} \\ |(a_1,\dots,a_{c_i})|=c_i-t_i, \\ a_{k-1} = 0 \implies a_{k} = 0, \\ 2 \leq k \leq c_i}}
(-1)^{t_i+L(a_1,\dots,a_{c_i})}
\binom{c_i}{a_1,\dots,a_{c_i}}. \nonumber
\end{align}
Since $c_i-a_1-\cdots -a_{l_i}=t_i$, using equation~(\ref{eq:MultinomilExpansion}) we see that multiplying ${\binom{t_j}{t_j-t'_j}}$ by $\binom{c_i}{a_1,\cdots,a_{c_i}}$ in (\ref{eq:InductionExpression}) when $i=j$ is the same as extending each
$a_1,\dots,a_{c_i}$ to $a_1,\dots,a_{c_i},t_j-t'_j$.
Which means we can rewrite (\ref{eq:InductionExpression}) as
\begin{align}\label{eq:InductionExpression2}
\sum_{(t'_1,\dots,t'_n) < (t_1,\dots,t_n) \leq (c_1,\dots,c_n)}
\prod_{1 \leq i \leq n}
\sum_{\substack{(a_1,\dots,a_{c_i}) \in \mathbb{Z}_{\geq 0} \\ |(a_1,\dots,a_{c_i})|=c_i-t_i, \\ a_{k-1} = 0 \implies a_{k} = 0, \\ 2 \leq k \leq c_i}}
(-1)^{t'_i+L(a_1,\dots,a_{c_i})}
\binom{c_i}{a_1,\dots,a_{c_i}, t_i-t'_i}
\end{align}
where we also take the signs into account noting that
\begin{equation*}
(-1)^{t_i-t'_i}(-1)^{t'_i+L(a_1,\dots,a_{c_i})}
=(-1)^{t'_i+L(a_1,\dots,a_{c_i})}.
\end{equation*}
Notice also that as $|(a_1,\dots,a_{l_i},t_i-t_i',0,\dots,0)|=c_i-t'_i$, once $t_i-t'_i$ is moved to the left of $a_k = 0$, the multinomial type coefficient terms appearing in (\ref{eq:InductionExpression2}) are the same as those appearing in
(\ref{eq:CancelCoef}).
It remains to check that the multiplicity of the sum of these coefficients in (\ref{eq:InductionExpression2}) agrees with (\ref{eq:CancelCoef}).
Each multinomial type coefficient in (\ref{eq:CancelCoef}) is determined by a choice of sequence $(a_1,\dots,a_{c_i})$ for $1\leq i \leq n$.
Once the product is expanded, the multinomial type coefficient terms in (\ref{eq:InductionExpression}) are the product of $n$ coefficients corresponding to $n$ sequences
\begin{equation*}
(a_1,\dots,a_{c_i})=(a_1,\dots,a_{l_i},0,\dots,0),
\end{equation*}
for each $1\leq i \leq n$ and some $1\leq l_i\leq c_i$.
The coefficient product terms in (\ref{eq:InductionExpression2}) of the same form correspond to $n$ sequences of the form
\begin{equation}\label{eq:SubSequences}
(a_1,\dots,a_{l_i},0,\dots,0) \text{ or } (a_1,\dots,a_{l_i-1},0,\dots,0)
\end{equation}
for $1 \leq i \leq n$, the second case corresponding to extending by $t_i-t'_i>0$ and the first to extending by $t_i-t'_i=0$.
Note that for at least some $i$ the second case must be chosen as $(t'_1,\dots,t'_n)<(t_1,\dots,t_n)$.
However, as we fix the order of the $n$ sequences all other possible combinations of choices occur exactly once.
The sign of each of the terms from (\ref{eq:CancelCoef}) inside the product is the product of $(-1)^{t'_i+L(a_1,\dots,a_{c_i})}=t'_i+l_i$ changes from those in (\ref{eq:InductionExpression2})
each time the first choice in (\ref{eq:SubSequences}) is taken as for non-zero $t_j-t'_j$, $L(a_1,\dots,a_{l_i}-1,t_j-t'_j,0,\dots,0) = L(a_1,\dots,a_{l_i}-1,0,\dots,0)+1$.
The number of ways to obtain a unique product of coefficients from (\ref{eq:CancelCoef}) in (\ref{eq:InductionExpression2}) is exactly all choices possible in (\ref{eq:SubSequences}).
This choice corresponds to picking subsets of an $n$ set that is not the entire set.
As this sign depends on the size of the chosen subset, the total sum of these terms is the alternating sum of the $n^{\text{th}}$ binomial coefficients (\ref{eq:binom}),
without the first term in the sum.
The result is therefore a single term of multiplicity $1$ up to sign.
This sign is the opposite of that of the first missing term from the alternating sum of binomial coefficients, which corresponds to the term in (\ref{eq:CancelCoef}) as required.
Finally it remains to consider the case when $(t'_1,\dots,t'_n)=(0,\dots,0)$.
Using the result of the previous part of the proof, substituting $t'_i=0$ in (\ref{eq:CancelCoef}), this will be the negative of
\begin{equation*}
\prod_{1 \leq i \leq n}
\sum_{\substack{(a_1,\dots,a_{c_i}) \in \mathbb{Z}_{\geq 0} \\ |(a_1,\dots,a_{c_i})|=c_i, \\ a_{k-1} = 0 \implies a_{k} = 0, \\ 2 \leq k \leq c_i}}
(-1)^{L(a_1,\dots,a_{c_i})}
\binom{c_i}{a_1,\dots,a_{c_i}}
=(*).
\end{equation*}
Since now $a_1+\dots+a_{c_i}=c_i$, the coefficients $\binom{c_i}{a_1,\dots,a_{c_i}}$ are genuine multinomial coefficients.
Hence, by collecting all sequence with the constant $L(a_1,\dots,a_{c_i})=j$ using equation~(\ref{eq:StirlingExpansion}) then Lemma~\ref{lem:StirlingIdentity} to rewrite
the equation above as
\begin{equation*}
(*) =
\prod_{1\leq i \leq n}\sum_{1\leq j \leq c_i}(-1)^j j!\stirling{c_i}{j}
=
\prod_{1\leq i \leq n}(-1)^{c_i} = (-1)^{c_1+\cdots+c_n}
\end{equation*}
which completes the proof.
\end{proof}
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We can now describe explicitly the image of $\bar{d}^2$ in the spectral sequence.
\begin{theorem}\label{thm:d^2Image}
The only non-zero differentials on the generators $x'_{2l}$ for $2\leq l\leq n+1$ are given by
\begin{equation*}
\bar{d}^{2(l-1)}(x'_{2(l-1)})
=\sum_{\substack{|(c_1,\dots,c_{l'})|=l}}{S^{(c_1,\dots,c_l)}}.
\end{equation*}
\end{theorem}
\begin{proof}
Applying Lemma~\ref{lem:d2BigS}, the image under $\bar{d}^2$ of the right hand side is the difference of complete homogeneous symmetric functions in $l$ variables with $\alpha$'s and $\beta$'s respectively.
By Remark~\ref{rmk:unique}, since no expression in only $\alpha$'s or $\beta$'s lies in the image of $\bar{d}^2$, the image of the difference of the homogeneous symmetric polynomials in $\alpha$'s and $\beta$'s must generate the image of $x'_{2l}$ under $\bar{d}^{2(l-1)}$.
\end{proof}
\subsection{Differentials for the free loop spectral sequence}\label{sec:diff}
Next we consider the map $\phi$ of the evaluation fibration of $SU(n+1)/T^n$ for $n \geq 1$
and the diagonal fibration studied in Section~\ref{sec:evalSS}
given by the following commutative diagram
\begin{equation*}\label{fig:fibcdSU}
\xymatrix{
{\Omega(SU(n+1)/T^n)} \ar[r]^(.5){} \ar[d]^(.45){id} & {\Lambda(SU(n+1)/T^n)} \ar[r]^{eval} \ar[d]^(.45){exp} & {SU(n+1)/T^n} \ar[d]^(.45){\Delta} \\
{\Omega(SU(n+1)/T^n)} \ar[r]^(.45){} & {Map(I,SU(n+1)/T^n)} \ar[r]^(.425){eval} & {SU(n+1)/T^n\times SU(n+1)/T^n}}
\end{equation*}
where $\exp$ is given by $\exp(\alpha)(t)=\alpha(e^{2\pi i t})$.
As $SU(n+1)/T^{n}$ is simply connected, the evaluation fibration induces a cohomology Leray-Serre spectral sequence $\{ E_r,d^r \}$.
Hence $\phi$ induces a map of spectral sequences $\phi^*:\{ \bar{E}_r,\bar{d}^r \} \to \{ E_r,d^r \}$.
More precisely for each $r\geq 2$ and $a,b \in \mathbb{Z}$, there is a commutative diagram
\begin{equation}\label{fig:phicd}
\xymatrix{
{\bar{E}_r^{a,b}} \ar[r]^(.4){d^r} \ar[d]^(.46){\phi^*} & {\bar{E}_r^{a+r,b-r+1}} \ar[d]^(.46){\phi^*} \\
{E_r^{a,b}} \ar[r]^(.4){\bar{d}^r} & {E_r^{a+r,b-r+1}}}
\end{equation}
where $\phi^*$ for each successive $r$ is the induced map on the homology of the previous page, beginning as the map induced on the tensor on the $\bar{E}_2$-pages
by the maps $id \colon \Omega(SU(n+1)/T^n)\to\Omega(SU(n+1)/T^n)$ and $\Delta \colon SU(n+1)/T^n \to SU(n+1)/T^n \times SU(n+1)/T^n$.
For the remainder of the section we use the notation
\begin{center}
$H^*(\Omega(SU(n+1)/T^n);\mathbb{Z})\cong \Gamma_{\mathbb{Z}}(x_2,x_4,\dots,x_{2n})\otimes\Lambda_{\mathbb{Z}}(y_1,\dots,y_{n})$
\end{center}
\begin{center}
and $\;\;\; H^*(SU(n+1)/T^n;\mathbb{Z})\cong \frac{\mathbb{Z}[\gamma_1,\dots,\gamma_{n+1}]}{\langle\sigma^\gamma_1,\dots,\sigma_{n+1}^\gamma\rangle}$,
\end{center}
where $|y_i|=1, |\gamma_j|=2, |x_{2i}|=2i$ for each $1\leq i\leq n,1\leq j\leq n+1$ and $\sigma^\gamma_1,\dots,\sigma_{n+1}^\gamma$
are the basis of elementary symmetric functions on $\gamma_i$.
We now determine all the differentials in $\{ E_r,d^r \}$.
\begin{theorem}\label{thm:allDiff}
For each integer $n\geq 1$ and for $2 \leq l \leq n+1$, the only non-zero differentials on generators of the $E_2$-page of $\{ E_r,d^r \}$ are up to class representative and sign are given by
\begin{equation*}
d^{2(l-1)}(x_{2(l-1)}) =
\sum_{\substack{|(c_1,\dots,c_{n})|=l, \\ 1\leq j \leq n, \; c_j \geq 1}}
{c_jy_{j}\gamma_1^{c_1}\cdots\gamma_j^{c_j-1}\cdots\gamma_n^{c_{n}}}.
\end{equation*}
\end{theorem}
\begin{proof}
The identity $id \colon \Omega(SU(n+1)/T^n)\to\Omega(SU(n+1)/T^n)$ induces the identity map on cohomology.
The diagonal map $\Delta \colon SU(n+1)/T^n \to SU(n+1)/T^n \times SU(n+1)/T^n$ induces the cup product on cohomology.
Hence by choosing appropriate generators of $\{ E_r,d^r \}$, we assume that
\begin{center}
$\phi^*(y'_i)=y_i,\;\;\phi^*(x'_i)=x_i\;\;$and$\;\;\phi^*(\alpha_i)=\phi^*(\beta_i)=\phi^*(u_i)=\gamma_i,\;\;$so$\;\;\phi^*(v_i)=0$.
\end{center}
For dimensional reasons, the only possibly non-zero differential on generators ${y}_i$ in $\{ E_r,d^r \}$ is $d^{2}$.
However for each $1\leq i\leq n$ using commutative diagram (\ref{fig:phicd}) and Lemma~\ref{lem:E^2_{*,1}d^2}, we have
\begin{center}
$d^2(y_i)=d^2(\phi^*(y'_i))=\phi^*(\bar{d}^2(y'_i))=\phi^*(v_i)=0$.
\end{center}
Additionally all differentials on generators $\gamma_i$, for each $1\leq i \leq n+1$, are zero for dimensional reasons.
Hence all elements of $E_2^{*,1}$ and $E_2^{*,0}$ survive to the $E_{\infty}$-page unless they are in the image of some differential
$d^r$ for $r\geq 2$.
Using commutative diagram (\ref{fig:phicd}), we have up to class representative and sign
\begin{equation*}
d^{2(l-1)}(x_{2(l-1)})=\phi^*(\bar{d}^{2(l-1)}(x'_{2(l-1)})).
\end{equation*}
Since $\phi^*(v_i)=0$, an expression for $\phi^*(\bar{d}^{2(l-1)}(x'_{2(l-1)}))$ is obtained form the terms in the expression of Theorem~\ref{thm:d^2Image} that do not contain $v_i$.
Therefore
\begin{align}\label{eq:ImediateDiff}
\nonumber
&d^{2(l-1)}(x_{2(l-1)}) = \\
&\sum_{\substack{|(c_1,\dots,c_{n})|=l, \\ 1\leq j \leq n, \; c_j \geq 1}}
{y_{j}\gamma_1^{c_1}\cdots\gamma_j^{c_j-1}\cdots\gamma_n^{c_{n}}}
\prod_{1 \leq i \leq n}
\sum_{\substack{(a_1,\dots,a_{c_i}) \in \mathbb{Z}_{\geq 0} \\ |(a_1,\dots,a_{c_i})|=c_i-\mathbb{I}_{i=j}, \\ a_{k-1} = 0 \implies a_{k} = 0, \\ 2 \leq k \leq c_i}}
(-1)^{\mathbb{I}_{i=j}+L(a_1,\dots,a_{c_i})}
\binom{c_i}{a_1,\dots,a_{c_i}}
\end{align}
where $\mathbb{I}_{i=j}$ is the indicator function equal if $1$ if $i=j$ and $0$ otherwise.
Notice that by applying (\ref{eq:StirlingExpansion}) to $(a_1,\dots,a_{c_i})$ with the same value of $L(a_1,\dots,a_{c_i})$ and then using Lemma~\ref{lem:StirlingIdentity}, when $i \neq j$, we have
\begin{equation*}
\sum_{\substack{(a_1,\dots,a_{c_i}) \in \mathbb{Z}_{\geq 0} \\ |(a_1,\dots,a_{c_i})|=c_i \\ a_{k-1} = 0 \implies a_{k} = 0, \\ 2 \leq k \leq c_i}}
(-1)^{L(a_1,\dots,a_{c_i})}
\binom{c_i}{a_1,\dots,a_{c_i}}
=
\sum_{1\leq L \leq c_i}(-1)^L\stirling{c_i}{L}
=
(-1)^{c_i}
.
\end{equation*}
Additionally, when $i=j$ and using (\ref{eq:MultinomilExpansion}) we have
\begin{align*}
&\sum_{\substack{(a_1,\dots,a_{c_i}) \in \mathbb{Z}_{\geq 0} \\ |(a_1,\dots,a_{c_i})|=c_i-1 \\ a_{k-1} = 0 \implies a_{k} = 0, \\ 2 \leq k \leq c_i}}
(-1)^{1+L(a_1,\dots,a_{c_i})}
\binom{c_i}{a_1,\dots,a_{c_i}}
\\
=&
\sum_{\substack{(a_1,\dots,a_{c_i}) \in \mathbb{Z}_{\geq 0} \\ |(a_1,\dots,a_{c_i})|=c_i-1 \\ a_{k-1} = 0 \implies a_{k} = 0, \\ 2 \leq k \leq c_i}}
(-1)^{1+L(a_1,\dots,a_{c_i})}
\binom{c_i}{1}
\binom{c_i-1}{a_1,\dots,a_{c_i}}
\\ =&
c_i\sum_{1\leq L \leq c_i-1}(-1)^{L+1} \stirling{c_i-1}{L}
=
c_i(-1)^{c_i}
\end{align*}
which together reduce (\ref{eq:ImediateDiff}) to
\begin{equation*}
d^{2(l-1)}(x_{2(l-1)}) =
\sum_{\substack{|(c_1,\dots,c_{n})|=l, \\ 1\leq j \leq n, \; c_j \geq 1}}
{(-1)^{\lceil \frac{n}{2} \rceil}c_jy_{j}\gamma_1^{c_1}\cdots\gamma_j^{c_j-1}\cdots\gamma_n^{c_{n}}}.
\end{equation*}
Since all terms have the same sign we choose the positive generator, obtaining the formula in the statement of the theorem.
\end{proof}
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\begin{remark}\label{rmk:DifOnDivPoly}
By applying $\phi^*$ to equation \eqref{eq:DifOnDivPoly} we obtain additionally for each $m\geq 2$ that
\begin{equation*}
d^{2(l-1)}((x_{2(l-1)}))
=
d^{2(l-1)}(x_{2(l-1)})(x_{2(l-1)})_{m-1}.
\end{equation*}
\end{remark}
\section{Spectral sequence computations with Gr{\"o}bner basis}\label{sec:SpectralGrobner}
The following proposition describes the application of Gr\"{o}bner basis to spectral sequences making use of the theory in Section~\ref{sec:Grobner}, motivates Theorem~\ref{thm:Ideals} and will be heavy used in Section~\ref{sec:LSU4/T3} of the paper.
A library of code for preforming computations outlined in the proposition can be found at \cite{Burfitt2021}.
The proposition applies straightforwardly to the Leray-Serre spectral sequence and is stated here for cohomology spectral sequences of algebras though could be reformulated through duality for a homology spectral sequence of algebras, swapping the base space with the fibre and rows for columns.
\begin{proposition}\label{thm:SpectralGrobner}
Let $\{ E_r, d^r \}$ be a first quadrant spectral sequence of algebras in coefficients $R$
for which the $E^2$-page is of the form $E_2^{p,q} = E_2^{p,0}\otimes E_2^{0,q}$.
Suppose also that $E_2^{*,0}$ can be expressed as the quotient of a polynomial algebra
\begin{equation*}
A = \frac{R[x_1,x_2,\dots,x_n]}{I}
\end{equation*}
for some ideal $I$ in $R[x_1,x_2,\dots,x_n]$.
For some $a\geq 0$,
let $S_a$ be the additive generators of $E_r^{0,a}\setminus\{x_1,x_2,\dots,x_n\}$.
The computation within each row may be broken down as follows.
The kernel of $d^r$ in $A$ on row $E_{r}^{*,a}$ is generated by
\begin{enumerate}
\item
the kernel of the map
\begin{equation*}
\phi_a^r \colon R[S_{a},x_1,x_2,\dots,x_n] \to R[S_{a-r+1},x_1,x_2,\dots,x_n]
\end{equation*}
induced by $d^r$ and
\item
the pre-image under $d^r$ of generators of the intersection
\begin{equation*}
\im(\phi_a^r) \cap S_{a-r+1}I
\end{equation*}
where $S_{a}I= \{ si \; | \; s\in S_{a-r+1}, i \in I \}$.
Treating $S_{a-r+1}I$ as an ideal generated by elements $si$ such that $s\in S_{a-r+1}$ and $i$ is a generator of $I$, the intersection can be computed by Gr\"{o}bner basis as an intersection of ideals
noting that any element that increases the degree of the polynomial element of $S_{a-r+1}$ is no longer on row $a$, hence is ignored as it is justified below.
\end{enumerate}
The algebra structure on the $E_{r+1}$-page is generated by
the union of row generators given above and $S_{a}I \cap E_r^{a,*}$ for each $a\geq 0$.
The relations for the algebra structure on $E_{r+1}$-page are given by the union
of $I,\: \im(d^b)$ for $2\leq b \leq r$ and relations of the first column $E_{2}^{*,0}$ intersected with the generators of the $E_{r+1}$-page algebra.
In addition when $R=\mathbb{Z}$, the type of torsion present in each row $E_r^{a,*}$ and their generating relations can be obtained from the reduced Gr\"{o}bner basis of the ideal
\begin{equation}\label{eq:GrobnerTorsionPart}
\langle I\cap E_r^{a,*},\: \im(d^2)\cap E_r^{a,*},\: \im(d^3)\cap E_r^{a,*}, \dots,\: \im(d^r)\cap E_r^{a,*} \rangle
\end{equation}
as the non-unit greatest common divisor of the absolute values of the coefficients of each element of the reduced Gr\"{o}bner basis.
In this case the coefficients greatest common divisors describe all possible torsion types present and their multiplicities.
\end{proposition}
For part (1) of the proposition, the kernel of a morphism of polynomial rings can also be obtained using Gr\"{o}bner basis, see for example \cite{Biase05}.
However such computations will be unnecessary in the present work.
During a Gr\"{o}bner basis computation of the intersection of ideals in part (2), expression containing elements in $E_r^{*,b}$ for $b>a$ may be considered, these can be discarded immediately, since reduction, $S$-polynomials and $G$-polynomials of such elements can never decease the row $b$ during the procedure without reducing the entire polynomial to zero.
Doing this greatly speeds up the execution of the algorithm.
\begin{proof}
For each $a\geq 0$ and $r\geq 2$,
the following diagram commutes
\begin{equation*}
\xymatrix{
{\ker(\phi_a^r)} \ar[r]^(.5){} \ar[d]^(.45){q} & {R[S_{a},x_1,x_2,\dots,x_n]} \ar[r]^(0.46){\phi_a^r} \ar[d]^(.45){q} & {R[S_{a-r+1},x_1,x_2,\dots,x_n]} \ar[d]^(.45){q} \\
{\ker(d^r)} \ar[r]^(.45){} & {E_r^{a,*}} \ar[r]^-(.5){d^r} & {E_r^{a-r+1,*}}}
\end{equation*}
where $q$ is the quotient map by $I$.
Elements of $\ker(\phi_a^r)$ remain in $\ker(d^r)$ after quotient map $q$. Other element of $R[S_{a-r+1},x_1,x_2,\dots,x_n]$ not in $\ker(\phi_a^r)$ that are in the kernel of $d^r$ under $q$ have non-trivial image under $\phi^r_n$ and so are contained in $S_{a-r+1}I$.
Hence parts (1) and (2) together describe the whole kernel of $d^r$ on row $E^{a,*}_{r}$.
The relations form $I$ and $\im(d^b)$ span all relations in the spectral sequence by construction and additional generators $S_{a-r+1}I \cap E_r^{a,*}$ are formally added to ensure that the all relations are contained in the span of the generators.
Finally, the torsion in the case $R=\mathbb{Z}$ can be seen to be correct by the following inductive argument to $E$-reduce any minimal generating set $T_1,\dots,T_m$ of the torsion relations to the torsion polynomials in the minimal Gr\"{o}bner basis $G$.
The leading term of $T_1$ must be divisible by the leading term of a unique element $g$ in $G$.
As $T_1$ can be reduced to $0$ by $G$, reducing its leading term to be equal to that of $g$ and then fully $E$-reducing it by $G\setminus g$ must leave exactly $g$.
We follow the same steps to reduce the rest of $T_2,\dots,T_m$ to element of $G$, except we first $E$-reduce each $T_i$ fully by all elements of $g$ already obtained form $T_1,\dots,T_{i-1}$ before beginning.
No element of $T_i$ can be reduced to zero by previously obtained element of $g$ as we assume that $T_1,\dots,T_m$ is minimal and there can be no additionally torsion element of $G$ not obtained form reduction of $T_1, \dots, T_m$ as it is assumed to generate the torsion relations.
\end{proof}
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Though not necessary in the course of this work, in some cases the procedure outlined in Proposition~\ref{thm:SpectralGrobner} can require an additional computational setup which can again be done algorithmicly as described in the flowing remark.
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\begin{remark}\label{rmk:Pre-ImageReduction}
Note that the pre-image of the Gr\"{o}bner basis of $\im(\phi_a^r) \cap S_{a-r+1}I$ in part (2) of Proposition~\ref{thm:SpectralGrobner} can be uniquely obtained computationally by fully reducing it by a Gr\"{o}bner basis of $\im(\phi_a^r)$ to $0$ while keeping track of the reductions.
The generators of $\im(\phi_a^{r})$ need not be a Gr\"{o}bner basis and since the reduction of $\im(\phi_a^r) \cap S_{a-r+1}I$ is in terms of a Gr\"{o}bner basis of $\im(\phi_a^r)$,
the resulting expression from the reduction is in terms of their Gr\"{o}bner basis not the original generators.
For the computations made in the paper we will only need to consider the case where the generators of $\im(\phi_a^{r})$ form a Gr\"{o}bner basis.
However an expression in term of the original generators can be deduced as follows.
Track the computation of the reduced Gr\"{o}bner basis of $\im(\phi_a^{r})$ to obtain an expression for the Gr\"{o}bner basis in terms of the original generators.
This expression is unique up to Syzygys of the Gr\"{o}bner basis which can be computed through a reduction procedure, see for example \cite{Erocal16},
the algorithm being equally valid over $\mathbb{Z}$.
\end{remark}
Though Proposition~\ref{thm:SpectralGrobner} provides a procedure for computations with spectral sequences of the appropriate form, the later calculations in this work are obtained under stricter assumptions that greatly enhance the computation efficacy of integral Gr\"{o}bner basis as detailed in the next remark.
\begin{remark}\label{rmk:HomGrobSimplification}
When the generators $x_1,x_2\dots,x_n$ of the algebra $A$ in Proposition~\ref{thm:SpectralGrobner} have the same degree, the representatives of generators of $\im(\phi_a^r)$ are homogeneous in their $x_1,x_2\dots,x_n$ components.
Reduction preserves homogeneity and
in addition, $S$-polynomial and $G$-polynomials of homogeneous polynomials used during the Gr\"{o}bner basis computation are again homogeneous polynomials and only ever increase the degree.
Assuming also that generators of $I$ are homogeneous and that elements of $A$ have bounded degree,
then when computing the Gr\"{o}bner basis with such polynomials to obtain $\im(\phi_a^r) \cap S_{a-r+1}I$ in part (2) of the proposition, we may discard any $S$-polynomial or $G$-polynomials of homogeneous degree greater than the maximal degree in $A$, greatly speeding up the computation.
\end{remark}
\section{Basis}\label{sec:basis}
We now develop theory to reveal the structure of the cohomology of the free loop space of $SU(n+1)/T^n$, which culminates in Theorem~\ref{thm:Ideals}.
To begin we consider a basis of $\mathbb{Z}[\gamma_1,\dots,\gamma_n]$ that resembles the image of the $d^2$ differential in Theorem~\ref{thm:allDiff},
as it becomes easier to study the $E_3$-page and subsequent pages of the spectral sequence, which in addition to making possible theoretical results also simplifies all computations in Section~\ref{sec:LSU4/T3}.
\begin{remark}\label{rmk:TildeBasis}
In $\mathbb{Z}[\gamma_1,\dots,\gamma_{n}]$, let $\bar{\gamma}=\gamma_1+\cdots+\gamma_n$ and $\tilde{\gamma}_i=\bar{\gamma}+\gamma_i$ for each $1\leq i\leq n$.
We rearrange the standard basis $\gamma_1,\dots,\gamma_n$ of $\mathbb{Z}[\gamma_1,\dots,\gamma_{n}]$ to $\gamma_1,\dots,\gamma_{n-1},\bar{\gamma}$
and then rearrange to $\tilde{\gamma}_1,\dots,\tilde{\gamma}_{n-1},\bar{\gamma}$,
by adding $\bar{\gamma}$ to all other basis elements.
Notice that the replacement $\gamma_i \mapsto \tilde{\gamma}_i$ for $1\leq i \leq n-1$, $\gamma_n \mapsto \bar{\gamma}$
could have instead been chosen as $\gamma_j \mapsto \bar{\gamma}$ for any $1\leq j \leq n$ and $\gamma_i \mapsto \tilde{\gamma}_i$ for any $i\neq j$.
Furthermore, replacing $\bar{\gamma}$ by $\tilde{\gamma}_n = (n+1)\bar{\gamma}-\tilde{\gamma}_1-\cdots-\tilde{\gamma}_{n-1}$ gives $\tilde{\gamma}_n$,
so $\tilde{\gamma}_1,\dots,\tilde{\gamma}_{n}$ forms a rational basis.
\end{remark}
\begin{proposition}\label{prop:basis}
Using the notation of equation~(\ref{eq:SipleRedusingHomogenious}),
we can rewrite $h^{n-l+2}_i$ for each $2\leq l \leq n+1$ in the basis of Remark~\ref{rmk:TildeBasis}, as
\begin{equation*}
h^{n-l+2}_l=\sum_{\substack{0\leq k \leq l \\
1\leq i_1 \leq \cdots \leq i_k \leq n-l+1}}
{(-1)^{l-k} \binom{n+1}{l-k} \tilde{\gamma}_{i_1}\cdots\tilde{\gamma}_{i_{k}}\bar{\gamma}^{l-k}}.
\end{equation*}
\end{proposition}
\begin{proof}
First note that in the basis of Remark~\ref{rmk:TildeBasis}, we can rewrite the original basis in terms of the new one by
\begin{equation}\label{eq:BaseExpression}
\gamma_i=\tilde{\gamma_i}-\bar{\gamma} \text{ for } 1\leq i\leq n-1, \;\;\; \gamma_n=n\bar{\gamma}-\sum_{i=1}^{n-1}{\tilde{\gamma_i}}.
\end{equation}
When $l=2$ using rearrangement (\ref{eq:BaseExpression}),
\begin{flalign}
h_2^{n}&=\sum_{a=0}^{2}{\big{(}(n\bar{\gamma}-\sum_{j=1}^{n-1}{\tilde{\gamma}_j})^{2-a}
\sum_{1\leq i_1\leq i_2 \leq n-1}{\prod^a_{k=1}{(\tilde{\gamma}_{i_k}-\bar{\gamma})}}\big{)}} \nonumber \\
&= (n\bar{\gamma}-\sum_{j=1}^{n-1}{\tilde{\gamma}_j})^{2}
+\sum^{n-1}_{a=1}{(n\bar{\gamma}-\sum_{j=1}^{n-1}{\tilde{\gamma}_j})(\tilde{\gamma}_a-\bar{\gamma})}
+\sum^{n-1}_{a=1}{(\tilde{\gamma}_{a}-\bar{\gamma})^2}
+\sum_{1\leq i_1 < i_2 \leq n-1}{(\tilde{\gamma}_{i_1}-\bar{\gamma})(\tilde{\gamma}_{i_2}-\bar{\gamma})}. \label{eq:l=2}
\end{flalign}
For $1\leq k,k_1, k_2\leq n-1$, $k_1 \neq k_2$,
we consider the terms of the form
\begin{equation*}
\bar{\gamma}^2, \; \tilde{\gamma}_k\bar{\gamma}, \; \tilde{\gamma}_k^2, \; \tilde{\gamma}_{k_1}\tilde{\gamma}_{k_2}
\end{equation*}
and count their occurrences in the summands of (\ref{eq:l=2}).
In total $n^2$ element of the form $\bar{\gamma}^2$ are produced by the first summand of (\ref{eq:l=2}),
minus $n(n-1)$ times in the second, $n-1$ in the third and $\binom{n-1}{2}$ in the last.
Hence in total
\begin{equation*}
n^2-n(n-1)+(n-1)+\binom{n-1}{2}=n+\binom{n-1}{1}+\binom{n-1}{2}=\binom{n}{1}+\binom{n}{2}=\binom{n+1}{2}.
\end{equation*}
In total $-2n$ elements of the form $\tilde{\gamma}_k\bar{\gamma}$ are produced in the first summand of (\ref{eq:l=2}),
$2n-1$ in the second, $-2$ in the third and $2-n$ in the last.
Hence in total
\begin{equation*}
-2n+(2n-1)-2+(2-n)=n+1=\binom{n+1}{1}.
\end{equation*}
The terms $\tilde{\gamma}_k^2$ appear once in the first summand of (\ref{eq:l=2}), once in the third and negative once in the second,
hence once in total.
The terms $\tilde{\gamma}_{k_1}\tilde{\gamma}_{k_2}$ appear twice in the first summand,
minus twice in the the second and once in the last,
hence once in total.
Therefore the conditions of the proposition are satisfied.
For $l\geq 3$, we first show that
\begin{equation*}
h^{n-l+2}_l=\sum_{\substack{0\leq k \leq l \\
1\leq i_1 \leq \cdots \leq i_k \leq n-l+2}}
{(-1)^{l-k} \binom{n+1}{l-k} \tilde{\gamma}_{i_1}\cdots\tilde{\gamma}_{i_{k}}\bar{\gamma}^{l-k}}.
\end{equation*}
where in the index on the sum here is $1\leq i_1 \leq \cdots \leq i_k \leq n-l+2$ rather than $1\leq i_1 \leq \cdots \leq i_k \leq n-l+1$ in the statement of the proposition.
The proposition then follows form the statement above by inductions. This is because we have already shown the case $l=2$ and we can obtain the expression for $h^{n-l+2}_l$ in the proposition for $l\geq 3$ from the one above by subtracting off $\tilde{\gamma}_{n-l+2}$ times the expression for $h^{n-l+1}_{l-1}$ in the statement of the proposition,
as this cancels all the summands containing a multiple $\tilde{\gamma}_{n-l+2}$.
Using rearrangement (\ref{eq:BaseExpression}),
\begin{equation}\label{eq:l>3}
h^{n-l+2}_l=\sum_{1\leq i_1\leq \cdots\leq i_l \leq n-l+2}{\prod_{k=1}^{l}{(\tilde{\gamma}_{i_k}-\bar{\gamma})}}.
\end{equation}
For any choice of $1\leq i_1\leq \cdots \leq i_k \leq n-l+2$ and non-negative integers $b,a_1,\dots,a_k$ such that $b+a_1+\cdots+a_k=l$,
the terms of the form
\begin{equation}\label{eq:ProdChoice}
\tilde{\gamma}_{i_1}^{a_1}\cdots\tilde{\gamma}_{i_k}^{a_k}\bar{\gamma}^{b}
\end{equation}
describe up to multiplicity all possible summand in the expansion of equation~(\ref{eq:l>3}).
Define the notation $h^{n-l+2}_l\{ \tilde{\gamma}_{i_1}^{a_1}\cdots\tilde{\gamma}_{i_k}^{a_k}\bar{\gamma}^{b} \}$ to be the multiplicity of the summand containing
$\tilde{\gamma}_{i_1}^{a_1}\cdots\tilde{\gamma}_{i_k}^{a_k}\bar{\gamma}^{b}$ in the expansion of equation~(\ref{eq:l>3}).
Using this notation and equation~(\ref{eq:l>3}) if we show that for each $n+1\geq l \geq 3$,
\begin{equation}\label{eq:BorckekenExpress}
h^{n-l+2}_l\{ \tilde{\gamma}_{i_1}^{a_1}\cdots\tilde{\gamma}_{i_k}^{a_k}\bar{\gamma}^{b} \} = (-1)^{b} \binom{n+1}{b}
\end{equation}
we would complete the proof of the proposition.
Considering each summand of equation~(\ref{eq:l>3}) in tern and counting the number of
$\tilde{\gamma}_{i_1}^{a_1}\cdots\tilde{\gamma}_{i_k}^{a_k}\bar{\gamma}^{b}$
produced in each product, we obtain
\begin{equation}\label{eq:ChoicExpanssion}
h^{n-l+2}_l\{ \tilde{\gamma}_{i_1}^{a_1}\cdots\tilde{\gamma}_{i_k}^{a_k}\bar{\gamma}^{b} \} =
(-1)^b
\sum_{\theta=0}^{b}{\multiset{n-l+2-k}{b-\theta}\sum_{\substack{\alpha_1+\cdots+\alpha_k=\theta \\ \alpha_j\geq 0}}
{\prod^{\theta}_{\beta=1}{\binom{a_\beta+\alpha_\beta}{\alpha_\beta}}}}.
\end{equation}
We proceed by induction on $n$ and prove (\ref{eq:BorckekenExpress}) for all $n\geq 1$ and $2\leq l\leq n+1$.
When $n=1$, the only valid value of $l$ is $2$ and $h^{n-l+2}_{l}=(\tilde{\gamma}_1-\bar{\gamma})^2$ whose expansions satisfies (\ref{eq:BorckekenExpress}).
Assume that (\ref{eq:BorckekenExpress}) holds for all $\phi\leq n$.
It is clear that if $b=0$ or $n+1$, then $h^{n-l+2}_{l}\{ \bar{\gamma}^{n+1} \}=(-1)^{n+1}$ and $h^{n-l+2}_{l}\{ \tilde{\gamma}_{i_1}^{a_1}\cdots\tilde{\gamma}_{i_k}^{a_k} \}=1$
for any choice of $a_1,\dots,a_{k}$ since in the expansion of equation~(\ref{eq:ChoicExpanssion}) there would be only one way to obtain the element.
For $1\leq b \leq n$, by induction
\begin{equation}\label{eq:Case(n,b)}
\binom{n}{b}=(-1)^{b}h^{n-l+1}_{l}\{ \tilde{\gamma}_{i_1}^{a_1}\cdots\tilde{\gamma}_{i_k}^{a_k}\bar{\gamma}^{b} \}
=\sum_{\theta=0}^{b}{\multiset{n-l+1-k}{b-\theta}\sum_{\substack{\alpha_1+\cdots+\alpha_k=\theta \\ \alpha_j\geq 0}}
{\prod^{\theta}_{\beta=1}{\binom{a_\beta+\alpha_\beta}{\alpha_\beta}}}}
\end{equation}
and
\begin{equation}\label{eq:Case(n,b-1)}
\binom{n}{b-1}=(-1)^{b-1}h^{n-l+2}_{l-1}\{ \tilde{\gamma}_{i_1}^{a_1}\cdots\tilde{\gamma}_{i_k}^{a_k}\bar{\gamma}^{b-1} \}
=\sum_{\theta=0}^{b-1}{\multiset{n-l+2-k}{b-1-\theta}\sum_{\substack{\alpha_1+\cdots+\alpha_k=\theta \\ \alpha_j\geq 0}}
{\prod^{\theta}_{\beta=1}{\binom{a_\beta+\alpha_\beta}{\alpha_\beta}}}}.
\end{equation}
For each $0\leq\theta\leq b-1$, the sum of values from (\ref{eq:Case(n,b)}) and (\ref{eq:Case(n,b-1)}) corresponds to the $\theta$ summand in the expression for
$h^{n-l+2}_l\{ \tilde{\gamma}_{i_1}^{a_1}\cdots\tilde{\gamma}_{i_k}^{a_k}\bar{\gamma}^{b} \}$
since the binomial expressions agree and the multiset expression sum to the correct result.
The only reaming summand in $h^{n-l+2}_l\{ \tilde{\gamma}_{i_1}^{a_1}\cdots\tilde{\gamma}_{i_k}^{a_k}\bar{\gamma}^{b} \}$ is the one corresponding to $\theta=b$.
However this agrees in (\ref{eq:ChoicExpanssion}) and (\ref{eq:Case(n,b)}) because $\multiset{n-l+2-k}{0}=\multiset{n-l+1-k}{0}=1$, with the binomial parts being identical.
\end{proof}
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\begin{remark}\label{rmk:NewBaisSymGrobner}
Note that for a fixed $n$ the set of expressions for $h^{n-l+2}_l$ with $2\leq l \leq n+1$ given in Proposition~\ref{prop:basis} remains a reduced Gr\"obner bases for the ideal they generate with respect to the lexicographic term order on variables $\tilde{\gamma}_1<\cdots<\tilde{\gamma}_n<\bar{\gamma}$.
\end{remark}
The next theorem shows that the change of basis interacts well with the differentials.
Recall that, the image of $d^m$ for $m\geq 2$ was determined in Theorem~\ref{thm:allDiff} and shown only to be non-trivial on terms containing $x_m$.
\begin{theorem}\label{thm:InductiveDiff}
For $2\leq l \leq n+1$,
\begin{equation*}
d^{2(l-1)}(x_{2(l-1)})=
\sum_{\substack{1 \leq i \leq n}}
{y_i(\tilde{\gamma}_i-\bar{\gamma})^{l-2}\tilde{\gamma}_i}.
\end{equation*}
After reduction by lower degree differentials this may be written as
\begin{equation*}
\sum_{\substack{1 \leq i \leq n}}
{y_i\tilde{\gamma}_i^{l-1}}.
\end{equation*}
\end{theorem}
\begin{proof}
Generalise the notation of Remark~\ref{rmk:TildeBasis} to reflect the form of all differentials on $x_{2(l-1)}$ as follows.
For each $1\leq i, \leq n$, write
\begin{equation*}
\tilde{\gamma}_i^{(j)}=
\sum_{\substack{|(c_1,\dots,c_{n})|=j}}
{(c_i+1)\gamma_1^{c_1}\cdots\gamma_i^{c_i}\cdots\gamma_n^{c_{n}}}.
\end{equation*}
In particular, we have $\tilde{\gamma}_i^{(1)}=\tilde{\gamma_i}$ and by Theorem~\ref{thm:allDiff}
\begin{equation*}
d^{2j}(x_{2j})=
\sum_{\substack{1 \leq i \leq n}}
{y_i\tilde{\gamma}_i^{(j)}}.
\end{equation*}
We next show that by quotienting out by symmetric polynomials,
\begin{equation}\label{eq:TildaRelation}
\tilde{\gamma}_i^{(j+1)}=\gamma_i\tilde{\gamma}_i^{(j)}
\end{equation}
and since by definition
\begin{equation*}
\gamma_i = \tilde{\gamma}_i-\bar{\gamma}
\end{equation*}
induction on $j$ then completes the proof of the first part of the theorem.
Notice that, using Definition~(\ref{defn:CompleteHomogeneous}) of the complete homogeneous symmetric polynomials
\begin{align*}
\tilde{\gamma}_i^{(j+1)} & =
\sum_{\substack{|(c_1,\dots,c_{n})|=j+1}}
{(c_i+1)\gamma_1^{c_1}\cdots\gamma_i^{c_i}\cdots\gamma_n^{c_{n}}} \\
& = h_{j+1} +
\sum_{\substack{|(c_1,\dots,c_{n})|=j+1,
\\ c_i \geq 1}}
{c_i\gamma_1^{c_1}\cdots\gamma_i^{c_i}\cdots\gamma_n^{c_{n}}} \\
& = h_{j+1} +
\gamma_i\sum_{\substack{|(c_1,\dots,c_{n})|=j+1}}
{c_i\gamma_1^{c_1}\cdots\gamma_i^{c_i-1}\cdots\gamma_n^{c_{n}}} \\
& = h_{j+1} + \gamma_i\tilde{\gamma}_i^{(j)}.
\end{align*}
This proves (\ref{eq:TildaRelation}).
The final statement of the theorem is proved by induction subtracting multiples of $\bar{\gamma}^m$ from the lower degree differentials.
\end{proof}
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\section{The integral cohomology of $\Lambda(SU(n+1)/T^n)$}\label{sec:Cohomology}
In this section we use the results in previous section to deduce structure in the final page of $\{ E^r,d^r \}$, the cohomology Leray-Serre spectral sequence of the evaluations fibration (\ref{eq:evalfib}).
This result alone performs part of the computation of the integral cohomology of the free loop space of the complete flag manifold of the special unitary group $SU(n+1)/T^n$ and is applied when $n=3$ in Section~\ref{sec:LSU4/T3}.
We first make the following notation and remark on the choice of basis up to sign.
For integers $1\leq t \leq j \leq n$ and $1\leq i_1 < \cdots < i_j \leq n$, write
\begin{equation}\label{eq:yHat}
\hat{y}_{i_1,\dots,i_j}
\end{equation}
to denote the product $y_{1}y_{2}\cdots y_{n}$ without the elements $y_{i_1},\dots,y_{i_j}$,
while
\begin{equation}\label{eq:yHatHat}
\hat{y}_{i_1,\dots,\hat{i}_t,\dots,i_j}
\end{equation}
denotes the product $y_{1}y_{2}\cdots y_{n}$ without the elements $y_{i_1},\dots,y_{i_j}$ except $y_{i_t}$ which is still included.
\begin{remark}\label{rmk:ySign}
By choosing the appropriate signs on generators $y_i$ for each $1\leq i \leq n$, we express the image of differentials given in Theorem~\ref{thm:InductiveDiff} by
\begin{equation*}
d^{2l}(x_{2l})=
\sum_{\substack{1 \leq i \leq n}}
{(-1)^{i+1}y_i(\tilde{\gamma}_i-\bar{\gamma})^{l-1}\tilde{\gamma}_i}
\end{equation*}
where $1\leq l \leq n$.
This choice is made to simplify multiplication of the differential by terms $y_1,\dots,y_n$ in the following way.
Recall that the generators $y_i$ generate an exterior algebra, in particular $y_i^2=0$.
So using Remark \ref{rmk:DifOnDivPoly}, for any $m\geq 1$, $1\leq j,l \leq n$ and $1\leq i_1 < \cdots < i_j \leq n$ we have
\begin{align*}
d^{2l}((x_{2l})_m\hat{y}_{i_1,\dots,i_j})
&=d^{2l}((x_{2l})_m)\hat{y}_{i_1,\dots,i_j} \\
&=(x_{2l})_{m-1}\sum_{t=1}^j{(-1)^{i_t+1}(-1)^{i_t+t-2}\hat{y}_{i_1,\dots,\hat{i}_t,\dots,i_j}(\tilde{\gamma}_{i_t}-\bar{\gamma})^{l-1}\tilde{\gamma}_{i_t}} \\
&=(x_{2l})_{m-1}\sum_{t=1}^j{(-1)^{t-1}\hat{y}_{i_1,\dots,\hat{i}_t,\dots,i_j}
(\tilde{\gamma}_{i_t}-\bar{\gamma})^{l-1}\tilde{\gamma}_{i_t}}
\end{align*}
where the additional $(-1)^{i_t+t-2}$ sign changes come from reordering the $y_i$.
The generator $y_t$ swaps places with $y_i$, $i_t-1$ times for $i<t$ changing the sign each time,
however $t-1$ of these $y_i$ are missing.
Therefore for the remainder of the section we assume the expression for the differential given above.
\end{remark}
The next theorem deduces for all $n\geq 2$ important structural information on the interaction between the images of differentials and the symmetric polynomials in a special case relevant to the spectral sequence $\{ E^r,\; d^r \}$.
In particular the second part of the theorem solves a particular case of part (2) of Proposition~\ref{thm:SpectralGrobner} and the first part describe the torsion generators in this case as detailed at the end of Proposition~\ref{thm:SpectralGrobner}.
Hence this result alone perform part of the calculations in Section~\ref{sec:LSU4/T3}.
\begin{theorem}\label{thm:Ideals}
Given integers
\begin{equation*}
1\leq i\leq n-1,\; 1\leq j,l \leq n,\; 2\leq j'\leq n+1 ,\; 1\leq k \leq n+1,\; 1\leq t < l,\; 1\leq i_1 \leq i_2 \leq \cdots \leq i_j \leq n
\end{equation*}
consider all ideals in $\mathbb{Z}[\tilde{\gamma}_i,\bar{\gamma},y_j]$ with $d^{2l}$ image as described in Remark~\ref{rmk:ySign}.
Then the following hold:
\begin{enumerate}
\item
There is a equality of ideals
\begin{equation*}\label{eq:BottemGrober}
\langle d^{2l}(x_{2l}\hat{y}_j),\; y_1\cdots y_n h_{j'}^{n-j'+2} \rangle
=
\langle y_1\cdots y_n\tilde{\gamma}_i,\; \binom{n+1}{k}y_1\cdots y_n\bar{\gamma}^k \rangle.
\end{equation*}
\item
For a fixed choice of $2 \leq l \leq n$, the intersection of ideals
\begin{equation*}\label{eq:IntersectBottem}
\langle d^{2l}(x_{2l}\hat{y}_j) \rangle
\cap
\langle d^{2t}(x_{2t}\hat{y}_j),\; y_1\cdots y_n h_{j'}^{n-j'+2} \rangle
\end{equation*}
is given by
\begin{align*}\label{eq:Bottem}
y_1\cdots y_n \langle & \left(\lcm \left(n+1,\binom{n+1}{j'}\right)
/\binom{n+1}{j'}
\right)
h_{j'}^{n-j'+2},
\tilde{\gamma}_i h_{j'}^{n-j'+2}, (n+1)\bar{\gamma}h_{j'}^{n-j'+2} \rangle
\end{align*}
if $l=1$ and
\begin{equation*}
\langle d^{2l}(x_{2l}\hat{y}_j) \rangle
\end{equation*}
otherwise.
\end{enumerate}
\end{theorem}
\begin{proof}
We begin by proving part $(2)$ in the case $l=1$, then extend to the general case.
Components of the proof of part $(2)$ are then used to prove part $(1)$ in the case $l=1$.
First rearranging the generator representatives, then using the second part of Theorem~\ref{thm:InductiveDiff} and Remark~\ref{rmk:TildeBasis}, we see that
\begin{align}\label{eq:d^2Ideal}
\langle d^{2}(x_2\hat{y}_j) \rangle
=& \langle (-1)^{i+1}d^2(x_2)\hat{y}_i,\; (-1)^{n+1}d^2(x_2)(\hat{y}_n+\sum_{1\leq k\leq n-1}(-1)^{i+1}\hat{y}_k)\rangle \nonumber
\\
=& \langle y_1\dots y_n \tilde{\gamma_i},\; (n+1)y_1\dots y_n \bar{\gamma} \rangle.
\end{align}
So when $l=1$ generators $y_1\cdots y_n\tilde{\gamma}_i h_{j'}^{n-j'+2}$ and $y_1\cdots y_n(n+1)\bar{\gamma}h_{j'}^{n-j'+2}$ lie on both sides of the intersection of ideals in part $(2)$,
hence are contained in the intersection.
By Proposition~\ref{prop:basis},
\begin{equation}\label{eq:SymBasis}
y_1,\dots,y_nh^{n-j'+2}_{j'}=y_1,\dots,y_n\sum_{\substack{0\leq t \leq j' \\ 1\leq i_1 \leq \cdots \leq i_t \leq n-j'+1}}
{(-1)^{j'-t} \binom{n+1}{j'-t} \tilde{\gamma}_{i_1}\cdots\tilde{\gamma}_{i_{t}}\bar{\gamma}^{j'-t}}.
\end{equation}
Notice that the terms of the sum with $t>0$,
are contained in $\langle y_1\dots y_n \tilde{\gamma}_i \rangle$, hence in $\langle d^{2l}(x_l\hat{y}_j) \rangle$ by (\ref{eq:d^2Ideal}).
Using again (\ref{eq:d^2Ideal}),
the left hand ideal of the intersection contains
the generator $(n+1)\bar{\gamma}$.
The summands of (\ref{eq:SymBasis}) with $t=0$, are divisible by $\bar{\gamma}$ not $\tilde{\gamma}_i$ and so are contained in the intersection
only when divisible by $n+1$.
Hence a scalar multiple of $h_{j'}^{n-{j'}+2}$ is in intersection
when multiplied by the least common multiple of the $n+1$ and $\binom{n+1}{j'}$ divided by $\binom{n+1}{j'}$ since all other terms in the summands of \eqref{eq:SymBasis} for $t>0$ are divisible by a $\tilde{\gamma}_i$.
Since all necessary degrees have been considered this completes the proof of prat $(2)$ when $l=1$.
Considering now $2 \leq l \leq n$, by Theorem~\ref{thm:InductiveDiff} combined with Remark~\ref{rmk:ySign}, we have
\begin{equation*}
d^{2l}(x_{2l})=
\sum_{\substack{1 \leq i \leq n}}
{(-1)^{i+1}y_i(\tilde{\gamma}_i-\bar{\gamma})^{l-1}\tilde{\gamma}_i}.
\end{equation*}
Hence applying the discussion in Remark~\ref{rmk:ySign}, we obtain
\begin{equation*}
d^{2l}(x_{2l}\hat{y}_j )=
{(-1)^{j+1}y_1\cdots y_n(\tilde{\gamma}_j-\bar{\gamma})^{l-1}\tilde{\gamma}_j}.
\end{equation*}
Therefore
\begin{equation*}
\langle d^{2l}(x_{2l}\hat{y}_j),\; y_1\cdots y_n h_{j'}^{n-j'+2} \rangle
=
\langle d^{2}(x_2\hat{y}_j),\; y_1\cdots y_n h_{j'}^{n-j'+2} \rangle
\end{equation*}
so the generators of $\langle \hat{y}_jd^{2l}(x_{2l}),\; y_1\cdots y_n h_{j'+1}^{n-j'+1} \rangle$ when $l>1$ may be omitted when considering the intersection of the ideals in part $(2)$ of the theorem.
In addition the equality of ideals in part $(1)$ also follows from (\ref{eq:d^2Ideal}) and (\ref{eq:SymBasis}).
When the generators of (\ref{eq:d^2Ideal}) are extended with those from (\ref{eq:SymBasis}),
using the discussion below (\ref{eq:SymBasis})
we see that the sum in (\ref{eq:SymBasis}) can be reduced to just $\binom{n+1}{k}y_1,\dots,y_n\bar{\gamma}$ for each $k=j'>1$.
This leaves the required set of ideal generators.
\end{proof}
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\begin{remark}\label{rmk:CasePre-image}
For the purposes of obtaining pre-images required in part (2) of Proposition~\ref{thm:SpectralGrobner}, we now express the intersection of ideas in part (2) of Theorem~\ref{thm:Ideals} when $2 \leq l \leq n$ in terms of $d^2(x_2)$ multiples free of terms divisible by symmetric polynomials.
As discussed in the proof of Theorem~\ref{thm:Ideals} $y_1\cdots y_n\tilde{\gamma}_i$ and $(n+1)y_1\cdots y_n\bar{\gamma}_n$ are in the image of the $d^2$ differential, so $y_1\cdots y_n\tilde{\gamma}_ih^{n-j'+2}_{j'}$ and $(n+1)y_1\cdots y_n\bar{\gamma}_nh^{n-j'+2}_{j'}$ need not be further considered. The remaining generators are
\begin{equation}\label{eq:HitGenerators}
\left(\lcm\left(n+1,\binom{n+1}{j'}\right)/\binom{n+1}{j'}\right)
y_1\cdots y_nh_{j'}^{n-j'+2}
\end{equation}
for each of which we want to obtain an generator of the preimage under $d^{2}$.
Recall that using Remarks~\ref{rmk:TildeBasis},~\ref{rmk:ySign} and Theorem~\ref{thm:InductiveDiff} we have that
\[
d^2(x_2\hat{y}_i)=
y_1\cdots y_n\tilde{\gamma}_i
\;\;\;\ \text{and} \;\;\;
d^2(x_2\hat{y}_n)=y_1\cdots y_n((n+1)\bar{\gamma}-\tilde{\gamma}_1-\cdots-\tilde{\gamma}_{n-1}).
\]
Hence we see that
\begin{align*}
& d^2 \Bigg{(}\lcm\left(n+1,\binom{n+1}{j'}\right)
x_2
\Bigg{(} (-1)^{j'}\frac{1}{n+1}
\bar{\gamma}^{j'-1}
(\hat{y}_1+\cdots+\hat{y}_{n-1}+\hat{y}_n)
\\ \nonumber & \;\;\;\; +
\frac{1}{\binom{n+1}{j'}}
\Bigg(
\sum_{\substack{1\leq t \leq j' \\ 1\leq i_1 \leq \cdots \leq i_t \leq n-j'+1}}
{(-1)^{j'-t} \binom{n+1}{j'-t}\hat{y}_{i_1} \tilde{\gamma}_{i_2}\cdots\tilde{\gamma}_{i_{t}}\bar{\gamma}^{j'-t}}
\Bigg) \Bigg{)}
\Bigg{)}
\\ & =
\left(\lcm\left(n+1,\binom{n+1}{j'}\right)/\binom{n+1}{j'}\right)
y_1\cdots y_n
\sum_{\substack{0\leq t \leq j' \\ 1\leq i_1 \leq \cdots \leq i_t \leq n-j'+1}}
{(-1)^{j'-t} \binom{n+1}{j'-t} \tilde{\gamma}_{i_1}\cdots\tilde{\gamma}_{i_{t}}\bar{\gamma}^{j'-t}}
\end{align*}
which using Proposition~\ref{prop:basis} we can see is the same as the expressions in \eqref{eq:HitGenerators}.
Therefore
\begin{align}\label{eq:PreImageHitGenerators}
&
\lcm\left(n+1,\binom{n+1}{j'}\right)
\Bigg(\frac{1}{n+1}
\bar{\gamma}^{j'-1}
(\hat{y}_1+\cdots+\hat{y}_{n-1}+\hat{y}_n)
\\ \nonumber & +
\frac{1}{\binom{n+1}{j'}}
\Bigg(
\sum_{\substack{1\leq t \leq j' \\ 1\leq i_1 \leq \cdots \leq i_t \leq n-j'+1}}
{(-1)^{t} \binom{n+1}{j'-t}\hat{y}_{i_1} \tilde{\gamma}_{i_2}\cdots\tilde{\gamma}_{i_{t}}\bar{\gamma}^{j'-t}}
\Bigg)\Bigg)
\end{align}
are representatives of the preimage under $d^2$ of generators in \eqref{eq:HitGenerators} as required.
We remove a multiple of $(-1)^{j'}$ as this only changes the sign of the generators.
\end{remark}
\section{The integral cohomology of $\Lambda(SU(4)/T^3)$}\label{sec:LSU4/T3}
Theorem~\ref{thm:Ideals} gives enough information to deduce the algebra structure for the final page of $\{ E^r,d^r \}$ in a special case within the cohomology Leray-Serre spectral sequence of the evaluations fibration (\ref{eq:evalfib}) converging to $H^*(\Lambda(SU(n+1)/T^n);\mathbb{Z})$.
Using Theorem~\ref{thm:Ideals}, Theorem~\ref{thm:InductiveDiff} and Proposition~\ref{thm:SpectralGrobner} we can deduce the final page completely in the case when $n=3$, which is considerably more complex than the rank $2$ case considered in \cite{Burfitt2018}.
\begin{theorem}\label{thm:CohomologySU4}
The $E^\infty$-page of the spectral sequence $\{ E^r,d^r \}$ of the evaluations fibration (\ref{eq:evalfib}) converging to $H^*(\Lambda(SU(4)/T^3);\mathbb{Z})$ is given by the algebra $A/I$, where
$A$ is the free graded commutative algebra
\begin{align*}
A = \Lambda_{\mathbb{Z}} ( &
\tilde{\gamma}_{1} ,\; \tilde{\gamma}_{2} ,\; \bar{\gamma} ,\;
y_1 ,\; y_2 ,\; y_3
,\;
(x_2)_{m_2}(x_4)_{m_4}(x_6)_{m_6}\tilde{\gamma}_2\tilde{\gamma}_1^2\bar{\gamma}^3,
(x_2)_{m_2}(x_4)_{m_4}(x_6)_{m_6}y_1y_2y_3
,\; \\ &
(x_2)_{m_2}(x_4)_{m_4}(x_6)_{m_6}(\tilde{\gamma}_1^2+\tilde{\gamma}_1\tilde{\gamma}_2+\tilde{\gamma}_2^2+4(\tilde{\gamma}_1+\tilde{\gamma}_2)\bar{\gamma}+6\bar{\gamma}^2)
,\; \\ &
(x_2)_{m_2}(x_4)_{m_4}(x_6)_{m_6}
(\tilde{\gamma}_1^3+4\tilde{\gamma}_1^2\bar{\gamma}+6\tilde{\gamma}_1\bar{\gamma}^2+4\bar{\gamma}^3)
,\;
(x_2)_{m_2}(x_4)_{m_4}(x_6)_{m_6}\bar{\gamma}^4
,\; \\ &
(x_2)_{m_2}(x_4)_{b_4}(x_6)_{b_6}(\hat{y}_1(2\tilde{\gamma}_1+2\tilde{\gamma}_2-5\bar{\gamma})+\hat{y}_2(2\tilde{\gamma}_2-5\bar{\gamma})+3\hat{y}_3\bar{\gamma})
,\; \\ &
(x_2)_{m_2}(x_4)_{b_4}(x_6)_{b_6}(\hat{y}_1(\tilde{\gamma}_1^2-4\tilde{\gamma}_1\bar{\gamma}+5\bar{\gamma}^2)-\hat{y}_1\bar{\gamma}^2-\hat{y}_3\bar{\gamma}^2)
,\;
(x_2)_{m_2}(x_4)_{b_4}(x_6)_{b_6}(\hat{y}_1+\hat{y}_2+\hat{y}_3)\bar{\gamma}^2
,\; \\ &
(x_4)_{m_4}(x_6)_{b_6}(\tilde{\gamma}_2\tilde{\gamma}_1^2-6\tilde{\gamma}_2\bar{\gamma}^2-6\tilde{\gamma}_1\bar{\gamma}^2-14\bar{\gamma}^3)
,\;
(x_4)_{m_4}(x_6)_{b_6}(\tilde{\gamma}_2\tilde{\gamma}_1\bar{\gamma}+3\tilde{\gamma}_2\bar{\gamma}^2+3\tilde{\gamma}_1\bar{\gamma}^2+6\bar{\gamma}^3)
,\; \\ &
(x_4)_{m_4}(x_6)_{b_6}\tilde{\gamma}_2\bar{\gamma}^3
,\;
(x_4)_{m_4}(x_6)_{b_6}(\tilde{\gamma}_1^2\bar{\gamma}+\tilde{\gamma}_1\bar{\gamma}^2)
,\;
(x_4)_{m_4}(x_6)_{b_6}\tilde{\gamma}_1\bar{\gamma}^3
,\; \\ &
(x_6)_{m_6}\tilde{\gamma}_1
,\;
(x_6)_{m_6}\tilde{\gamma}_2
,\;
(x_6)_{m_6}\bar{\gamma}
,\; \\ &
(x_2)_{m_2}(x_4)_{b_4}(x_6)_{b_6}(y_1\tilde{\gamma}_1-y_2\tilde{\gamma}_2-y_3(\tilde{\gamma}_2+\tilde{\gamma}_1+4\bar{\gamma}))
, \; \\ &
(x_4)_{m_4}(x_6)_{a_6}y_1\tilde{\gamma}_1^2-y_2\tilde{\gamma}_2^2+y_3(\tilde{\gamma}_2+\tilde{\gamma}_1+4\bar{\gamma})^2
, \; \\ &
(x_6)_{m_6}y_1\tilde{\gamma}_1^3-y_2\tilde{\gamma}_2^3-y_3(\tilde{\gamma}_2+\tilde{\gamma}_1+4\bar{\gamma})^3
, \; \\ &
(x_2)_{m_2}(x_4)_{b_4}(x_6)_{b_6}(
y_1(\tilde{\gamma}_2^2+4\tilde{\gamma}_2\bar{\gamma}+6\bar{\gamma}^2)
-y_2(\tilde{\gamma}_1^2+4\tilde{\gamma}_1\bar{\gamma}+6\bar{\gamma}^2)
-y_3(\tilde{\gamma}_2^2+4\tilde{\gamma}_2\bar{\gamma}-4\bar{\gamma}^2))
,\; \\ &
(x_2)_{m_2}(x_4)_{b_4}(x_6)_{b_6}(
y_1(\tilde{\gamma}_2\bar{\gamma}^3+\tilde{\gamma}_1\bar{\gamma}^3)
+y_2\tilde{\gamma}_1\bar{\gamma}^3
-y_3\tilde{\gamma}_2\bar{\gamma}^3
,\; \\ &
(x_2)_{m_2}(x_4)_{b_4}(x_6)_{b_6}(
y_1(2\tilde{\gamma}_2\bar{\gamma}^2+2\tilde{\gamma}_1\bar{\gamma}^2+8\bar{\gamma}^3)
-y_2(3\tilde{\gamma}_1^2\bar{\gamma}+10\tilde{\gamma}_1\bar{\gamma}^2+10\bar{\gamma}^3)
\\ & \;\;\;\;\;\;\;\;\;\;\;\;
+y_3(2\tilde{\gamma}_2\tilde{\gamma}_1^2+8\tilde{\gamma}_2\tilde{\gamma}_1\bar{\gamma}+10\tilde{\gamma}_2\bar{\gamma}^2+5\tilde{\gamma}_1^2\bar{\gamma}+20\tilde{\gamma}_1\bar{\gamma}^2+22\bar{\gamma}^3))
,\; \\ &
(x_2)_{m_2}(x_4)_{b_4}(x_6)_{b_6}(
y_2\tilde{\gamma}_1^2\bar{\gamma}^3
+y_3\tilde{\gamma}_1^2\bar{\gamma}^3)
,\; \\ &
(x_2)_{m_2}(x_4)_{b_4}(x_6)_{b_6}(
y_2(3\tilde{\gamma}_1^2\bar{\gamma}^2+12\tilde{\gamma}_1\bar{\gamma}^3)
\\ & \;\;\;\;\;\;\;\;\;\;\;\;
-y_3(2\tilde{\gamma}_2\tilde{\gamma}_1^2\bar{\gamma}+8\tilde{\gamma}_2\tilde{\gamma}_1\bar{\gamma}^2+12\tilde{\gamma}_2\bar{\gamma}^3+5\tilde{\gamma}_1^2\bar{\gamma}^2+20\tilde{\gamma}_1\bar{\gamma}^3))
,\; \\ &
(x_2)_{m_2}(x_4)_{b_4}(x_6)_{b_6}
y_3(\tilde{\gamma}_2\tilde{\gamma}_1^2\bar{\gamma}^2+4\tilde{\gamma}_2\tilde{\gamma}_1\bar{\gamma}^3+4\tilde{\gamma}_1^2\bar{\gamma}^3)
,\; \\ &
(x_4)_{m_4}(x_6)_{b_6}(y_1(\tilde{\gamma}_2+3\bar{\gamma})-y_2(\tilde{\gamma}_1+3\bar{\gamma})+y_3(2\tilde{\gamma}_2+2\tilde{\gamma}_1+6\bar{\gamma}))
,\; \\ &
(x_4)_{m_4}(x_6)_{b_6}(y_1\tilde{\gamma}_1+y_2(\tilde{\gamma}_1+3\bar{\gamma})-y_3(2\tilde{\gamma}_2+\tilde{\gamma}_1+5\bar{\gamma}))
,\; \\ &
(x_4)_{m_4}(x_6)_{b_6}(y_1\bar{\gamma}^2+y_3(\tilde{\gamma}_1^2+3\tilde{\gamma}_1\bar{\gamma}+3\bar{\gamma}^2))
,\; \\ &
(x_4)_{m_4}(x_6)_{b_6}(2y_1\bar{\gamma}-y_2(2\tilde{\gamma}_1+4\bar{\gamma})+y_3(2\tilde{\gamma}_2+2\tilde{\gamma}_1+6\bar{\gamma}))
,\; \\ &
(x_4)_{m_4}(x_6)_{b_6}(y_2(\tilde{\gamma}_2+\tilde{\gamma}_1+3\bar{\gamma})-y_3(\tilde{\gamma}_2+\bar{\gamma}))
,\; \\ &
(x_4)_{m_4}(x_6)_{b_6}(y_2\tilde{\gamma}_1^2-y_3(5\tilde{\gamma}_1^2+12\tilde{\gamma}_1\bar{\gamma}+12\bar{\gamma}^2))
,\; \\ &
(x_4)_{m_4}(x_6)_{b_6}(y_2\tilde{\gamma}_1\bar{\gamma}+y_3(3\tilde{\gamma}_1^2+7\tilde{\gamma}_1\bar{\gamma}+6\bar{\gamma}^2))
,\; \\ &
(x_4)_{m_4}(x_6)_{b_6}(y_2\bar{\gamma}^2-y_3(\tilde{\gamma}_1^2+2\tilde{\gamma}_1\bar{\gamma}+\bar{\gamma}^2))
,\; \\ &
(x_4)_{m_4}(x_6)_{b_6}y_3(\tilde{\gamma}_2\bar{\gamma}+\tilde{\gamma}_1\bar{\gamma}+4\bar{\gamma}^2)
,\; \\ &
(x_4)_{m_4}(x_6)_{b_6}y_3(\tilde{\gamma}_1^2\bar{\gamma}-2\bar{\gamma}^3)
,\;
(x_4)_{m_4}(x_6)_{b_6}y_3(\tilde{\gamma}_1\bar{\gamma}^2+2\bar{\gamma}^3)
,\; \\ &
(x_6)_{m_6}(y_1-y_2)
,\;
(x_6)_{m_6}y_3
)
\end{align*}
with $|\tilde{\gamma}_{j}|=|\bar{\gamma}|=2$, $|(x_k)_{m_k}|=km_k$ and $|(x_k)_{a_k}|=ka_k$, $I$ is the ideal
\begin{align*}
I = [ &
(x_2)_{a_2}
((x_2)_1^b-!(x_2)_b)s_2
,\; \\ &
(x_4)_{a_4}
((x_4)_1^b-!(x_4)_b)s_4
,\; \\ &
(x_6)_{a_6}
((x_6)_1^b-!(x_6)_b)s_6
,\; \\ &
(x_2)_{a_2}(x_4)_{a_4}(x_6)_{a_6}(\tilde{\gamma}_1^2+\tilde{\gamma}_1\tilde{\gamma}_2+\tilde{\gamma}_2^2+4(\tilde{\gamma}_1+\tilde{\gamma}_2)\bar{\gamma}+6\bar{\gamma}^2)
,\; \\ &
(x_2)_{a_2}(x_4)_{a_4}(x_6)_{a_6}
(\tilde{\gamma}_1^3+4\tilde{\gamma}_1^2\bar{\gamma}+6\tilde{\gamma}_1\bar{\gamma}^2+4\bar{\gamma}^3)
,\; \\ &
(x_2)_{a_2}(x_4)_{a_4}(x_6)_{a_6}\bar{\gamma}^4
,\; \\ &
(x_2)_{a_2}(x_4)_{a_4}(x_6)_{a_6}
(y_1\tilde{\gamma}_1+y_2\tilde{\gamma}_2+y_3(4\bar{\gamma}-\tilde{\gamma}_1-\tilde{\gamma}_2))
,\; \\ &
(x_2)_{a_2}(x_4)_{a_4}(x_6)_{a_6}
(y_1\tilde{\gamma}_1^2+y_2\tilde{\gamma}_2^2+y_3(4\bar{\gamma}-\tilde{\gamma_1}-\tilde{\gamma}_2)^2)
,\; \\ &
(x_2)_{a_2}(x_4)_{a_4}(x_6)_{a_6}
(y_1\tilde{\gamma}_1^3+y_2\tilde{\gamma}_2^3+y_3(4\bar{\gamma}-\tilde{\gamma}_1-\tilde{\gamma}_2)^3)
]
\end{align*}
for elements $s_k\in S_k$ given by
\begin{equation*}
S_k = \{ a/{(x_k)_{m_k}} \; | \; a \text{ is a generator of } A \text{ divisible by some } (x_k)_{m_k} \}
\end{equation*}
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where $b\geq 1$, $k=2,\:4,\:6$ and $m_k,a_k\geq 0$ with the additional condition that some $m_k\geq 1$ when appearing in a generator of $A$.
Furthermore, the cohomology algebra $H^*(\Lambda(SU(4)/T^3);\mathbb{Z})$ is isomorphic as a module
to the algebra $A/I$ up to order of $2$-torsion and $4$-torsion.
In addition there are no multiplicative extension problem on
the sub-algebra generated by $\gamma_1, \: \gamma_2, \: \gamma_3$, the sub-algebra generated by $y_1, \: y_2, \: y_3$
and no multiplicative extension on elements $y_i\gamma_j$ for $1\leq i,j\leq 3$.
\end{theorem}
\begin{proof}
Throughout the proof all indices lie in the same ranges given in the statement of the theorem.
All examples of code used for computation can be found in \cite{Burfitt2021}.
Consider the cohomology Leray-Serre spectral sequence $\{ E_r,d^r \}$ associated to the evaluation fibration of $SU(4)/T^3$ studied in Section~\ref{sec:FreeLoopSU(n+1)/Tn},
\begin{equation*}
\Omega(SU(4)/T^{3})\to\Lambda(SU(4)/T^3)\to SU(4)/T^3.
\end{equation*}
We first consider the algebra structure of the $E_{\infty}$-page of the spectral sequence,
then consider the implications for the cohomology algebra.
By Theorem~\ref{thm:H*SU/T} and using the basis of Remark~\ref{rmk:TildeBasis}, the integral cohomology of the base space $SU(4)/T^3$ is given by
\begin{equation*}
\frac{\mathbb{Z}[\tilde{\gamma}_1,\:\tilde{\gamma}_2,\:\bar{\gamma}]}{\langle h_{2}^{3},\: h_{3}^{2},\: h_{4}^{1} \rangle}
\end{equation*}
where $|\tilde{\gamma}_j|=|\bar{\gamma}|=2$.
Using Proposition~\ref{prop:basis} after changing the sign of the $\bar{\gamma}$ generator, we have
\begin{align}\label{eq:SU4SymetricQuotient}
& h_{2}^{3} =
\tilde{\gamma}_2^2+\tilde{\gamma}_1\tilde{\gamma}_2+\tilde{\gamma}_1^2+4(\tilde{\gamma}_2+\tilde{\gamma}_1)\bar{\gamma}+6\bar{\gamma}^2
,\; \nonumber \\ &
h_{3}^{2} =
\tilde{\gamma}_1^3+4\tilde{\gamma}_1^2\bar{\gamma}+6\tilde{\gamma}_1\bar{\gamma}^2+4\bar{\gamma}^3
\; \\ \text{and } &
h_{4}^{1} = \bar{\gamma}^4. \nonumber
\end{align}
Therefore, the maximal degree of elements of the algebra is $6$.
From~(\ref{eq:BaseLoopFlag}), the integral cohomology of the fibre $\Omega(SU(4)/T^3)$ is given by
\begin{equation}\label{eq:SU4LoopFibre}
\Lambda_\mathbb{Z}(y_1,\:y_2,\:y_3)\otimes\Gamma_\mathbb{Z}[x_2,\:x_4,\:x_6]
\end{equation}
where $|y_i|=1$ and $|x_k|=k$.
Applying Theorem~\ref{thm:GrobnerOver} with the Gr\"obner basis from Theorems \ref{thm:monomial sum}, the additive generators on the $E_2$-page of the spectral sequence are given by representative elements of the form
\begin{equation}\label{eq:SU4Representatives}
(x_2)_{a_2}(x_4)_{a_4}(x_6)_{a_6}y_{\alpha_1}\cdots y_{\alpha_l}P
\end{equation}
where $0\leq l \leq 3$, $1\leq \alpha_1< \cdots < \alpha_{l}\leq 3$ and $P\in \mathbb{Z}[\tilde{\gamma}_1,\:\tilde{\gamma}_2,\:\bar{\gamma}]$ is a monomial of degree at most $6$.
By Theorem~\ref{thm:allDiff}, the only non-zero differentials are $d^2,\;d^4$ and $d^6$
which are non-zero only on the generators $x_2,\:x_4$ and $x_6$, respectively.
Therefore the spectral sequence converges by the seventh page.
Using Theorem~\ref{thm:InductiveDiff} and substituting $\tilde{\gamma}_3$ for $-(4\bar{\gamma}+\gamma_1+\gamma_2)$ in the basis of Remark~\ref{rmk:TildeBasis} and the sign change on $\bar{\gamma}$ made in (\ref{eq:SU4SymetricQuotient}), the images of the differentials up to sign are generated by
\begin{align}\label{eq:SU4Difs}
d^2(x_2) &= y_1\tilde{\gamma}_1-y_2\tilde{\gamma}_2-y_3(4\bar{\gamma}+\gamma_1+\gamma_2)
, \nonumber \\
d^4(x_4) &= y_1\tilde{\gamma}_1^2-y_2\tilde{\gamma}_2^2+y_3(4\bar{\gamma}+\tilde{\gamma}_1+\tilde{\gamma}_2)^2
\\
\text{and } d^6(x_6) &=
y_1\tilde{\gamma}_1^3-y_2\tilde{\gamma}_2^3-y_3(4\bar{\gamma}+\gamma_1+\gamma_2)^3.
\nonumber
\end{align}
At this point we immediately obtain a number of the generators and relations occurring in $A$ and $I$ of the statement of the theorem.
The monomial generators
\begin{equation*}
\tilde{\gamma}_1,\:\tilde{\gamma}_2,\:\bar{\gamma},\: y_1,\:y_2,\:y_3,
(x_2)_{m_2}(x_4)_{m_4}(x_6)_{m_6}\tilde{\gamma}_2\tilde{\gamma}_1^2\bar{\gamma}^3
\text{ and }
(x_2)_{m_2}(x_4)_{m_4}(x_6)_{m_6}y_1y_2y_3
\end{equation*}
occur in $E_2^{*,0}$ or $E_2^{0,*}$ and are always in the kernel of the differentials, so are algebra generators of the $E_\infty$-page.
All relations on the $E_7$-page coming from the divided polynomial relations in
$H^*(\Omega(SU(4)/T^3);\mathbb{Z})$ given in (\ref{eq:SU4LoopFibre}) of the form
$(x_k)_1^m-m!(x_k)_m$,
the symmetric relations in (\ref{eq:SU4SymetricQuotient}) and
images of the differentials (\ref{eq:SU4Difs})
hold on the $E_\infty$-page and therefore are in $I$.
It is also necessary to include the multiple of these relations by $(x_2)_{a_2}(x_4)_{a_4}(x_6)_{a_6}$
to ensure that they occur as generators of the algebra $A$.
In addition, we add to the relations $((x_k)_1^m-m!(x_k)_m)s_k$, where the multiple of an element $s_k$ from the set
$S_k$ ensures all generators that occur as a multiple of some $(x_k)_{a_k}$ appear in $I$.
We have considered all possible relations occurring on the $E_{\infty}$-page of the spectral sequence,
so it remains to determine all generators of $A$.
Any additional generators arise as elements whose image under the differentials obtained using (\ref{eq:SU4Difs}) lie inside the ideal generated by the symmetric relations~(\ref{eq:SU4SymetricQuotient}).
To this end we need only consider elements whose image is generated by elements from (\ref{eq:SU4Representatives}) with $l=1,2,3$.
The case $l=3$ coincides with Theorem~\ref{thm:Ideals}.
Since up to $(x_2)_{a_2}(x_4)_{a_4}(x_6)_{a_6}$ multiples, the image of $d^2$ in this case can be rearranged as
\begin{equation*}
y_1y_2y_2\tilde{\gamma}_1,
\; y_1y_2y_2\tilde{\gamma}_2,
\; 4y_1y_2y_2\bar{\gamma}
\end{equation*}
there are no generators corresponding to part (1) of Proposition~\ref{thm:SpectralGrobner} in this case.
In addition in the course of the proof of Theorem~\ref{thm:Ideals}
it is shown that the images of $d^4$ and $d^6$ in this case are contained in the image of $d^2$ so their domain on these rows lies entirely in the kernel.
Part (2) of Proposition~\ref{thm:SpectralGrobner} coincides with the results of part (2) of Theorem~\ref{thm:Ideals},
so using the expression \eqref{eq:PreImageHitGenerators} from Remark \ref{rmk:CasePre-image} we add to the generator of $A$, the following expressions
\begin{equation*}
\hat{y}_1(2\tilde{\gamma}_1+2\tilde{\gamma}_2-5\bar{\gamma})+\hat{y}_2(2\tilde{\gamma}_2-5\bar{\gamma})+3\hat{y}_3\bar{\gamma}
,\;
\hat{y}_1(\tilde{\gamma}_1^2-4\tilde{\gamma}_1\bar{\gamma}+5\bar{\gamma}^2)-\hat{y}_1\bar{\gamma}^2-\hat{y}_3\bar{\gamma}^2
,\;
(\hat{y}_1+\hat{y}_2+\hat{y}_3)\bar{\gamma}^2
.
\end{equation*}
As noted in Remark~\ref{rmk:NewBaisSymGrobner}, the generators (\ref{eq:SU4SymetricQuotient}) are a Gr{\"o}bner basis with respect to the lexicographic minimal ordering given by
\begin{equation*}
\tilde{\gamma}_2 > \tilde{\gamma}_1 > \bar{\gamma}.
\end{equation*}
Hence using Theorem~\ref{thm:GrobnerOver}, we see that any element
on the $E_2$-page
may be reduced by elements of (\ref{eq:SU4SymetricQuotient}) to a unique form and this is zero if and only if it is a multiple of an element of the symmetric ideal. In addition, from the leading terms of (\ref{eq:SU4SymetricQuotient}) it is clear that we need only consider up to
\begin{equation*}
\tilde{\gamma}_2\tilde{\gamma}_1^2\bar{\gamma}^3
\end{equation*}
multiples of the image of the differentials given in (\ref{eq:SU4Difs}).
We now consider elements in the rows of the spectral sequence corresponding to elements in (\ref{eq:SU4Representatives}) when $l=1$.
In this case the image of $d^2,d^4$ and $d^6$ on $x_2,x_4$ and $x_6$, respectively are generated by a single element, so there are no generators of the kernel corresponding to part (1) of Proposition~\ref{thm:SpectralGrobner} that we have not already included.
Applying part (2) of Proposition~\ref{thm:SpectralGrobner} to rows in this case,
we make all Gr\"obner bases computations up to degree $6$ in variables $\tilde{\gamma}_1,\: \tilde{\gamma}_2\:, \bar{\gamma}$ and degree $1$ in variables $y_1,\:y_2,\:y_3$.
For the general case it is sufficient to consider the (\ref{eq:SU4SymetricQuotient}) reduced forms of $\tilde{\gamma}_1,\:\tilde{\gamma}_2,\:\bar{\gamma}$ multiples of the images of differentials given in (\ref{eq:SU4Difs}), as other rows will be multiples of theses by elements of $\Gamma_{\mathbb{Z}[x_2.x_4,x_6]}$.
In computer computations, we use the extend lexicographic monomial ordering
\begin{equation*}
y_1 > y_2 > y_3 > \tilde{\gamma}_2 > \tilde{\gamma}_1 > \bar{\gamma}.
\end{equation*}
In the case of $\phi^2_2$ on multipels of $x_2$, using Gr\"obner bases to compute the intersection of the ideals generated by $\phi^2_2(x_2)$ and (\ref{eq:SU4SymetricQuotient}) gives an ideal generated by the three elements $\phi^2_2(x_2)h^3_2$, $\phi^2_2(x_2)h^2_3$ and $\phi^2_2(x_2)h^4_1$,
all of which are are multiples of elements of (\ref{eq:SU4SymetricQuotient}), hence no additional generators need be added to $A$ in this case.
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Computing the Gr\"obner bases to obtain the intersection of the ideals generated by $\phi^4_4$ on multiples of $x_4$ and (\ref{eq:SU4SymetricQuotient}) gives an ideal generated by elements $\phi^4_4(x_4)h^2_3$, $\phi^4_4(x_4)h^4_1$ and
\begin{align*}
& \phi^4_4(x_4)(\tilde{\gamma}_2\tilde{\gamma}_1^2-6\tilde{\gamma}_2\bar{\gamma}^2-6\tilde{\gamma}_1\bar{\gamma}^2-14\bar{\gamma}^3), \\
& \phi^4_4(x_4)(\tilde{\gamma}_2\tilde{\gamma}_1\bar{\gamma}+3\tilde{\gamma}_2\bar{\gamma}^2+3\tilde{\gamma}_1\bar{\gamma}^2+6\bar{\gamma}^3), \\
& \phi^4_4(x_4)\tilde{\gamma}_2\bar{\gamma}^3, \\
& \phi^4_4(x_4)(\tilde{\gamma}_1^3+2\tilde{\gamma}_1\bar{\gamma}^2+4\bar{\gamma}^3), \\
& \phi^4_4(x_4)(\tilde{\gamma}_1^2\bar{\gamma}+\tilde{\gamma}_1\bar{\gamma}^2), \\
& \phi^4_4(x_4)\tilde{\gamma}_1\bar{\gamma}^3.
\end{align*}
Since
\begin{equation*}
h_2^3 = (\tilde{\gamma}_1^3+2\tilde{\gamma}_1\bar{\gamma}^2+4\bar{\gamma}^3)+4(\tilde{\gamma}_1^2\bar{\gamma}+\tilde{\gamma}_1\bar{\gamma}^2)
\end{equation*}
we need not add any generators to $A$ as multiples $x_4(\tilde{\gamma}_1^3+2\tilde{\gamma}_1\bar{\gamma}^2+4\bar{\gamma}^3)$.
It can again be checked by Gr\"obner bases applying part (2) of Proposition~\ref{thm:SpectralGrobner} that $(x_4)_{m_4}(x_6)_{b_6}$ multiples of
\begin{align*}
\tilde{\gamma}_2\tilde{\gamma}_1^2-6\tilde{\gamma}_2\bar{\gamma}^2-6\tilde{\gamma}_1\bar{\gamma}^2-14\bar{\gamma}^3,\:
\tilde{\gamma}_2\tilde{\gamma}_1\bar{\gamma}+3\tilde{\gamma}_2\bar{\gamma}^2+3\tilde{\gamma}_1\bar{\gamma}^2+6\bar{\gamma}^3,\:
\tilde{\gamma}_2\bar{\gamma}^3,\:
\tilde{\gamma}_1^2\bar{\gamma}+\tilde{\gamma}_1\bar{\gamma}^2
\text{ and } \tilde{\gamma}_1\bar{\gamma}^3
\end{align*}
are in the kernel of $d^6$, therefore they are added to $A$ as generators since the spectral sequence converges on the $E_7$-page.
Here part (1) of Proposition~\ref{thm:SpectralGrobner} need not be considered as we have already confirmed that everything lies in the kernel.
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Finally the Gr\"obner bases of the intersection of the ideals generated by $\phi^6_6(x_6)$ and (\ref{eq:SU4SymetricQuotient}) gives an ideal generated by elements
\begin{equation*}
d^6(x_6)\tilde{\gamma}_2,\: d^6(x_6)\tilde{\gamma}_1,\: d^6(x_6)\bar{\gamma}
\end{equation*}
all of whose $(x_6)_{m_6}$ multiples are added to $A$ as generators.
Lastly we consider elements in the rows of the spectral sequence corresponding to the elements in (\ref{eq:SU4Representatives}) when $l=2$.
In the case of $\phi^2_2$ on multiples of $y_ix_2$ for $i=1,2,3$
having image generated by
\begin{align*}
y_1d^2(x_2) & = -y_1y_2\tilde{\gamma}_2 -y_1y_3(\tilde{\gamma}_2+\tilde{\gamma}_1+4\bar{\gamma}),
\\
y_2d^2(x_2) & = -y_1y_2\tilde{\gamma}_1 -y_2y_3(\tilde{\gamma}_2+\tilde{\gamma}_1+4\bar{\gamma})
\\
\text{and } y_3d^2(x_2) & = -y_1y_3\tilde{\gamma}_1 +y_2y_3\tilde{\gamma}_2.
\end{align*}
We first note that
\begin{equation*}
y_1d^2(x_2)\tilde{\gamma}_1-y_2d^2(x_2)\tilde{\gamma}_2-y_3d^2(x_2)(\tilde{\gamma}_2+\tilde{\gamma}_1+4\bar{\gamma}) = 0.
\end{equation*}
Similarly we have the following relations of the $d^4$ and $d^6$ differentials,
\begin{align*}
& y_1d^2(x_2)\tilde{\gamma}_1^2-y_2d^2(x_2)\tilde{\gamma}_2^2+y_3d^2(x_2)(\tilde{\gamma}_2+\tilde{\gamma}_1+4\bar{\gamma})^2 = 0
\\ \text{and} \;\;\; & y_1d^2(x_2)\tilde{\gamma}_1^3-y_2d^2(x_2)\tilde{\gamma}_2^3-y_3d^2(x_2)(\tilde{\gamma}_2+\tilde{\gamma}_1+4\bar{\gamma})^3 = 0.
\end{align*}
Therefore
$(x_2)_{m_2}(x_4)_{b_4}(x_6)_{b_6}$ multiples of $y_1\tilde{\gamma}_1-y_2\tilde{\gamma}_2-y_3(\tilde{\gamma}_2+\tilde{\gamma}_1+4\bar{\gamma})$,
$(x_4)_{m_4}(x_6)_{b_6}$ multiples of $y_1\tilde{\gamma}_1^2-y_2\tilde{\gamma}_2^2+y_3(\tilde{\gamma}_2+\tilde{\gamma}_1+4\bar{\gamma})^2$
and
$(x_6)_{m_6}$ multiples of $y_1\tilde{\gamma}_1^3-y_2\tilde{\gamma}_2^3-y_3(\tilde{\gamma}_2+\tilde{\gamma}_1+4\bar{\gamma})^3$
are added to $A$.
It can be checked by computing the Syzygys of the ideals spanned by the image of the three sets of differentials that these are the only additional generators not already included in $A$ corresponding to part (1) of Proposition~\ref{thm:SpectralGrobner}.
Considering part (2) of Proposition~\ref{thm:SpectralGrobner}, using Gr\"obner bases to compute the intersection of the ideals generated by $\phi^2_2y_i(x_2)$ and (\ref{eq:SU4SymetricQuotient}) gives an ideal generated by elements corresponding to
$y_i\phi^2_2(x_2)h_3^2,\: y_i\phi^2_2(x_2)h_2^3,\: y_i\phi^2_2(x_2)h_1^4$ and
\begin{align*}
& -y_1\phi^2_2(x_2)(\tilde{\gamma}_2^2+4\tilde{\gamma}_2\bar{\gamma}+6\bar{\gamma}^2)
+y_2\phi^2_2(x_2)(\tilde{\gamma}_1^2+4\tilde{\gamma}_1\bar{\gamma}+6\bar{\gamma}^2)
+y_3\phi^2_2(x_2)(\tilde{\gamma}_2^2+4\tilde{\gamma}_1^2
-4\bar{\gamma}^2)
,\\
& -y_1\phi^2_2(x_2)(\tilde{\gamma}_2\bar{\gamma}^3+\tilde{\gamma}_1\bar{\gamma}^3)
-y_2\phi^2_2(x_2)\tilde{\gamma}_1\bar{\gamma}^3
+y_3\phi^2_2(x_2)\tilde{\gamma}_2\bar{\gamma}^3
,\\
& -y_1\phi^2_2(x_2)(2\tilde{\gamma}_2\bar{\gamma}^2+2\tilde{\gamma}_1\bar{\gamma}^2+8\bar{\gamma}^3)
+y_2\phi^2_2(x_2)(3\tilde{\gamma}_1^2\bar{\gamma}+10\tilde{\gamma}_1\bar{\gamma}^2+10\bar{\gamma}^3) \\ & \;\;\;\;\;\;\;\;\;\;\;\;
-y_3\phi^2_2(x_2)(2\tilde{\gamma}_2\tilde{\gamma}_1^2+8\tilde{\gamma}_2\tilde{\gamma}_1\bar{\gamma}+10\tilde{\gamma}_2\bar{\gamma}^2+5\tilde{\gamma}_1^2\bar{\gamma}+20\tilde{\gamma}_1\bar{\gamma}^2+22\bar{\gamma}^3)
,\\
& -y_2\phi^2_2(x_2)\tilde{\gamma}_1^2\bar{\gamma}^3-y_3\phi^2_2(x_2)\tilde{\gamma}_1^2\bar{\gamma}^3
,\\
& -y_2\phi^2_2(x_2)(3\tilde{\gamma}_1^2\bar{\gamma}^2+12\tilde{\gamma}_1\bar{\gamma}^3)
+y_3\phi^2_2(x_2)(2\tilde{\gamma}_2\tilde{\gamma}_1^2\bar{\gamma}+8\tilde{\gamma}_2\tilde{\gamma}_1\bar{\gamma}^2+12\tilde{\gamma}_2\bar{\gamma}^3+5\tilde{\gamma}_1^2\bar{\gamma}^2+20\tilde{\gamma}_1\bar{\gamma}^3)
,\\
& -y_3\phi^2_2(x_2)(\tilde{\gamma}_2\tilde{\gamma}_1^2\bar{\gamma}^2+4\tilde{\gamma}_2\tilde{\gamma}_1\bar{\gamma}^3+4\tilde{\gamma}_1^2\bar{\gamma}^3)
\end{align*}
all of whose corresponding $d^2$ preimage $(x_2)_{m_2}(x_4)_{b_4}(x_6)_{b_6}$ multiples are added to $A$ as generators up to sign as it can again be check with Gr\"obner bases that all generators remain in the kernels of the $d^4$ and $d^6$ differentials.
Using Gr\"obner bases to compute the intersection of the ideals generated by $y_i\phi^4_4(x_4)$ and (\ref{eq:SU4SymetricQuotient}) gives an ideal generated by elements corresponding to
$y_3\phi^4_4(x_4)h_1^4$ and
\begin{align}
& \label{eq:kerd4row21}
-y_1\phi^4_4(x_4)(\tilde{\gamma}_2+3\bar{\gamma})+y_2\phi^4_4(x_4)(\tilde{\gamma}_1+3\bar{\gamma})-y_3\phi^4_4(x_4)(2\tilde{\gamma}_2+2\tilde{\gamma}_1+6\bar{\gamma})
,\\ & \label{eq:kerd4row22}
-y_1\phi^4_4(x_4)\tilde{\gamma}_1-y_2\phi^4_4(x_4)(\tilde{\gamma}_1+3\bar{\gamma})+y_3\phi^4_4(x_4)(2\tilde{\gamma}_2+\tilde{\gamma}_1+5\bar{\gamma})
,\\ & \label{eq:kerd4row23}
-y_1\phi^4_4(x_4)\bar{\gamma}^2-y_3\phi^4_4(x_4)(\tilde{\gamma}_1^2+3\tilde{\gamma}_1\bar{\gamma}+3\bar{\gamma}^2)
,\\ & \label{eq:kerd4row24}
-2y_1\phi^4_4(x_4)\bar{\gamma}+y_2\phi^4_4(x_4)(2\tilde{\gamma}_1+4\bar{\gamma})-y_3\phi^4_4(x_4)(2\tilde{\gamma}_2+2\tilde{\gamma}_1+6\bar{\gamma})
,\\ & \label{eq:kerd4row25}
-y_2\phi^4_4(x_4)(\tilde{\gamma}_2+\tilde{\gamma}_1+3\bar{\gamma})+y_3\phi^4_4(x_4)(\tilde{\gamma}_2+\bar{\gamma})
,\\ & \label{eq:kerd4row26}
-y_2\phi^4_4(x_4)\tilde{\gamma}_1^2+y_3\phi^4_4(x_4)(5\tilde{\gamma}_1^2+12\tilde{\gamma}_1\bar{\gamma}+12\bar{\gamma}^2)
,\\ & \label{eq:kerd4row27}
-y_2\phi^4_4(x_4)\tilde{\gamma}_1\bar{\gamma}-y_3\phi^4_4(x_4)(3\tilde{\gamma}_1^2+7\tilde{\gamma}_1\bar{\gamma}+6\bar{\gamma}^2)
,\\ & \label{eq:kerd4row28}
-y_2\phi^4_4(x_4)\bar{\gamma}^2+y_3\phi^4_4(x_4)(\tilde{\gamma}_1^2+2\tilde{\gamma}_1\bar{\gamma}+\bar{\gamma}^2)
,\\ & \label{eq:kerd4row29}
-y_3\phi^4_4(x_4)(\tilde{\gamma}_2^2-\tilde{\gamma}_1^2-5\tilde{\gamma}_1\bar{\gamma}-10\bar{\gamma}^2)
,\\ & \label{eq:kerd4row210}
-y_3\phi^4_4(x_4)(\tilde{\gamma}_2\tilde{\gamma}_1+2\tilde{\gamma}_1^2+5\tilde{\gamma}_1\bar{\gamma})
,\\ & \label{eq:kerd4row211}
-y_3\phi^4_4(x_4)(\tilde{\gamma}_2\bar{\gamma}+\tilde{\gamma}_1\bar{\gamma}+4\bar{\gamma}^2)
,\\ & \label{eq:kerd4row212}
-y_3\phi^4_4(x_4)\tilde{\gamma}_1^3
,\\ & \label{eq:kerd4row213}
-y_3\phi^4_4(x_4)(\tilde{\gamma}_1^2\bar{\gamma}-2\bar{\gamma}^3)
,\\ & \label{eq:kerd4row214}
-y_3\phi^4_4(x_4)(\tilde{\gamma}_1\bar{\gamma}^2+2\bar{\gamma}^3).
\end{align}
Notice that
\begin{align*}
-y_2\phi^4_4(x_4)h^2_3 = &
\tilde{\gamma}_2(\ref{eq:kerd4row25})+(\ref{eq:kerd4row26})+\bar{\gamma}(\ref{eq:kerd4row25})+3(\ref{eq:kerd4row27})+3(\ref{eq:kerd4row28})+(\ref{eq:kerd4row29})+2(\ref{eq:kerd4row211})
, \\
-y_3\phi^4_4(x_4)h^2_3 = &
(\ref{eq:kerd4row29})+(\ref{eq:kerd4row210})+4(\ref{eq:kerd4row211})
, \\
-y_3\phi^4_4(x_4)h^3_2 = &
(\ref{eq:kerd4row212})+4(\ref{eq:kerd4row213})+6(\ref{eq:kerd4row214}).
\end{align*}
Hence as can again be checked that all of these generator are in the kernel of $d^6$, the $(x_4)_{m_4}(x_6)_{b_6}$ multiples of $d^4$ preimages of all except
(\ref{eq:kerd4row29}), (\ref{eq:kerd4row210}) and (\ref{eq:kerd4row212})
are added to $A$ as generators up to sign.
Using Gr\"obner bases to compute the intersection of the ideals generated by $\phi^6_6y_i(x_6)$ and (\ref{eq:SU4SymetricQuotient}) gives an ideal generated by the elements corresponding to
\begin{align*}
&
-y_1\phi^6_6(x_6)+y_2\phi^6_6y_i(x_6)
,\;
-y_2\phi^6_6(x_6)\tilde{\gamma}_2
,\;
-y_2\phi^6_6(x_6)\tilde{\gamma}_1
,\;
-y_2\phi^6_6(x_6)\bar{\gamma}
\text{ and }
-y_3\phi^6_6(x_6).
\end{align*}
Only the first and last of whose $(x_6)_{m_6}$ multiples are added to $A$ as generators up to sign, as the others are already products of exiting generators.
By computing the Gr\"obner bases corresponding to (\ref{eq:GrobnerTorsionPart}) of Proposition~\ref{thm:SpectralGrobner}, we determine that only $2$-torsion and $4$-torsion occurs on the $E_\infty$-page of the spectral sequence.
Hence module structure of the integral cohomology algebra of $SU(4)/T^3$ up to torsion type is determined by looking at the spectral sequence in modulo $2$ coefficients in the same way as in \cite[Theorem~5.2]{Burfitt2018} where the modulo $3$ spectral sequence is considered.
The only remaining additive extension problem is whether the $4$-torsion on the $E_\infty$-page is $2$-torsion or $4$-torsion in $H^*(\Lambda (SU(4)/T^3);\mathbb{Z})$.
The multiplicative extension problems on certain subalgebras are also determined in the same way as in the proof of \cite[Theorem~5.2]{Burfitt2018}.
\end{proof}
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\end{document}
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\begin{document}
\title{Information geometry of warped product spaces}
\begin{abstract}
Information geometry is an important tool to study statistical models. There are some important examples in statistical models which are regarded as warped products. In this paper, we study
information geometry of warped products. We consider the case where the warped product and its fiber space are equipped with dually flat connections and, in the particular case of a cone, characterize the connections on the base space $\mathbb{R}_{>0}$. The resulting connections turn out to be the $\alpha$-connections with $\alpha = \pm{1}$.
\end{abstract}
\tableofcontents
\section{Introduction}
Recently, the study of spaces consisting of probability measures is getting more attention. As tools to investigate such spaces, there are two famous theories in geometry:
information geometry and Wasserstein geometry. Information geometry is mainly concerned with finite dimensional statistical models and Wasserstein geometry is concerned with infinite dimensional spaces of probability measures.
We can compare these two geometries, for example, on Gaussian distributions.
This paper concerns information geometry of a warped product and, in particular, on a cone, which is a kind of warped product of the line $\mathbb{R}_{>0}$ and a manifold.
Under some natural assumptions, we characterize connections on the line, with which warped products are constructed.
The assumption we set is different from that of \cite{leo} and matches examples of statistical models.
Examples of warped product metrics include the denormalizations of the Fisher metric, the Bogoliubov-Kubo-Mori metric and the Fisher metric on the Takano Gaussian space, which is a set of multivariate Gaussian distributions with restricted parameters.
Besides these examples, there are some more statistical models represented as warped products.
In \cite{takatsu2}, it was shown that the Wasserstein Gaussian space, which is the set of multivariate Gaussian distributions on $\mathbb{R}^n$ with mean zero equipped with the $L^2$-Wasserstein metric, has a cone structure and in \cite{location} the relations between Fisher metrics of location scale models and warped product metrics are studied.
It seems that warped products get more attention in the field of statistical models than before.
Although information geometry is studied on real manifolds, the theory of statistical manifolds is studied in the field of affine geometry and statistical structures on complex manifolds get more attention as in \cite{furuhata1}.
Also in this field, warped products are important since they play an important role in the theory of submanifolds in complex manifolds, for example, CR submanifold theory as in \cite{chen1}.
There are many researches extending the theories of CR submanifolds in K\"{a}hler manifolds to submanifolds in holomorphic statistical manifolds as in \cite{chen2}.
Statistical structures in \cite{furuhata2} and the structures cultivated in this paper are slightly different because we do not care the compatibility of statistical structures and complex structures. This compatibility is expressed in the definition of holomorphic statistical structures in \cite{furuhata1}.
This paper is organized as follows. In Section 2, we briefly review information geometry. Section 3 is devoted to some formulas in warped products. Then in Section 4, we study cones and consider necessary conditions for making both the cone and the fiber space to
be dually flat. This necessary condition states that there are only two possible connections on the line. The following theorem is one of our main results.
\begin{theorem*}\rm{(Theorem 4.1)}
Under Assumption \ref{assume2}, we have
\begin{equation*}
D_{\partial_t}\partial_t = \frac{1}{t}\frac{\partial}{\partial t} \mbox{ or } -\frac{1}{t}\frac{\partial}{\partial t},
\end{equation*}
where $t$ is the natural coordinate on the line $\mathbb{R}_{>0}$, which is the base space of the warped product.
\end{theorem*}
By observing examples, these two connections turn out to be the $\alpha$-connections with $\alpha = \pm{1}$.
An analogous characterization for the Takano Gaussian space is also considered in Section 5. In Section 6, we discuss dually flat connections on the Wasserstein Gaussian space. We remark that, although it is known in \cite{tayebi} that there is no dually flat proper doubly warped Finsler manifold, what they actually proved is that some coordinates cannot be dual affine coordinates.
Thus our claims do not contradict their claim. We also discuss this point in Section 6.
In Section 7, we study two-dimensional warped products as an appendix.
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\section{Introduction}
Recently, the study of spaces consisting of probability measures is getting more attention. As tools to investigate such spaces, there are two famous theories in geometry:
information geometry and Wasserstein geometry. Information geometry is mainly concerned with finite dimensional statistical models and Wasserstein geometry is concerned with infinite dimensional spaces of probability measures.
We can compare these two geometries, for example, on Gaussian distributions.
This paper concerns information geometry of a warped product and, in particular, on a cone, which is a kind of warped product of the line $\mathbb{R}_{>0}$ and a manifold.
Under some natural assumptions, we characterize connections on the line, with which warped products are constructed.
The assumption we set is different from that of \cite{leo} and matches examples of statistical models.
Examples of warped product metrics include the denormalizations of the Fisher metric, the Bogoliubov-Kubo-Mori metric and the Fisher metric on the Takano Gaussian space, which is a set of multivariate Gaussian distributions with restricted parameters.
Besides these examples, there are some more statistical models represented as warped products.
In \cite{takatsu2}, it was shown that the Wasserstein Gaussian space, which is the set of multivariate Gaussian distributions on $\mathbb{R}^n$ with mean zero equipped with the $L^2$-Wasserstein metric, has a cone structure and in \cite{location} the relations between Fisher metrics of location scale models and warped product metrics are studied.
It seems that warped products get more attention in the field of statistical models than before.
Although information geometry is studied on real manifolds, the theory of statistical manifolds is studied in the field of affine geometry and statistical structures on complex manifolds get more attention as in \cite{furuhata1}.
Also in this field, warped products are important since they play an important role in the theory of submanifolds in complex manifolds, for example, CR submanifold theory as in \cite{chen1}.
There are many researches extending the theories of CR submanifolds in K\"{a}hler manifolds to submanifolds in holomorphic statistical manifolds as in \cite{chen2}.
Statistical structures in \cite{furuhata2} and the structures cultivated in this paper are slightly different because we do not care the compatibility of statistical structures and complex structures. This compatibility is expressed in the definition of holomorphic statistical structures in \cite{furuhata1}.
This paper is organized as follows. In Section 2, we briefly review information geometry. Section 3 is devoted to some formulas in warped products. Then in Section 4, we study cones and consider necessary conditions for making both the cone and the fiber space to
be dually flat. This necessary condition states that there are only two possible connections on the line. The following theorem is one of our main results.
\begin{theorem*}\rm{(Theorem 4.1)}
Under Assumption \ref{assume2}, we have
\begin{equation*}
D_{\partial_t}\partial_t = \frac{1}{t}\frac{\partial}{\partial t} \mbox{ or } -\frac{1}{t}\frac{\partial}{\partial t},
\end{equation*}
where $t$ is the natural coordinate on the line $\mathbb{R}_{>0}$, which is the base space of the warped product.
\end{theorem*}
By observing examples, these two connections turn out to be the $\alpha$-connections with $\alpha = \pm{1}$.
An analogous characterization for the Takano Gaussian space is also considered in Section 5. In Section 6, we discuss dually flat connections on the Wasserstein Gaussian space. We remark that, although it is known in \cite{tayebi} that there is no dually flat proper doubly warped Finsler manifold, what they actually proved is that some coordinates cannot be dual affine coordinates.
Thus our claims do not contradict their claim. We also discuss this point in Section 6.
In Section 7, we study two-dimensional warped products as an appendix.
\section{Preliminaries}
\subsection{Information geometry}\label{infogeo}
We briefly review the basics of information geometry, we refer to \cite{amari} for further reading.
Let $(M,g)$ be a Riemannian manifold and $\nabla$ be an affine connection of $M$.
$\mathfrak{X}(M)$ denotes the set of $C^\infty$ vector fields on $M$.
We define another affine connection $\nabla^*$ by
\begin{equation*}
Xg(Y,Z) = g(\nabla_XY,Z) + g(Y,\nabla^*_XZ)
\end{equation*}
for $X,Y,Z\in\mathfrak{X}(M)$. We call $\nabla^*$ the \emph{dual connection} of $\nabla$. We define the torsion and the curvature of $\nabla$ by
\begin{equation*}
T(X,Y) := \nabla_XY - \nabla_YX - [X,Y],\quad
R(X,Y)Z := [\nabla_X,\nabla_Y]Z - \nabla_{[X,Y]}Z,
\end{equation*}
respectively.
If $R$ satisfies
\begin{equation*}
R(X,Y)Z = k\{g(Y,Z)X - g(X,Z)Y\}
\end{equation*}
for some $k\in\mathbb{R}$ and all $X,Y,Z\in\mathfrak{X}(M)$, $(M,g,\nabla)$ is called a space of constant curvature $k$. We summarize some important facts on $\nabla$ and $\nabla^*$ in the following.
\begin{proposition}\label{levi}
Let $\nabla$ and $\nabla^*$ be dual affine connections of $M$. If two of the following conditions hold true, then the other two of them also hold true:
\begin{itemize}
\item $\nabla$ is torsion free,
\item $\nabla^*$ is torsion free,
\item $\nabla g$ is a symmetric tensor,
\item $\frac{\nabla + \nabla^*}{2}$ is the Levi-Civita connection of $g$.
\end{itemize}
\end{proposition}
\begin{proposition}\label{curvature_prop}
Let $(M,g,\nabla,\nabla^*)$ be a Riemannian manifold with dual affine connections.
The curvature with respect to $\nabla$ vanishes if and only if the curvature with respect to $\nabla^*$ vanishes.
\end{proposition}
Let $(M,g,\nabla,\nabla^*)$ be a Riemannian manifold with dual affine connections. If the torsion and the curvature with respect to $\nabla$
and those of $\nabla^*$ all vanish, then we say that $(M,g,\nabla,\nabla^*)$ is \emph{dually flat}.
For a local coordinate system $(U; x_1,\cdots, x_n)$, if the Christoffel symbols $\{\Gamma^k_{ij}\}$ of $\nabla$
vanish, we call it \emph{$\nabla$-affine coordinates}.
\begin{proposition}
Let $(M,g,\nabla,\nabla^*)$ be a Riemannian manifold with dual affine connections. If it is dually flat, then there exist $\nabla$-affine coordinates $(x_i)$ and $\nabla^*$-affine coordinates $(y_j)$ such that
\begin{equation*}
g\left(\frac{\partial}{\partial x_i},\frac{\partial}{\partial y_j}\right) = \delta_{ij}.
\end{equation*}
\end{proposition}
The coordinates $\{(x_i),(y_j)\}$ above are called \emph{dual affine coordinates}.
Using dual affine coordinates, we can construct the canonical divergence [2, \S 3.4].
Next, we introduce the Fisher metric and $\alpha$-connections.
Consider a family $\mathcal{S}$ of probability distributions on a finite set $\mathcal{X}$.
Suppose that $\mathcal{S}$ is parameterized by $n$ real-valued variables $[\xi^1,\ldots,\xi^n]$ so that
\begin{equation*}
\mathcal{S} := \{p_\xi = p(x;\xi)\mid \xi = [\xi^1,\ldots,\xi^n]\in\Xi\},
\end{equation*}
where $\Xi$ is an open subset of $\mathbb{R}^n$.
For $\alpha \in\mathbb{R}, u>0, x\in\mathcal{X}$ and $\xi\in\Xi$, we put
\begin{equation*}
L^{(\alpha)}(u) := \begin{cases}
\frac{2}{1-\alpha}u^{\frac{1-\alpha}{2}} & (\alpha\neq 1),\\
\log u & (\alpha = 1),
\end{cases}
\quad
l^{(\alpha)}(x;\xi) := L^{(\alpha)}(p(x;\xi)).
\end{equation*}
Then, we define the \emph{Fisher metric} $g$ as
\begin{equation*}
g_{ij}(\xi) := \int\partial_il^{(\alpha)}(x;\xi)\partial_jl^{(-\alpha)}(x;\xi)\, dx,
\end{equation*}
and \emph{$\alpha$-connections} $\nabla^{(\alpha)}$ as
\begin{equation*}
\Gamma^{(\alpha)}_{ij,k}(\xi) := \int\partial_i\partial_j l^{(\alpha)}(x;\xi)\partial_kl^{(-\alpha)}(x;\xi)\, dx,
\end{equation*}
where $g(\nabla^{(\alpha)}_{\partial_i}\partial_j,\partial_k) = \Gamma^{(\alpha)}_{ij,k}$.
Note that the Fisher metric does not depend on $\alpha$. We set
\begin{equation*}
\tilde{\mathcal{S}} := \{\tau p_\xi \mid \xi \in\Xi , \tau > 0\},
\end{equation*}
and call it the \emph{denormalization} of $\mathcal{S}$.
In \cite{amari}, the Fisher metric and connections on $\tilde{\mathcal{S}}$ are defined as follows. An extension $\tilde{l}$ of $l$ is defined as
\begin{equation*}
\widetilde{l}^{(\alpha)} = \widetilde{l}^{(\alpha)}(x;\xi,\tau) := L^{(\alpha)}(\tau p(x;\xi)).
\end{equation*}
Using this $\tilde{l}$, we define the metric and connections on $\widetilde{\mathcal{S}}$ by
\begin{equation}\label{netric_denormalization}
\tilde{g}_{ij}(\xi) := \int\partial_i\tilde{l}^{(\alpha)}\partial_j\tilde{l}^{(-\alpha)}\, dx,\quad\tilde{\Gamma}^{(\alpha)}_{ij,k} = \int\partial_i\partial_j \tilde{l}^{(\alpha)}\partial_k\tilde{l}^{(-\alpha)}\, dx.
\end{equation}
\subsection{Quantum information geometry}
Information geometry of density matrices is called quantum information geometry.
The set of density matrices $\mathcal{D}$ is defined as
\begin{equation*}
\mathcal{D} := \{\rho\in\mathbb{P}(n) | \mathrm{Tr}(\rho) = 1\},
\end{equation*}
where $\mathbb{P}(n)$ is the set of $n\times n$ positive definite Hermitian matrices.
Parameterizing elements of $\mathcal{D}$ as $\rho_\xi$ by $\xi\in\Xi$, the $m$-representation of the natural basis is written as
\begin{equation*}
(\partial_i)^{(m)} = \partial_i\rho.
\end{equation*}
The \emph{mixture connection} $\nabla^{(m)}$ is a connection such that
\begin{equation*}
(\nabla^{(m)}_{\partial_i}\partial_j)^{(m)} = \partial_i\partial_j\rho.
\end{equation*}
We set
\begin{equation*}
\mathcal{MON} := \left\{f:\mathbb{R}_{>0}\rightarrow \mathbb{R}_{>0} | f \mbox{ is operator monotone}, \,f(1) = 1, f(t) = tf\left(\frac{1}{t}\right)\right\}.
\end{equation*}
The \emph{monotone metric} for $f\in\mathcal{MON}$ is expressed as
\begin{equation*}
g^f_\rho(X,Y) = \mbox{Tr}\left\{X^*\frac{1}{(2\pi i)^2}\oint\oint c(\xi,\eta) \frac{1}{\xi-\rho} Y \frac{1}{\eta-\rho} \, d\xi d\eta \right\},
\end{equation*}
where $c(x,y) = 1/(yf(x/y))$ and $\xi(t), \eta(t)$ are paths surrounding the positive spectrum of $\rho$. The monotone metric for $f(x) = (x-1)/\log x$ is called the Bogoliubov-Kubo-Mori (BKM) metric. We refer to \cite{amari} and \cite{dit} for further reading.
It is known that the BKM metric enjoys the following remarkable property.
\begin{proposition}
$(\mathcal{D},\rm{BKM})$ equipped with the mixture connection is dually flat.
\end{proposition}
We can find a proof of this proposition in [2, Theorem 7.1], and the proof does not use the condition that the matrices considered have trace 1. Thus
we can prove the proposition below in completely the same way.
\begin{proposition}
$(\mathbb{P}(n),\rm{BKM})$ equipped with the mixture connection is dually flat.
\end{proposition}
\subsection{Takano Gaussian space}\label{takano_gauss}
In this subsection, we explain some results from \cite{takano}. We consider multivariate Gaussian distributions
\begin{equation*}
p(x;\xi) = \frac{1}{(\sqrt{2\pi}\sigma)^n}\prod_{i=1}^n \exp\left\{-\frac{(x_i-m_i)^2}{2\sigma^2}\right\},
\end{equation*}
where $\xi = (\sigma,m_1,\ldots , m_n)\in L^{(n+1)}, \,L^{(n+1)} := \mathbb{R}_{>0}\times \mathbb{R}^n$.
By a straightforward calculation, we obtain the Fisher metric $G$ as
\begin{equation*}
G_{\sigma\sigma} = \frac{2n}{\sigma^2},\quad G_{\sigma i} = G_{i \sigma} = 0,\quad G_{ij} = \frac{1}{\sigma^2}\delta_{ij},
\end{equation*}
where $\partial_\sigma = \partial/\partial\sigma$ and $\partial_i = \partial/\partial m_i$, i.e.,
\begin{equation*}
ds^2 = \frac{1}{\sigma^2}(2nd\sigma^2 + dm_1^2 + \cdots + dm_n^2).
\end{equation*}
Its $\alpha$-connections are
\begin{equation*}
\Gamma^{(\alpha)}_{ij,k} = 0, \quad \Gamma_{ij,\sigma}^{(\alpha)} = \frac{1-\alpha}{\sigma^3}\delta_{ij},\quad \Gamma^{(\alpha)}_{i\sigma,k} = -\frac{1 + \alpha}{\sigma^3}\delta_{ik},
\end{equation*}
\begin{equation*}
\Gamma^{(\alpha)}_{i\sigma,\sigma} = 0,\quad \Gamma^{(\alpha)}_{\sigma\sigma,i} = 0,\quad \Gamma^{(\alpha)}_{\sigma\sigma,\sigma} = -(1 + 2\alpha)\frac{2n}{\sigma^3},
\end{equation*}
and
\begin{equation*}
\nabla^{(\alpha)}_{\partial_i}\partial_j = \frac{1-\alpha}{2n\sigma}\delta_{ij}\partial_\sigma,\quad \nabla^{(\alpha)}_{\partial_i}\partial_\sigma = \nabla^{(\alpha)}_{\partial_\sigma}\partial_i = -\frac{1 + \alpha}{\sigma}\partial_i,\quad \nabla^{(\alpha)}_{\partial_\sigma}\partial_\sigma = -\frac{1 + 2\alpha}{\sigma}\partial_\sigma.
\end{equation*}
In \cite{takano}, they call \emph{$\alpha$-flat} if the curvature tensor with respect to the $\alpha$-connection vanishes identically and the following fact is proved.
\begin{proposition}
$(L^{(n + 1)},ds^2,\nabla^{(\alpha)})$ is a space of constant curvature $-\frac{(1-\alpha)(1 + \alpha)}{2n}$. In particular,
$(L^{(n + 1)},ds^2)$ is $(\pm{1})$-flat.
\end{proposition}
For simplicity, we call $(L^{(n + 1)},ds^2)$ the \emph{Takano Gaussian space} in this paper.
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\section{Warped products}\label{cone}
In this section, we calculate dual affine connections on warped products.
\subsection{Koszul formula}
Let $\nabla$, $\nabla^*$ be torsion free dual affine connections on $(M,g)$.
For $X,Y,Z,W \in \mathfrak{X}(M)$, let us first see a kind of Koszul formula for $\nabla$. Summing up
\begin{eqnarray*}
Xg(Y,Z) &=& g(\nabla_XY,Z) + g(Y,\nabla^*_XZ)\nonumber,\\
Yg(X,Z) &=& g(\nabla_YX,Z) + g(X,\nabla^*_YZ)\nonumber,\\
-Zg(X,Y) &=& -g(\nabla_ZX,Y) - g(X,\nabla^*_ZY),
\end{eqnarray*}
we get
\begin{eqnarray*}
Xg(Y,Z) + Yg(X,Z) - Zg(X,Y) &=& g(\nabla_XY,Z) + g(\nabla_YX,Z) + g(Y,\nabla^*_XZ - \nabla_ZX) + g(X,\nabla^*_YZ - \nabla^*_ZY).
\end{eqnarray*}
Recalling that we consider torsion free affine connections, we have
\begin{equation}\label{star}
2g(\nabla_XY,Z) = Xg(Y,Z) + Yg(X,Z) - Zg(X,Y) + g([X,Y],Z) - g(Y,\nabla^*_XZ-\nabla_ZX) - g(X,[Y,Z]).
\end{equation}
In order to have a further look on $(\nabla^*_XZ-\nabla_ZX)$, we put
\begin{equation*}
a(X,W) := \nabla^*_XW - \nabla_WX,
\end{equation*}
and calculate
\begin{eqnarray*}
a(X,W) - a(W,X) &=& (\nabla^*_XW-\nabla_WX) - (\nabla^*_WX - \nabla_XW) = 2[X,W],\nonumber\\
a(X,W) + a(W,X) &=& (\nabla^*_XW - \nabla_XW) + (\nabla^*_WX - \nabla_WX) = -2\left(P_XW + P_WX\right),
\end{eqnarray*}
where we put
\begin{equation*}
P := \frac{\nabla-\nabla^*}{2}.
\end{equation*}
Let us collect some properties of $P$.
\begin{lemma}
Let $f$ be an arbitrary $C^\infty$ function on $M$. For any $X,Y,Z\in\mathfrak{X}(M)$, we have the following equations:
\begin{equation}
\label{aa}
P_XY = P_YX,
\end{equation}
\begin{equation}
\label{bb}
g(P_XY,Z) = g(Y,P_XZ),
\end{equation}
\begin{equation}
\label{cc}
P_{fX} Y = fP_XY,\quad P_XfY = fP_XY.
\end{equation}
\begin{proof}
For (\ref{aa}),
\begin{equation*}
P_XY - P_YX = \frac{\nabla_XY-\nabla_YX}{2} - \frac{\nabla^*_XY - \nabla^*_YX}{2} = \frac{1}{2}([X,Y]-[X,Y]) = 0.
\end{equation*}
For (\ref{bb}),
\begin{eqnarray*}
g\left(\frac{\nabla_X-\nabla^*_X}{2}Y,Z\right) &=& \frac{1}{2}\left(g(\nabla_XY,Z)-g(\nabla^*_XY,Z)\right)\nonumber\\
&=& \frac{1}{2}\left(Xg(Y,Z)-g(Y,\nabla^*_XZ)\right)-\frac{1}{2}(Xg(Y,Z)-g(Y,\nabla_XZ))\nonumber\\
&=& \frac{1}{2}\left(g(Y,\nabla_XZ)-g(Y,\nabla^*_XZ)\right)\nonumber\\
&=& g(Y,P_XZ).
\end{eqnarray*}
For (\ref{cc}), the first equation is clear and we also observe
\begin{equation*}
P_X(fY) = \frac{\nabla_X(fY) - \nabla^*_X(fY)}{2} = Xf\frac{Y - Y}{2} + f\frac{\nabla_XY - \nabla^*_XY}{2} = fP_XY.
\end{equation*}
\end{proof}
\end{lemma}
By the above lemma, we can express $a$ using $P$ as
\begin{equation*}
a(X,W) + a(W,X) = -2(P_XW + P_WX) = -4P_XW,\quad a(X,W) = [X,W] - 2P_XW.
\end{equation*}
Substituting this into (\ref{star}), we obtain the following Koszul formula:
\begin{equation}
2g(\nabla_XY,Z) = Xg(Y,Z) + Yg(X,Z) - Zg(X,Y) + g([X,Y],Z) - g(Y,[X,Z] - 2P_XZ) - g(X,[Y,Z]).\label{starstar}
\end{equation}
\subsection{O'Neill formulas for affine connections on warped products}
Let $(B,g_B),(F,g_F)$ be Riemannian manifolds, $f$ be a positive $C^{\infty}$-function on $B$,
$M := B\times_f F$ be the warped product of them equipped with the metric $G := g_B + f^2 g_F$, and $D,D^*$ be torsion free dual affine connections on $M$.
Denote by $\mathcal{L}(F)$, $\mathcal{L}(B)$ the sets of lifts of vector fields on $F$ to $M$, $B$ to $M$, respectively.
Let $X,Y,Z\in\mathcal{L}(B)$, $U,V,W\in\mathcal{L}(F)$ in the sequel.
We will assume the following.
\begin{assume}\label{assume1}
$D_XY\in\mathcal{L}(B)$ for all $X,Y \in\mathcal{L}(B)$.
\end{assume}
\begin{lemma}\label{3_2}
Under Assumption \ref{assume1}, we have
\begin{equation*}
G(D^*_XY,V) = 0,\quad G(P_XV,Y) = 0.
\end{equation*}
\end{lemma}
\begin{proof}
The first equation follows from Assumption \ref{assume1} and the fact that $(D + D^*)/2$ is the Levi-Civita connection (recall Proposition \ref{levi}).
To see the second equation, since $G(Y,V) = G(X,V) = 0$ and $[X,V] = [Y,V] = 0$, we have
\begin{eqnarray*}
2G(D_XY,V) &=& XG(Y,V) + YG(X,V) - VG(X,Y) + G([X,Y],V) - G(Y,[X,V]-2P_XV) - G(X,[Y,V])\nonumber\\
&=& -VG(X,Y) + G([X,Y],V) + G(Y,2P_XV)\nonumber\\
&=& G(Y,2P_XV).
\end{eqnarray*}
By combining this with $2G(D_XY,V) = 0$ by Assumption \ref{assume1}, the second equation holds.
\end{proof}
Let us modify some formulas on warped products in \cite{oneil} for the Levi-Civita connections to those for affine connections.
First we express $D_VX$.
On the one hand, $G(D_XV,Y)= 0$ by the Koszul formula (\ref{starstar}) and Lemma \ref{3_2}. On the other hand, it follows from (\ref{bb}) and (\ref{starstar}) that
\begin{eqnarray*}
2G(D_XV,W) &=& XG(V,W) + VG(X,W) - WG(X,V) + G([X,V],W) - G(V,[X,W]-2P_XW) - G(X,[V,W])\nonumber\\
&=& XG(V,W) + 2G(V,P_XW)\nonumber\\
&=& XG(V,W) + 2G(P_XV,W).
\end{eqnarray*}
Since
\begin{equation*}
XG(V,W) = 2\frac{Xf}{f}G(V,W)
\end{equation*}
by the definition of $G$ and $D$ is torsion free, we obtain
\begin{equation}\label{alpha}
D_VX = D_XV =\frac{Xf}{f}V + P_XV.
\end{equation}
Next we consider $D_VW$. Observe that
\begin{equation*}\label{gamma}
G(D_VW,X) = -G(W,D^*_VX) = -G\left(W,\frac{Xf}{f}V-P_XV\right).
\end{equation*}
Using $Xf = G(\mbox{grad }f,X)$, (\ref{aa}) and (\ref{bb}), we have
\begin{equation*}
G(D_VW,X) = G\left(-\frac{G(V,W)}{f}\mbox{grad }f + P_VW,X\right).
\end{equation*}
Thus we obtain
\begin{equation}\label{beta}
\mbox{Hor }D_VW = -\frac{G(V,W)}{f}\mbox{grad }f + \mbox{Hor }P_VW,
\end{equation}
where $\mbox{Hor}$ denotes the projection to $TB$.
\begin{remark}
As another way to reach these formulas, we can use the fact that $(D + D^*)/2$ is the Levi-Civita connection and formulas in \cite{oneil}.
For example,
\begin{equation*}
\left(\frac{D + D^*}{2}\right)_XV = \frac{Xf}{f}V
\end{equation*}
implies
\begin{equation*}
D_XV = \frac{Xf}{f}V + \frac{D_XV - D^*_XV}{2} = \frac{Xf}{f}V + P_XV.
\end{equation*}
\end{remark}
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\section{Cones}
In this section, we specialize our study of warped products to cones.
We fix our framework and assumptions (including Assumption \ref{assume1}).
\begin{assume}\label{assume2}
Let $B = \mathbb{R}_{>0}$ with the Euclidean metric $g_B$ such that $g_B(\frac{\partial}{\partial t},\frac{\partial}{\partial t}) = 1$, $f(t) = t$ and $(\widetilde{\nabla},\widetilde{\nabla}^*)$ be dually flat affine connections on $(F,g_F)$.
Let $D$, $D^*$ be dually flat affine connections on $B\times_f F$ and $G$ be its warped product metric. We assume that $D$ satisfies
\begin{itemize}
\item $D_XY \mbox{ is horizontal, } i.e. , \,D_XY\in\mathcal{L}(B)$ for all $X,Y\in\mathcal{L}(B)$,
\item $\mbox{Ver }(D_VW) = \mbox{Lift }(\widetilde{\nabla}_VW)$ for all $V,W\in\mathcal{L}(F)$,
\end{itemize}
where Ver is the projection to $TF$.
\end{assume}
\begin{remark}
Denote the curvature with respect to $D$ by $R$ and the curvature with respect to $\widetilde{\nabla}$ by ${}^FR$. Denote their duals by $R^*$ and $^FR^{*}$.
Note that by Proposition \ref{curvature_prop}, we have $R^* = {}^FR^* = 0$ when $R = {}^FR = 0$.
\end{remark}
In the following arguments in this section, we assume this assumption without mentioning.
We shall study what $R = {}^FR =0$ means and characterize admissible connections on $B$ (Theorem 4.1).
\subsection{Calculations of $G(R(U,V)V,U)$}\label{second}
We first consider in vertical directions.
We are going to calculate the Gauss equation (the relatioin between $R$ and $^FR$) for affine connections on $F$ and $M$ in a similar way to \cite{oneil}.
For $U,V,W,Q\in\mathcal{L}(F)$, it follows from Assumption 4.1 that
\begin{eqnarray*}
G(D_UD_VW,Q) &=& G(D_U(\mbox{Ver }D_VW),Q) + G(D_U(\mbox{Hor }D_VW),Q)\nonumber\\
&=& G(\widetilde{\nabla}_U\widetilde{\nabla}_VW,Q) + \left\{UG(I\hspace{-.1em}I(V,W),Q) - G(I\hspace{-.1em}I(V,W),D^*_UQ)\right\}\nonumber\\
&=& G(\widetilde{\nabla}_U\widetilde{\nabla}_VW,Q) - G(I\hspace{-.1em}I(V,W),\mbox{Hor }D^*_UQ)\nonumber\\
&=& G(\widetilde{\nabla}_U\widetilde{\nabla}_VW,Q) -G(I\hspace{-.1em}I(V,W),I\hspace{-.1em}I^*(U,Q)),
\end{eqnarray*}
where $I\hspace{-.1em}I(W,X) := \mbox{Hor }{D}_WX$, which is an affine version of the second fundamental form on $F$. Thus we have
\begin{eqnarray}\label{gauss1}
G(R(U,V)W,Q) &=& G({}^FR(U,V)W,Q) - G(I\hspace{-.1em}I(V,W),I\hspace{-.1em}I^*(U,Q)) + G(I\hspace{-.1em}I(U,W),I\hspace{-.1em}I^*(V,Q)).
\end{eqnarray}
Recall that ${}^FR = 0$ by Assumption \ref{assume2}. Observe from (\ref{beta}) that
\begin{eqnarray*}
G(I\hspace{-.1em}I(V,W),I\hspace{-.1em}I^*(U,Q)) &=& G\left(-\frac{G(V,W)}{f}\mbox{grad}f + \mbox{Hor}(P_VW), -\frac{G(U,Q)}{f}\mbox{grad}f - \mbox{Hor}(P_UQ)\right)\nonumber \\
&=& \frac{\|\mbox{grad}f\|^2}{f^2}G(V,W)G(U,Q) - G(\mbox{Hor}(P_VW),\mbox{Hor}(P_UQ))\nonumber \\
&&\qquad+ \frac{G(V,W)G(\mbox{grad}f,P_UQ)}{f} - \frac{G(U,Q)G(\mbox{grad}f,P_VW)}{f},
\end{eqnarray*}
and similarly
\begin{eqnarray*}
G(I\hspace{-.1em}I(U,W),I\hspace{-.1em}I^*(V,Q)) &=& \frac{\|\mbox{grad}f\|^2}{f^2}G(U,W)G(V,Q) - G(\mbox{Hor}(P_UW),\mbox{Hor}(P_VQ)) \nonumber \\
&&\qquad+ \frac{G(U,W)G(\mbox{grad}f,P_VQ)}{f} - \frac{G(V,Q)G(\mbox{grad}f,P_UW)}{f}.
\end{eqnarray*}
Substituting these and letting $W=V$ and $Q = U$,
\begin{eqnarray*}
G(R(U,V)V,U) &=& - \frac{\|\mbox{grad }f\|^2}{f^2}\left\{G(V,V)G(U,U) - G(U,V)^2\right\}\nonumber\\
&&\qquad + G(\mbox{Hor}(P_VV),\mbox{Hor}(P_UU)) -G(\mbox{Hor}(P_UV),\mbox{Hor}(P_VU)) - \frac{G(V,V)G(\mbox{grad }f,P_UU)}{f} \nonumber\\
&&\qquad + \frac{G(U,U)G(\mbox{grad }f,P_VV)}{f}+ \frac{G(U,V)G(\mbox{grad }f,P_VU)}{f} - \frac{G(V,U)G(\mbox{grad }f,P_UV)}{f}\nonumber\\
&=& - \frac{\|\mbox{grad }f\|^2}{f^2}\left\{G(V,V)G(U,U) - G(U,V)^2\right\}\nonumber\\
&&\qquad + G(\mbox{Hor}(P_VV),\mbox{Hor}(P_UU)) -G(\mbox{Hor}(P_VU),\mbox{Hor}(P_VU)) - \frac{G(V,V)G(\mbox{grad }f,P_UU)}{f} \nonumber\\
&&\qquad + \frac{G(U,U)G(\mbox{grad }f,P_VV)}{f},
\end{eqnarray*}
where we used (\ref{aa}).
Recalling $R = 0$ by Assumption \ref{assume2}, for any $U,V\in\mathcal{L}(F)$, we find
\begin{eqnarray*}
& &\frac{\|\mbox{grad }f\|^2}{f^2}\left\{G(V,V)G(U,U) - G(U,V)^2\right\}- G(\mbox{Hor}(P_VV),\mbox{Hor}(P_UU)) +G(\mbox{Hor}(P_VU),\mbox{Hor}(P_VU)) \nonumber\\
&&\qquad+ \frac{G(V,V)G(\mbox{grad }f,P_UU)}{f} - \frac{G(U,U)G(\mbox{grad }f,P_VV)}{f} = 0.
\end{eqnarray*}
Focusing on the symmetric and anti-symmetric parts in $U$ and $V$, and recalling $f(t) = t$, we obtain the following two equations:
\begin{equation}
\label{a}
G(V,V)G(\mbox{grad }f,P_UU) = G(U,U)G(\mbox{grad }f,P_VV),
\end{equation}
\begin{equation}
\label{b}
\frac{1}{t^2}\left\{G(V,V)G(U,U) - G(U,V)^2\right\}- G(\mbox{Hor}(P_VV),\mbox{Hor}(P_UU)) +G(\mbox{Hor}(P_VU),\mbox{Hor}(P_VU)) = 0.
\end{equation}
To characterize admissible connections on $B$, we prepare some lemmas.
\begin{lemma}\label{lem4_1}
We have
\begin{equation*}
{\rm{Hor}}\left(P_{\frac{U}{\|U\|}}\frac{U}{\|U\|}\right) = {\rm{Hor}}\left(P_{\frac{V}{\|V\|}}\frac{V}{\|V\|}\right)
\end{equation*}
for any $U,V\in\mathcal{L}(F)$.
\end{lemma}
\begin{proof}
It follows from (\ref{a}) that
\begin{equation*}
G\left(\mbox{grad }f,\mbox{Hor}P_{\frac{U}{\|U\|}}\frac{U}{\|U\|}\right) = G\left(\mbox{grad }f,\mbox{Hor}P_{\frac{V}{\|V\|}}\frac{V}{\|V\|}\right)
\end{equation*}
and we get
\begin{equation*}
\mbox{Hor}\left(P_{\frac{U}{\|U\|}}\frac{U}{\|U\|}\right) = \mbox{Hor}\left(P_{\frac{V}{\|V\|}}\frac{V}{\|V\|}\right).
\end{equation*}
\end{proof}
Hereafter, fix an arbitrary $x\in F$. Let $(\xi_i)$ be normal coordinates around $x$ on $F$ and denote $\partial_i = \frac{\partial}{\partial\xi_i}$.
\begin{lemma}\label{lem4_2}
We have
\begin{equation*}
({\rm{Hor}}P_{\partial_i}\partial_i)_{(x,t)} = \left(t\frac{\partial}{\partial t}\right) \mbox{ or } \left(- t\frac{\partial}{\partial t}\right).
\end{equation*}
\end{lemma}
\begin{proof}
Put $U = \partial_i$ and $V = \partial_j(i\neq j)$. Then Lemma \ref{lem4_1} implies
\begin{equation*}
\frac{\mbox{Hor } P_{(U + V)}(U + V)}{\|U + V\|^2} = \frac{\mbox{Hor }P_UU}{\|U\|^2} = \frac{\mbox{Hor }P_VV}{\|V\|^2}.
\end{equation*}
Note that, for all $t > 0$,
\begin{equation*}
\|U + V\|^2_{(x,t)} = \|U\|^2_{(x,t)} + \|V\|^2_{(x,t)} = 2t^2.
\end{equation*}
This yields
\begin{equation*}
\left\{\mbox{Hor}P_{(U + V)}(U + V)\right\}_{(x,t)} = 2\left(\mbox{Hor}P_UU\right)_{(x,t)} = 2(\mbox{Hor }P_VV)_{(x,t)},
\end{equation*}
and hence
\begin{equation}\label{delta}
(\mbox{Hor}P_UV)_{(x,t)} = 0.
\end{equation}
Combining this with (\ref{b}), we have for all $t$,
\begin{equation*}
t^2= G_{(x,t)}(\mbox{Hor}P_VV,\mbox{Hor}P_UU) = G_{(x,t)}(\mbox{Hor}P_UU,\mbox{Hor}P_UU).
\end{equation*}
This proves the claim.
\end{proof}
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\subsection{Calculations of $G(R(V,X)X,V)$}\label{coneconnection}
Now, we put $X = \frac{\partial}{\partial t}$ and define $k$ by $D_XX = k(t)\frac{\partial}{\partial t}$.
\begin{lemma}\label{lem4_3}
Let $\partial_i = \frac{\partial}{\partial\xi_i}$ as in Lemma \ref{lem4_2}.
We have
\begin{equation*}
(P_X\partial_i)_{(x,t)} = \left(\frac{1}{t}\partial_i\right)_{(x,t)} \mbox{ or } \left(- \frac{1}{t}\partial_i\right)_{(x,t)}
\end{equation*}
for all $t > 0$.
\end{lemma}
\begin{proof}
Put $V = \partial_i$. Recall that $(\mbox{Hor }P_VV)_{(x,t)} = t\frac{\partial}{\partial t} \mbox{ or } \left(-t\frac{\partial}{\partial t}\right)$ by Lemma \ref{lem4_2}.
First, we consider the case $(\mbox{Hor}P_VV)_{(x,t)} = t\frac{\partial}{\partial t}$.
We deduce from (\ref{alpha}) that
\begin{eqnarray*}
G_{(x,t)}(D^*_VD_X^*X,V) &=& G\left(-k(t)\left(\frac{1}{t}V-P_XV\right),V\right)\nonumber\\
&=& -k(t)t + k(t)t\nonumber\\
&=& 0
\end{eqnarray*}
for all $t > 0$, where the second equality follows since $G(P_XV,V) = G(P_VX,V) = G(X,P_VV) = t$ by (\ref{aa}) and (\ref{bb}).
We similarly find from (\ref{alpha}) that
\begin{eqnarray*}
G_{(x,t)}(D^*_XD^*_VX,V) &=& XG\left(\frac{1}{t}V - P_XV,V\right) - G\left(\frac{V}{t}- P_XV,D_XV\right)\nonumber\\
&=& \frac{\partial}{\partial t}\left(\frac{1}{t}t^2 - t\right) - G\left(\frac{V}{t} - P_XV,\frac{V}{t} + P_XV\right)\nonumber\\
&=& -G\left(\frac{V}{t},\frac{V}{t}\right) + G(P_XV,P_XV)\nonumber\\
&=& -1 + G(P_XV,P_XV).
\end{eqnarray*}
Therefore we obtain for all $t > 0$, since $R^* = 0$,
\begin{equation*}
G(P_XV,P_XV) = 1.
\end{equation*}
Next we consider the case $(\mbox{Hor}P_VV)_{(x,t)} = -t\frac{\partial}{\partial t}$.
We have
\begin{eqnarray*}
G_{(x,t)}(D_V(D_XX),V) &=& G\left(k(t)\left(\frac{1}{t}V+P_XV\right),V\right)\nonumber\\
&=& tk(t) - tk(t)\nonumber\\
&=& 0
\end{eqnarray*}
and
\begin{eqnarray*}
G_{(x,t)}(D_X(D_VX),V) &=& G\left(D_X\left(\frac{1}{t}V + P_XV\right),V\right)\nonumber\\
&=& XG\left(\frac{1}{t}V + P_XV,V\right) - G\left(\frac{V}{t} + P_XV,\frac{V}{t} - P_XV\right)\nonumber\\
&=&- 1 + G(P_XV,P_XV).
\end{eqnarray*}
Since $R = 0$, we have $G_{(x,t)}(P_XV,P_XV) = 1$.
For $U = \partial_j(i\neq j)$, using $(\mbox{Hor}P_VU)_{(x,t)} = 0$ in (\ref{delta}), (\ref{aa}) and (\ref{bb}), we have
\begin{equation*}
G_{(x,t)}(P_XV,U) = G_{(x,t)}(P_VX,U) = G_{(x,t)}(X,P_VU) = 0.
\end{equation*}
Moreover $G(P_XV,X) = G(V,P_XX) = 0$. Therefore $(P_XV)_{(x,t)}$ and $V_{(x,t)}$ are linearly dependent for all $t > 0$, which proves the claim.
\end{proof}
The next result is the aim of this section, which is a characterization of connections on the line $B$.
\begin{theorem}\label{mainthm}
Under Assumption \ref{assume2}, we have
\begin{equation*}
k(t) = \frac{1}{t} \mbox{ or } \left(-\frac{1}{t}\right).
\end{equation*}
\end{theorem}
\begin{proof}
Put $V = \partial_i$.
When $(\mbox{Hor}P_VV)_{(x,t)} = -t\frac{\partial}{\partial t}$, we have
\begin{equation*}
G_{(x,t)}(P_XV,V) = G_{(x,t)}(P_VX,V) = G_{(x,t)}(X,P_VV) = -t.
\end{equation*}
Combining this with Lemma \ref{lem4_3}, we find
\begin{equation*}
(P_VX)_{(x,t)} = -\frac{1}{t}V.
\end{equation*}
We similarly find that $P_VX = \frac{1}{t}V$ if $\mbox{Hor }P_VV = t\frac{\partial}{\partial t}$.
Hence, we need to consider only the following two cases.
First, we consider the case $(\mbox{Hor}P_VV)_{(x,t)} = t\frac{\partial}{\partial t}$ and $(P_VX)_{(x,t)} = \frac{1}{t}V$. We have
\begin{eqnarray*}
G(D_V(D_XX),V) &=& G\left(k(t)\left(\frac{1}{t}V + P_V\frac{\partial}{\partial t}\right),V\right) = 2tk(t),
\end{eqnarray*}
and
\begin{eqnarray*}
G(D_X(D_VX),V) &=& XG(D_VX,V) - G(D_VX,D^*_XV)\nonumber\\
&=& \frac{\partial}{\partial t}\left\{G\left(\frac{1}{t}V + P_XV,V\right)\right\}-G\left(\frac{V}{t} + P_XV,\frac{V}{t}-P_XV\right)\nonumber\\
&=& \frac{\partial}{\partial t}(t + t) - 1 + 1 = 2.
\end{eqnarray*}
Hence by $R = 0$, we obtain $k(t) = \frac{1}{t}$.
Next, we consider the case $(\mbox{Hor}P_VV)_{(x,t)} = -t\frac{\partial}{\partial t}$ and $(P_VX)_{(x,t)} = -\frac{1}{t}V$.
We similarly have
\begin{eqnarray*}
G(D_V^*D_X^*X,V) &=& -k(t)G\left(\frac{1}{t}V - P_V\frac{\partial}{\partial t},V\right) = -2tk(t),
\end{eqnarray*}
and
\begin{eqnarray*}
G(D_X^*D_V^*X,V) &=& \frac{\partial}{\partial t}(t + t) - G\left(\frac{V}{t} - P_XV,\frac{V}{t} + P_XV\right) = 2-1 + 1 = 2.
\end{eqnarray*}
Therefore $k(t) = -\frac{1}{t}$.
\end{proof}
From the above proof, we obtain that $\mbox{Hor }(P_{\partial_i}\partial_i) = t\frac{\partial}{\partial t}$ and $P_{\partial_i}X = \frac{1}{t}\partial_i$ if $k(t) = \frac{1}{t}$, and
that $\mbox{Hor }(P_{\partial_i}\partial_i) = -t\frac{\partial}{\partial t}$ and $P_{\partial_i}X = -\frac{1}{t}\partial_i$ if $k(t) = -\frac{1}{t}$.
\subsection{Example 1: Denormalization}
Here we consider the denormalization (recall Subsection \ref{infogeo}) as an example of warped product with affine connections.
Since we can prove the isometry to a warped product in the same way as in Subsection {\ref{bkmcone}}, we omit a
detailed proof and see an explicit expression of an isometry between the warped product and the denormalization.
Let $\widetilde{\mathcal{S}}$ be the set of positive finite measures on a finite set $\mathcal{X}$. We define a map $h$ as
\begin{eqnarray*}
h:\mathbb{R}_{>0}\times \mathcal{S} &\rightarrow& \widetilde{\mathcal{S}}\nonumber\\
(t,p) &\longmapsto& t^2p/4.
\end{eqnarray*}
We pull back $\tilde{g}$ on $\widetilde{\mathcal{S}}$ in (\ref{netric_denormalization}) by $h$ and define the induced metric $G$ on $\mathbb{R}_{>0}\times \mathcal{S}$. This $(\mathbb{R}_{>0}\times \mathcal{S},G)$ is a warped product and
$c(t) := t^2p/4$ is a line of constant speed 1.
Let $\{\xi_1,\ldots,\xi_n\}$ be a coordinate system of $\mathcal{S}$. We adopt $\{\tau,\xi_1,\ldots,\xi_n\}$ as a coordinate system of $\tilde{\mathcal{S}}$ and
denote its natural basis by $\tilde{\partial}_i = \frac{\partial}{\partial\xi_i}$ and $\tilde{\partial}_\tau = \frac{\partial}{\partial \tau}$.
For a vector field $X = X^i\partial_i\in \mathfrak{X}(\mathcal{S})$, we define $\tilde{X} := X^i\tilde{\partial_i}\in \mathfrak{X}(\widetilde{\mathcal{S}})$.
We can set affine connections on the denormalization as in Subsection \ref{infogeo}. In \cite{amari}, it is expressed as follows. For $X,Y\in \mathfrak{X}(\mathcal{S})$,
\begin{eqnarray*}
\widetilde{\nabla}^{(\alpha)}_{\widetilde{X}}\widetilde{Y} &=& \widetilde{(\nabla^{(\alpha)}_XY)} - \frac{1 + \alpha}{2}\langle\widetilde{X},\widetilde{Y}\rangle\tilde{\partial}_\tau,\nonumber\\
\widetilde{\nabla}^{(\alpha)}_{\tilde{\partial}_\tau}\widetilde{X} &=& \widetilde{\nabla}^{(\alpha)}_{\widetilde{X}}\tilde{\partial}_\tau = \frac{1-\alpha}{2}\frac{1}{\tau}\widetilde{X},\nonumber\\
\widetilde{\nabla}^{(\alpha)}_{\tilde{\partial}_\tau}\tilde{\partial}_\tau &=& -\frac{1 + \alpha}{2}\frac{1}{\tau}\tilde{\partial}_\tau.
\end{eqnarray*}
Also, the metric $\widetilde{g}$ is expressed as
\begin{equation*}
\widetilde{g}_{ij} = \tau g_{ij},\quad \widetilde{g}_{i\tau} = 0,\quad\widetilde{g}_{\tau\tau} = \frac{1}{\tau}.
\end{equation*}
We can check that this connection satisfies Assumption \ref{assume2} by direct calculations, that is to say, the $\alpha$-connection on the denormalization is compatible with the warped product structure and their curvatures vanish at $\alpha = \pm{1}$.
Let us see that the results we obtained in Subsections \ref{second} and \ref{coneconnection} are also obtained in this situation.
We set $\tau = t^2/4$. Note that $\|\partial_\tau\| := \sqrt{\tilde{g}(\partial_\tau,\partial_\tau)}= 1/\sqrt{\tau}$. We have, by omitting the tilde for simplicity,
\begin{eqnarray*}
D_{\frac{\partial_\tau}{\|\partial_\tau\|}}X &=& \frac{1}{\|\partial_\tau\|}\frac{1-\alpha}{2}\frac{1}{\tau} X = \frac{1-\alpha}{2}\frac{1}{\sqrt{\tau}}X = \frac{1-\alpha}{t}X,\nonumber\\
P_X\frac{\partial_\tau}{\|\partial_\tau\|} &=& P_{\frac{\partial_\tau}{\|\partial_\tau\|}}X = \frac{1}{2}\left(\frac{1-\alpha}{t}-\frac{1 + \alpha}{t}\right)X = -\frac{\alpha}{t}X,\nonumber\\
D_{\frac{\partial_\tau}{\|\partial_\tau\|}}\frac{\partial_\tau}{\|\partial_\tau\|} &=& \sqrt{\tau}\left\{(\partial_\tau\sqrt{\tau})\partial_\tau + \frac{1}{\|\partial_\tau\|}\left(-\frac{1+\alpha}{2}\frac{1}{\tau}\partial_\tau\right)\right\} = -\frac{\alpha}{2}\partial_\tau = -\frac{\alpha}{t}\frac{\partial_\tau}{\|\partial_\tau\|},
\end{eqnarray*}
where $D = \widetilde{\nabla}^{(\alpha)}$.
When $\alpha = \pm{1}$, these equations are compatible with the connections in Subsection \ref{coneconnection}.
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\subsection{Example 2: BKM cone}\label{bkmcone}
Next, we consider $\mathbb{P}(n)$ equipped with the extended BKM metric (recall Subsection 2.2), which we call the BKM cone.
We first show that $\mathbb{P}(n)$ with the extended monotone metric (not only the BKM metric) has a warped product structure.
\begin{proposition}
$\mathbb{P}(n)$ equipped with the extended monotone metric is a warped product. Precisely, there exists an isometry as follows:
\begin{eqnarray*}
\mathbb{R}_{>0}\times_{l(t) = t}\mathcal{D} &\rightarrow& (\mathbb{P}(n),g^f)\nonumber\\
(t,\rho) &\mapsto& \frac{t^2\rho}{4},
\end{eqnarray*}
where $g^f$ is an arbitrary monotone metric.
\end{proposition}
\begin{proof}
For simplicity, we calculate $2\times 2$ matrices as in \cite{dit}. The following argument can be easily extended to the $n\times n$ case.
For an arbitrary $\rho \in\mathcal{D}$, there exists a unitary matrix $U$ such that $U\rho U^* = \rho_0$, where
$\rho_0 = \mbox{diag}[x,y]$ for some $x,y\in\mathbb{R}$.
We set
\begin{equation*}
X_1 = \begin{pmatrix}
2 & 0\\
0 & 0
\end{pmatrix},\quad X_2 = \begin{pmatrix}
0 & 0\\
0 & 2
\end{pmatrix},\quad
X_3 = \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix},\quad
X_4 = \begin{pmatrix}
0 & i\\
-i & 0
\end{pmatrix}.
\end{equation*}
These $X_1,X_2,X_3,X_4$ form an orthogonal basis of every tangent space of $(\mathbb{P}(2),g^f)$.
Let us calculate the length of these vectors at $\rho_0$:
\begin{eqnarray*}
g_{\rho_0}^f(X_1,X_1) &=& \frac{1}{(2\pi i)^2}\mathrm{Tr}\oint\oint c(\xi,\eta)\begin{pmatrix}
2 & 0\\
0 & 0
\end{pmatrix}\begin{pmatrix}
\frac{1}{\xi-x} & 0\\
0 & \frac{1}{\xi-y}
\end{pmatrix}\begin{pmatrix}
2 & 0\\
0 & 0
\end{pmatrix}\begin{pmatrix}
\frac{1}{\eta-x} & 0\\
0 & \frac{1}{\eta-y}
\end{pmatrix}\, d\xi d\eta\nonumber\\
&=& \frac{1}{(2\pi i)^2}\mathrm{Tr}\oint\oint c(\xi,\eta)\begin{pmatrix}
\frac{4}{(\xi - x)(\eta - x)} & 0\\
0 & 0
\end{pmatrix}\, d\xi d\eta\nonumber\\
&=& 4c(x,x),\nonumber\\
g^f_{\rho_0}(X_2,X_2) &=& 4c(y,y),\nonumber\\
g_{\rho_0}^f(X_3,X_3) &=& \frac{1}{(2\pi i)^2}\mathrm{Tr}\oint\oint c(\xi,\eta)\begin{pmatrix}
\frac{1}{(\xi-y)(\eta-x)} & 0\\
0 & \frac{1}{(\xi-x)(\eta-y)}
\end{pmatrix}\, d\xi d\eta\nonumber\\
&=& 2c(x,y),\nonumber\\
g_{\rho_0}^f(X_4,X_4) &=& \frac{1}{(2\pi i)^2}\mathrm{Tr}\oint\oint c(\xi,\eta)\begin{pmatrix}
\frac{1}{(\xi-y)(\eta-x)} & 0\\
0 & \frac{1}{(\xi-x)(\eta-y)}
\end{pmatrix}\, d\xi d\eta\nonumber\\
&=& 2c(x,y).
\end{eqnarray*}
These calculations show that, for any $k> 0$ and any tangent vectors $X$ and $Y$, we have
\begin{equation*}
g^f_{k\rho}(X,Y) = g^f_{k\rho_0}(UXU^*,UYU^*) = \frac{1}{k}g^f_{\rho_0}(UXU^*,UYU^*) = \frac{1}{k}g^f_\rho(X,Y),
\end{equation*}
where we used the fact that $g^f_{U\rho U^*}(UXU^*,UYU^*) = g^f_\rho(X,Y)$.
Thus, we obtain
\begin{equation}\label{mono1}
g^f_{k\rho}(kX,kX) = kg^f_\rho(X,X).
\end{equation}
We define $h$ as
\begin{eqnarray*}
h:\mathbb{R}_{>0}\times\mathcal{D} &\rightarrow& \mathbb{P}(2)\nonumber\\
(t,\rho) &\mapsto& \frac{t^2\rho}{4}.
\end{eqnarray*}
We pull back $g^f$ on $\mathbb{P}(n)$ by $h$ and define $G$ on $\mathbb{R}_{>0}\times \mathcal{D}$.
We show that $G$ is a warped product metric on $\mathbb{R}_{>0}\times\mathcal{D}$.
We consider the lines
\begin{equation*}
\gamma(t) := \frac{t^2\rho}{4},\quad \gamma_0(t) := \frac{t^2\rho_0}{4}.
\end{equation*}
Then we find
\begin{eqnarray*}
G_{(t,\rho)}\left(\frac{\partial}{\partial t},\frac{\partial}{\partial t}\right) &=& g_{\gamma(t)}^f(\gamma'(t),\gamma'(t))\nonumber\\
&=& g^f_{U\gamma(t)U^*}(U\gamma'(t)U^*,U\gamma'(t)U^*)\nonumber\\
&=&g^f_{\gamma_0(t)}(\gamma_0'(t),\gamma_0'(t))\nonumber\\
&=& g_{\gamma_0(t)}^f\left(\begin{pmatrix}
tx/2 & 0\\
0 & 0
\end{pmatrix},\begin{pmatrix}
tx/2 & 0\\
0 & 0
\end{pmatrix}\right)+g_{\gamma_0(t)}^f\left(\begin{pmatrix}
0 & 0\\
0 & ty/2
\end{pmatrix},\begin{pmatrix}
0 & 0\\
0 & ty/2
\end{pmatrix}\right)\nonumber\\
&=& \left(\frac{tx}{2}\right)^2 \frac{4}{t^2x} + \left(\frac{ty}{2}\right)^2 \frac{4}{t^2y}\nonumber\\
&=& \mathrm{Tr}\rho.
\end{eqnarray*}
Hereafter, we assume $x + y = 1$. We now get
\begin{equation}\label{warp1}
G_{(t,\rho)}\left(\frac{\partial}{\partial t},\frac{\partial}{\partial t}\right) = 1.
\end{equation}
Let us calculate $dh$. Since $h$ is expressed by the natural coordinates of $\mathbb{R}_{>0}\times \mathcal{D}$ and $\mathbb{P}(2)$ as
\begin{equation*}
\left(t,\begin{pmatrix}
x & z + iw\\
z - iw & 1-x
\end{pmatrix}\right)\longmapsto\frac{t^2}{4}\begin{pmatrix}
x & z + iw\\
z - iw & 1-x
\end{pmatrix} = \begin{pmatrix}
a & b + ic\\
b - ic & d
\end{pmatrix},
\end{equation*}
we have
\begin{equation*}
(Jh)_\rho = \begin{pmatrix}
tx/2 & t^2/4 & 0 & 0 \\
tz/2 & 0 & t^2/4 & 0 \\
tw/2 & 0 & 0 & t^2/4 \\
t(1-x)/2 & -t^2/4 & 0 & 0 \\
\end{pmatrix}.
\end{equation*}
The pull-back metric $G$ satisfies
\begin{eqnarray*}
G\left(\left(\frac{\partial}{\partial z}\right)_{(t,\rho)},\left(\frac{\partial}{\partial z}\right)_{(t,\rho)}\right) = g^f\left(\frac{t^2}{4}\left(\frac{\partial}{\partial b}\right)_{\frac{t^2}{4}\rho},\frac{t^2}{4}\left(\frac{\partial}{\partial b}\right)_{\frac{t^2}{4}\rho}\right),
\end{eqnarray*}
where
\begin{equation*}
\left(\frac{\partial}{\partial b}\right)_{\frac{t^2}{4}\rho} := \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}.
\end{equation*}
Using (\ref{mono1}), we obtain
\begin{eqnarray*}
g^f\left(\frac{t^2}{4}\left(\frac{\partial}{\partial b}\right)_{\frac{t^2}{4}\rho},\frac{t^2}{4}\left(\frac{\partial}{\partial b}\right)_{\frac{t^2}{4}\rho}\right) &=& t^2g^f_{\rho/4}\left(\frac{1}{4}\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}
,\frac{1}{4}\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}\right)\nonumber\\
&=& t^2G\left(\left(\frac{\partial}{\partial z}\right)_{(1,\rho)},\left(\frac{\partial}{\partial z}\right)_{(1,\rho)}\right).
\end{eqnarray*}
Hence,
\begin{equation}\label{warp2}
G_{(t,\rho)}\left(\frac{\partial}{\partial z},\frac{\partial}{\partial z}\right) = t^2G_{(1,\rho)}\left(\frac{\partial}{\partial z},\frac{\partial}{\partial z}\right).
\end{equation}
The same equation holds for $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial w}$.
Combining this with \eqref{warp1}, we see that $G$ is a warped product metric with the warping function $l(t) = t$.
\end{proof}
For $n=2$, if we take trivial coordinates of $\mathcal{D}$ such as
\begin{equation*}
\rho(x,y,z) = \begin{pmatrix}
x & y + iz\\
y - iz & 1-x
\end{pmatrix},
\end{equation*}
the mixture connection is an affine connection, for which $\{x,y,z\}$ is affine coordinates.
For example, we have the following calculation for the mixture connection ${\nabla}^{(m)}$. Set $X = \frac{\partial}{\partial x}$, $Y = \phi\frac{\partial}{\partial y}$, for an arbitrary function $\phi$ on $\mathcal{D}$. Then we have
\begin{equation*}
(\nabla^{(m)}_XY)_\rho = \left\{\nabla^{(m)}_{\frac{\partial}{\partial x}}\left(\phi\frac{\partial}{\partial y}\right) \right\}_{\rho}= \left(\frac{\partial \phi}{\partial x}\frac{\partial}{\partial y}\right)_{\rho} = \frac{\partial \phi}{\partial x}\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} = \frac{\partial}{\partial x}\left(\phi\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}\right) = X(Y\rho).
\end{equation*}
Similarly, if we take the coordinates of $\mathbb{P}(2)$ such as
\begin{equation*}
\widetilde{\rho} = \begin{pmatrix}
\alpha & \beta + i\gamma\\
\beta - i \gamma & \zeta
\end{pmatrix},
\end{equation*}
then the connection $D$ whose affine coordinate system is $\{\alpha,\beta,\gamma,\zeta\}$ satisfies $(D_XY)_{\widetilde{\rho}} = X(Y\widetilde{\rho})$.
For another coordinates such as
\begin{equation*}
\widetilde{\rho} = \tau\begin{pmatrix}
x & y + iz\\
y-iz & 1-x
\end{pmatrix},
\end{equation*}
we have $D_{\partial_\tau}\partial_\tau = 0$ and $D_{\partial_x}\partial_x = D_{\partial_y}\partial_y = D_{\partial_z}\partial_z = 0$. Combining this with $\widetilde{\nabla}_{\partial_x}\partial_x = \widetilde{\nabla}_{\partial_y}\partial_y = \widetilde{\nabla}_{\partial_z}\partial_z = 0$, we see that the connection $D$ defined above satisfies Assumption \ref{assume2}.
\begin{remark}
In \cite{grasseli}, quantum $\alpha$-connections on the set of positive definite matrices and their dually flatness are studied. Also for the quantum $\alpha$-connections, we can check the same compatibility between connections and the warped product structure as that of the classical denormalization, which we studied in Subsection 4.3.
\end{remark}
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\section{Connections on the Takano Gaussian space}
In this section, we consider the Takano Gaussian space (recall Subsection 2.3) and show an analogue to Theorem 4.1.
Let $(B,g_B),(F,g_F)$ be Riemannian manifolds and $\nabla^F$ denotes the Levi-Civita connection of $F$.
We furnish $M := B \times F$ with a metric $G$ such that
\begin{equation*}
G := f^2 g_B + b^2 g_F,
\end{equation*}
where $f$, $b$ are positive functions on $B$. This is the same situation as in the Takano Gaussian space.
Denote by $\nabla$ the Levi-Civita connection on $(M,G)$, and by $\mathcal{L}(F)$, $\mathcal{L}(B)$ the sets of lifts of tangent vector fields of $F$ to $M$, $B$ to $M$, respectively.
Simple calculations show that $\nabla_XY$ is horizontal for any $X,Y\in\mathcal{L}(B)$ and $\mbox{Ver}\nabla_VW = \mbox{Lift}(\nabla^F_VW)$ for any $V,W\in\mathcal{L}(F)$.
From now on, we set $M:=L^{n+1}$, $F:= \{(m_1,\ldots,m_n) | m_i\in\mathbb{R}\}$ and $B := \{\sigma\in\mathbb{R}_{>0}\}$. Let $G$ be the Fisher metric on $M$. Let $D$ be an arbitrary affine connection on $M$. We define the affine connection $\widetilde{\nabla}^F$ on the fiber space $F$ by the natural projection of $D$.
We fix our framework.
\begin{assume}\label{assume_5}
We assume that $D$ satisfies
\begin{itemize}
\item $D_XY \mbox{ is horizontal, } i.e. ,\, D_XY\in\mathcal{L}(B)$ for any $X,Y\in\mathcal{L}(B)$,
\item $\mbox{Ver }(D_VW) = \mbox{Lift }(\widetilde{\nabla}^F_VW)$ for any $V,W\in\mathcal{L}(F)$.
\end{itemize}
Let $R$ be the curvature with respect to $D$, $^FR$ be the curvature with respect to $\widetilde{\nabla}^F$, and $R^*$ and $^FR^{*}$ be their duals.
We also assume $R = {}^FR =0$.
\end{assume}
\begin{remark}
If we take an $\alpha$-connection of the Takano Gaussian space as $D$, we see that $(F,\widetilde{\nabla}^F)$ is dually flat from the expression of the Christoffel symbols of the Takano Gaussian space in Subsection \ref{takano_gauss}.
\end{remark}
In the following arguments in this section, we assume this assumption without mentioning.
As we saw in Subsection 4.1, the following equations hold:
\begin{equation}
\label{c}
G(V,V)G(\mbox{grad }b,P_UU) = G(U,U)G(\mbox{grad }b,P_VV),
\end{equation}
\begin{equation}
\label{d}
\frac{\|\mbox{grad }b\|^2}{b^2}\left\{G(V,V)G(U,U) - G(U,V)^2\right\}- G(\mbox{Hor}(P_VV),\mbox{Hor}(P_UU)) +G(\mbox{Hor}(P_VU),\mbox{Hor}(P_VU)) = 0,
\end{equation}
where $b(\sigma) = \frac{\sqrt{2n}}{\sigma}$.
In the following arguments, we set $X =\partial_\sigma$, $U= \partial_i,V = \partial_j$, where $\partial_\sigma = \frac{\partial}{\partial\sigma}$ and $\partial_i = \frac{\partial}{\partial m_i}$.
To characterize the connections on the line, we prepare some lemmas.
\begin{lemma}\label{prop5_4}
We have
\begin{equation*}
({\rm{Hor}}P_UU)_{(x,\sigma)} = {\frac{1}{2n\sigma}}\partial_\sigma \mbox{ or } \left(- {\frac{1}{2n\sigma}}\partial_\sigma\right).
\end{equation*}
\end{lemma}
\begin{proof}
In the same way as Lemma \ref{lem4_1}, we have
\begin{equation}\label{lem5_1}
{\rm{Hor}}P_UU = {\rm{Hor}}P_VV.
\end{equation}
Applying (\ref{c}) to $U + V$ and $V$, we obtain
\begin{equation*}
G(U + V,U + V)G(\mbox{grad }b,P_VV) = G(V,V)G(\mbox{grad }b,P_{(U + V)}( U + V)).
\end{equation*}
Substituting $G(U + V ,U + V) = 2/\sigma^2$ and $G(V,V) = 1/\sigma^2$ to the equation above, we get
\begin{equation*}
2\mbox{Hor }P_VV = \mbox{Hor }P_{(U + V)}(U + V).
\end{equation*}
Hence,
\begin{equation*}
\mbox{Hor }(P_UU + P_VV + 2 P_UV) = 2\mbox{Hor }P_VV.
\end{equation*}
Together with (\ref{lem5_1}), we get
\begin{equation*}
(\mbox{Hor}P_UV)_{(x,\sigma)} = 0.
\end{equation*}
Combining this with (\ref{d}) and
\begin{equation*}
\frac{\|\mbox{grad }b\|^2}{b^2} = \frac{G(\mbox{grad }b,\mbox{grad }b)}{b^2} = \frac{G\left(G^{\sigma\sigma}\partial_\sigma\left(\frac{1}{\sigma}\right)\partial_\sigma,G^{\sigma\sigma}\partial_\sigma\left(\frac{1}{\sigma}\right)\partial_\sigma\right)}{\left(\frac{1}{\sigma^2}\right)} = \left(\frac{\sigma}{2n}\right)^2G(\partial_\sigma,\partial_\sigma) = \frac{1}{2n},
\end{equation*}
we have, for all $t > 0$,
\begin{equation*}
\frac{1}{2n}\frac{1}{\sigma^2}\frac{1}{\sigma^2} =G_{(x,\sigma)}(\mbox{Hor}P_VV,\mbox{Hor}P_UU)= G_{(x,\sigma)}(\mbox{Hor}P_UU,\mbox{Hor}P_UU),
\end{equation*}
which proves the claim.
\end{proof}
We define $k,l$ by $D_XX = k(\sigma)\frac{\partial}{\partial\sigma}$ and $D^*_XX = l(\sigma)\frac{\partial}{\partial\sigma}$.
Combining $G(\partial_\sigma,\partial_\sigma) = \frac{2n}{\sigma^2}$ with
$ \partial_\sigma G(\partial_\sigma,\partial_\sigma) = G(D_{\partial_\sigma}\partial_\sigma,\partial_\sigma) + G(\partial_\sigma,D^*_{\partial_\sigma}\partial_\sigma)$,
we obtain
\begin{equation*}
-\frac{4n}{\sigma^3} = \frac{2n}{\sigma^2}(k(\sigma) + l(\sigma)).
\end{equation*}
Hence,
\begin{equation}\label{star5}
-\frac{2}{\sigma} = k(\sigma) + l(\sigma)
\end{equation}
holds.
\begin{theorem}\label{takano_2}
Under Assumption \ref{assume_5}, the connection on the line is
\begin{equation*}
D_{\partial_\sigma}\partial_\sigma = \frac{1}{\sigma}\partial_\sigma \mbox{ or } D_{\partial_\sigma}\partial_\sigma = -\frac{3}{\sigma}\partial_\sigma.
\end{equation*}
\end{theorem}
\begin{proof}
We only check the case of
$\mbox{Hor}P_VV = \frac{1}{2n\sigma}\partial_\sigma$, because the other case of $\mbox{Hor}P_VV = -\frac{1}{2n\sigma}$ follows from the completely same argument.
According to the O'Neill formula (\ref{alpha}) for affine connections, we have
\begin{equation*}
D_VX = D_XV = \frac{\partial_\sigma b}{b}V + P_VX = -\frac{1}{\sigma}V + P_XV.
\end{equation*}
Using this, we calculate $G(R^*(V,X)X,V)$. We have
\begin{eqnarray*}
G(D^*_VD^*_XX,V) &=& G\left(l(\sigma)\left\{-\frac{1}{\sigma}V-P_XV\right\},V\right)\nonumber\\
&=& l(\sigma)\left\{-\frac{1}{\sigma}G(V,V) - G(X,P_VV)\right\}\nonumber\\
&=& l(\sigma)\left(-\frac{1}{\sigma}\frac{1}{\sigma^2} - G\left(\partial_\sigma,\frac{1}{2n\sigma}\partial_\sigma\right)\right)\nonumber\\
&=& l(\sigma)\left(-\frac{1}{\sigma^3} - \frac{1}{2n\sigma}\frac{2n}{\sigma^2}\right)\nonumber\\
&=& l(\sigma)\left(-\frac{2}{\sigma^3}\right),
\end{eqnarray*}
and
\begin{eqnarray*}
G(D^*_XD^*_VX,V) &=& XG(D^*_VX,V) - G(D^*_VX,D_XV)\nonumber\\
&=& \partial_\sigma G\left(-\frac{1}{\sigma}V-P_XV,V\right) - G\left(\frac{V}{\sigma},\frac{V}{\sigma}\right) + G(P_XV,P_XV)\nonumber\\
&=& \partial_\sigma\left(-\frac{1}{\sigma}G(V,V)-G(X,P_VV)\right) - \frac{1}{\sigma^2}G(V,V) + G(P_XV,P_XV)\nonumber\\
&=& \frac{5}{\sigma^4} + G(P_XV,P_XV).
\end{eqnarray*}
Since $R^* = 0$, we obtain
\begin{equation}\label{5thm_eq_1}
l(\sigma)\left(-\frac{2}{\sigma^3}\right) = \frac{5}{\sigma^4} + G(P_XV,P_XV).
\end{equation}
Next, let us calculate $G(R(V,X)X,V)$. We have
\begin{eqnarray*}
G(D_VD_XX,V) &=& G\left(k(\sigma)(D_VX),V\right)\nonumber\\
&=& k(\sigma)G\left(-\frac{1}{\sigma}V + P_XV,V\right)\nonumber\\
&=& k(\sigma)\left(-\frac{1}{\sigma^3} + G(P_XV,V)\right)\nonumber\\
&=& k(\sigma)\left(-\frac{1}{\sigma^3} + \frac{1}{2n\sigma}\frac{2n}{\sigma^2}\right) = 0,
\end{eqnarray*}
and
\begin{eqnarray*}
G(D_XD_VX,V) &=& XG(D_VX,V) - G(D_VX,D^*_XV)\nonumber\\
&=& \partial_\sigma G\left(-\frac{1}{\sigma}V + P_XV,V\right) - G\left(\frac{V}{\sigma},\frac{V}{\sigma}\right) + G(P_XV,P_XV)\nonumber\\
&=& \partial_\sigma\left(-\frac{1}{\sigma}\frac{1}{\sigma^2} + G(X,P_VV)\right) - \frac{1}{\sigma^2}\frac{1}{\sigma^2} + G(P_XV,P_XV)\nonumber\\
&=& \partial_\sigma\left(-\frac{1}{\sigma^3} + \frac{1}{\sigma^3}\right) - \frac{1}{\sigma^4} + G(P_XV,P_XV) = -\frac{1}{\sigma^4} + G(P_XV,P_XV).
\end{eqnarray*}
Since $R = 0$, we have
\begin{equation*}
0 = -\frac{1}{\sigma^4} + G(P_XV,P_XV).
\end{equation*}
Combining this with (\ref{5thm_eq_1}) and (\ref{star5}), we obtain
\begin{equation*}
l(\sigma)= -\frac{3}{\sigma},\quad k(\sigma) = \frac{1}{\sigma}.
\end{equation*}
The other case is shown in the same way.
\end{proof}
Note that these connections coincide with the $\alpha $-connections at $\alpha = \pm{1}$ in the Takano Gaussian space (recall Subsection 2.3).
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\section{Discussion: Wasserstein Gaussian space}
By \emph{Wasserstein Gaussian space}, we mean the set of multivariate Gaussian distributions on $\mathbb{R}^n$ with mean zero equipped with the $L^2$-Wasserstein metric.
When we started investigating warped products in information geometry, we thought that we would be able to find dually flat connections on the Wasserstein Gaussian space and calculate its canonical divergence.
The scenario we thought was the following. In \cite{takatsu}, it is proved that the Wasserstein Gaussian space has a cone structure. Recently in \cite{fujiwara2}, it is proved that we can find dually flat connections on
the space of density matrices equipped with the monotone metric.
We can apply this result because the SLD metric and the Wasserstein metric on Gaussian distributions are essentialy the same on $\mathcal{D}$ \cite{bures}.
We thought that once we study dually flat affine connections on warped products, we would be able to extend the dually flat connections on the fiber space to the warped product in a natural way.
However, it turned out that it is difficult to draw dual affine coordinates of the dually flat connections on warped products we made. Thus, we do not know how to calculate the canonical divergence.
Let us explain the difficulty in this section.
In the previous sections, we discussed necessary conditions for warped products and fiber spaces to be dually flat.
First, we show that it is also a sufficient condition for the Wasserstein Gaussian space. The question is, when we extend connections on the fiber space to the warped product, whether the warped product with those connections becomes dually flat or not.
According to the arguments in Section 4, we now define the connection $D$ on a cone $M = \mathbb{R}_{>0}\times_{f} F$, where $f(t) = t$ and $(F,g_F)$ is a Riemannian manifold. We consider the situation that the fiber space is equipped with a dually flat affine connection $\widetilde{\nabla}$.
Let $G := g_B + f^2g_F$ be the warped product metric on $M$.
Denote the base space by $(\mathbb{R}_{>0},g_B)$ with a coordinate $\{t\in\mathbb{R}_{>0}\}$ such that $g_B(\frac{\partial}{\partial t},\frac{\partial}{\partial t}) = 1$.
Let $X:= \frac{\partial}{\partial t}$ and $\{U_i\}_{i=1}^n\subset\mathcal{L}(F)$ be a basis of $\mathcal{L}(F)$ such that $[U_i,U_j] = 0$ for any $i,j\in\{1,\ldots,n\}$. Following Theorem \ref{mainthm} and Lemma \ref{lem4_3}, define the connection $D$ by
\begin{equation*}
D_XX = \frac{1}{t}\frac{\partial}{\partial t},\quad D_XV = D_VX := \frac{2}{t}V,\quad
\begin{cases}
\mbox{Hor } D_VW := 0,\\
\mbox{Ver } D_VW := \mbox{Lift} (\widetilde{\nabla}_VW),
\end{cases}
\end{equation*}
where $V,W$ are arbitrary vectors in $\{U_i\}_{i=1}^n$.
\begin{proposition}
$(M,D,D^*)$ is a dually flat space.
\end{proposition}
\begin{proof}
We only have to check that the curvature vanishes with respect to $D$. Let $U,V,W,Q$ be arbitrary vectors in $\{U_i\}_{i=1}^n$.
Note that we have
\begin{eqnarray}\label{last1}
\mbox{Hor}D_UV = 0,
\end{eqnarray}
and
\begin{equation}\label{last2}
D^*_XU = D^*_UX = \frac{1}{t}U - \frac{1}{t}U = 0.
\end{equation}
We first check that $R(U,V)W$ vanishes.
From (\ref{last1}) and (\ref{last2}), we have
\begin{eqnarray*}
G(R(U,V)W,X) &=& G(D_UD_VW,X) - G(D_VD_UW,X)\nonumber\\
&=& \left\{UG(D_VW,X) - G(D_VW,D^*_UX)\right\} - \left\{VG(D_UW,X) - G(D_UW,D^*_VX)\right\}\nonumber\\
&=& UG(\mbox{Hor}D_VW,X) - VG(\mbox{Hor}D_UW,X) = 0.
\end{eqnarray*}
Recall from Subsection 4.1 and (\ref{beta}) that
\begin{eqnarray*}
G(I\hspace{-.1em}I(V,W),I\hspace{-.1em}I^*(U,Q)) &=& \frac{\|\mbox{grad}f\|^2}{f^2}G(V,W)G(U,Q) - G(\mbox{Hor}(P_VW),\mbox{Hor}(P_UQ)) \nonumber\\
&&\qquad+ \frac{G(V,W)G(\mbox{grad}f,P_UQ)}{f} - \frac{G(U,Q)G(\mbox{grad}f,P_VW)}{f}\nonumber\\
&=& \frac{1}{t^2}G(V,W)G(U,Q) - G\left(\frac{G(V,W)}{t}\frac{\partial}{\partial t},\frac{G(U,Q)}{t}\frac{\partial}{\partial t}\right) \nonumber\\
&&\qquad+ \frac{1}{t}G(V,W)G\left(\frac{\partial}{\partial t},\frac{G(U,Q)}{t}\frac{\partial}{\partial t}\right) - \frac{1}{t}G(U,Q)G\left(\frac{\partial}{\partial t},\frac{G(V,W)}{t}\frac{\partial}{\partial t}\right)\nonumber\\
&=& 0,
\end{eqnarray*}
thus we have $G(R(U,V)W,Q) = G(^FR(U,V)W,Q) = 0$ by (\ref{gauss1}). Hence, we have $R(U,V)W = 0$.
Next, we check that $R(X,U)V$ vanishes. From (\ref{last1}) and (\ref{last2}), we have
\begin{eqnarray*}
G(R(X,U)V,X) &=& G(D_XD_UV- D_UD_XV,X)\nonumber\\
&=& \{XG(D_UV,X) - G(D_UV,D^*_XX)\} - UG(D_XV,X) \nonumber\\
&=& -UG\left(\frac{2}{t}V,X\right) = 0.
\end{eqnarray*}
We also have
\begin{eqnarray*}
G(R(X,U)V,Q) &=& \{XG(D_UV,Q) - G(D_UV,D^*_XQ)\} - \{UG(D_XV,Q) - G(D_XV,D^*_UQ)\}\nonumber\\
&=& 2t\{g_F(\widetilde{\nabla}_UV,Q) - Ug_F(V,Q) + g_F(V,\widetilde{\nabla}^*_UQ)\}\nonumber\\
&=& 0.
\end{eqnarray*}
Hence, $R(X,U)V = 0$. In a similar way, we can check $R(U,X)X = 0$.
\end{proof}
\begin{remark}
We denote the Wasserstein Gaussian space over $\mathbb{R}^n$ by the \emph{$n\times n$ Wasserstein Gaussian space} since its elements are represented by $n\times n$ covariance matrices.
For the $2\times 2$ Wasserstein Gaussian space, we remark that the existence of a dually flat affine connection $\widetilde{\nabla}$ is guaranteed by \cite{fujiwara2}.
\end{remark}
Thus the above proposition implies that we can furnish the $2\times 2$ Wasserstein Gaussian space with dually flat affine connections.
Though we think it necessary to draw dual affine coordinates to calculate the canonical divergence, it turned out to be difficult.
This is because, for example, $D_XU$ does not vanish, which means that the trivial extension of affine coordinates on the fiber space does not give affine coordinates of the warped product.
Here, trivial extension means $\{t,\xi_1,\ldots,\xi_n\}$ for affine coordinates $\{\xi_1,\cdots,\xi_n\}$ of the fiber space and the coordinate $\{t\}$ of the line.
\begin{remark}\label{finsler1}
In \cite{tayebi}, it is claimed that there is no dually flat proper doubly warped Finsler manifolds.
Let us restrict their argument to Riemannian manifolds.
For two manifolds $M_1$ and $M_2$ and their doubly warped product $(M_1\times M_2,G)$,
let $(x_i)$ and $(u_\alpha)$ be coordinates of $M_1$ and $M_2$, respectively.
Then their claim asserts that the coordinates $((x_i),(u_\alpha))$ on $(M_1\times M_2,G)$ cannot be affine coordinates for any dually flat connections on $M_1\times M_2$ unless $G$ is the product metric.
\end{remark}
For further understanding the relation between affine coordinates and their connections, let us observe the $2\times 2$ BKM cone (recall Subsection 4.4).
Let $\bar{\nabla}$ be an affine connection whose affine coordinate is $\{a,b,c,d\}$, with which $2\times 2$ matrices are expressed as
\begin{equation*}
\begin{pmatrix}
a & c + ib\\
c -ib & d
\end{pmatrix}.
\end{equation*}
This $\bar{\nabla}$ is a dually flat affine connection on the BKM cone.
Let $\bar{D}$ be an affine connection whose affine coordinate is $\{t,\alpha,\beta,\gamma\}$, with which $2\times 2$ matrices are expressed as
\begin{equation*}
t\begin{pmatrix}
\alpha & \beta + i\gamma\\
\beta - i\gamma& 1-\alpha
\end{pmatrix}.
\end{equation*}
Relations of these coordinates are
\begin{equation*}
t = a + d,\quad\alpha = \frac{a}{a + d}, \quad1-\alpha = \frac{d}{a + d},
\end{equation*}
\begin{equation*}
\frac{\partial}{\partial\alpha} = \begin{pmatrix}
t & 0\\
0 & -t
\end{pmatrix} = (a+d)\left(\frac{\partial}{\partial a} - \frac{\partial}{\partial d}\right),
\end{equation*}
\begin{equation*}
\frac{\partial}{\partial t} = \begin{pmatrix}
\alpha & 0\\
0 & 1-\alpha
\end{pmatrix} = \frac{a}{a + d} \frac{\partial}{\partial a} + \frac{d}{a + d}\frac{\partial}{\partial d}.
\end{equation*}
Using these relations, we calculate
\begin{eqnarray*}
\bar{\nabla}_{\frac{\partial}{\partial\alpha}}\frac{\partial}{\partial t} &=& \bar{\nabla}_{(a + d)\left(\frac{\partial}{\partial a}-\frac{\partial}{\partial d}\right)}\left(\frac{a}{a + d} \frac{\partial}{\partial a} + \frac{d}{a + d}\frac{\partial}{\partial d}\right)\nonumber\\
&=& (a+d)\left(\frac{\partial}{\partial a}-\frac{\partial}{\partial d}\right)\left(\frac{a}{a + d}\right)\frac{\partial}{\partial a} + (a+d)\left(\frac{\partial}{\partial a}-\frac{\partial}{\partial d}\right)\left(\frac{d}{a + d}\right)\frac{\partial}{\partial d}\nonumber\\
&=& (a+d)\left(\frac{d}{(a + d)^2}+\frac{a}{(a+d)^2}\right)\frac{\partial}{\partial a} + (a+d)\left(-\frac{d}{(a+d)^2}-\frac{a}{(a+d)^2}\right)\frac{\partial}{\partial d}\nonumber\\
&=& \frac{\partial}{\partial a} -\frac{\partial}{\partial d}\neq 0.
\end{eqnarray*}
On the other hand
\begin{equation*}
\bar{D}_{\frac{\partial}{\partial\alpha}}\frac{\partial}{\partial t} = 0.
\end{equation*}
Hence, $\bar{\nabla}$ and $\bar{D}$ are different.
\\
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\section{Appendix}
Main contributions of this appendix are following two points.
\begin{itemize}
\item We study an example of warped product whose dually flat connections are not realized as ($\pm{1}$)-connections of $\alpha$-connections.
\item We study dually flat connections compatible with the structure of a two-dimensional warped product.
\end{itemize}
\subsection{Preliminaries for elliptic distributions}
As described in \cite{elliptic}, a $p$-dimensional random variable $X$ is said to have an elliptic distribution with parameters $\mu^{\mathrm{T}} = (\mu_1,\cdots, \mu_p)$ and $\Psi$, a $p\times p$ positive definite matrix, if its density is
\begin{equation*}
p_h(x|\mu,\Psi) = \frac{h\{(x-\mu)^{\mathrm{T}}\Psi^{-1}(x-\mu)\}}{\sqrt{\det\Psi}}
\end{equation*}
for some function $h$. We say that $X$ has an $EL_p^h(\mu,\Psi)$ distribution.
We consider the class of one-dimensional elliptic distributions $EL_1^h(\mu,\sigma^2)$, where $\theta = (\mu,\sigma)$. We set $Z$ as an $EL_1^h(0,1)$ random variable and $ W = \{d \log h(Z^2)\}/d(Z^2)$.
We also set
\begin{equation*}
a = E(Z^2W^2),\quad b = E(Z^4W^2), \quad d = E(Z^6W^3).
\end{equation*}
The Fisher metric of elliptic distributions is
\begin{eqnarray}\label{fisher}
ds^2 = \frac{4a d\mu^2 + (4b-1)d\sigma^2}{\sigma^2}.
\end{eqnarray}
We denote the Fisher metric $ds^2$ as $G_F$.
\begin{example}
Gaussian distribution, Student's t distribution and Cauchy distributions are examples of elliptic distributions. Their constants are given in Table \ref{table:data_type}, which is calculated in \cite{elliptic}.
\begin{table}[hbtp]
\caption{Important constants}
\label{table:data_type}
\centering
\begin{tabular}{lccc}
\hline
& Gauss & Cauchy & Student's t \\
\hline \hline
$a$ & $\frac{1}{4}$ & $\frac{1}{8}$ & $\frac{k + 1}{4(k + 3)}$ \\
$b$ & $\frac{3}{4}$ & $\frac{3}{8}$ & $\frac{3(k + 1)}{4(k + 3)}$\\
$d$ & $\frac{-15}{8}$ & $-\frac{5}{16}$ & $-\frac{15(k + 1)^2}{8(k + 3)(k + 5)}$\\
\hline
\end{tabular}
\end{table}
\end{example}
\subsection{Elliptic distributions as warped products}
We set
\begin{equation*}
t := \sqrt{4b-1}\log\sigma.
\end{equation*}
Since
\begin{equation*}
G_F\left(\frac{\partial}{\partial t},\frac{\partial}{\partial t}\right) = G_F\left(\frac{\partial\sigma}{\partial t}\frac{\partial}{\partial \sigma},\frac{\partial\sigma}{\partial t}\frac{\partial}{\partial \sigma}\right) = \frac{\sigma^2}{4b-1}G_F\left(\frac{\partial}{\partial\sigma},\frac{\partial}{\partial\sigma}\right) = 1,
\end{equation*}
we have
\begin{equation*}
G_F = dt^2 + f(t)^2d\mu^2,
\end{equation*}
where
\begin{equation*}
f(t) := \sqrt{4a}\exp\left(-\frac{t}{\sqrt{4b-1}}\right).
\end{equation*}
\subsection{Calculations of $R(V,X,X,V)$}
For the parameter spaces $M$ of elliptic distributions, we set new assumptions.
\begin{assume}\label{assume_appendix}
Let $B = \mathbb{R}_{>0}$ with the Euclidean metric $g_B$ such that $g\left(\frac{\partial}{\partial t},\frac{\partial}{\partial t}\right) = 1$, $f(t) = \sqrt{4a}\exp\left(-\frac{t}{\sqrt{4b-1}}\right)$ and $F = \mathbb{R}$ with the Euclidean metric
and $\widetilde{\nabla},\widetilde{\nabla}^*$ be a dually flat affine connections on $\mathbb{R}$ with the Euclidean metric $g_\mu$ such that $g_\mu\left(\frac{\partial}{\partial\mu},\frac{\partial}{\partial\mu}\right) = 1$. For an arbitrary connection $D$ on $B\times_f F$, we assume that $D$ satisfies
\begin{itemize}
\item $D_XY \mbox{ is horizontal, } i.e. ,\, D_XY\in\mathcal{L}(B)$ for any $X,Y\in\mathcal{L}(B)$,
\item $\mbox{Ver }(D_VW) = \mbox{Lift }(\widetilde{\nabla}^F_VW)$ for any $V,W\in\mathcal{L}(F)$.
\end{itemize}
We also assume $R = 0$ where $R$ is the curvature of $M$ with respect to $D$.
\end{assume}
We next calculate the curvature $R$ under Assumption \ref{assume_appendix}. We set the notations $X = \frac{\partial}{\partial t}, V = \frac{\partial}{\partial \mu}$ and $k,l$ as
\begin{equation*}
D_{\frac{\partial}{\partial t}}\frac{\partial}{\partial t} = k(t)\frac{\partial}{\partial t}
\end{equation*}
and
\begin{equation*}
P_XV = l(t,\mu)\frac{\partial}{\partial \mu},
\end{equation*}
where $P_XV := \frac{1}{2}(D_XV - D^*_XV)$.
Since
\begin{equation*}
\frac{Xf}{f} = \frac{\sqrt{4a}\left(-\frac{1}{\sqrt{4b-1}}\right)\exp\left(-\frac{t}{\sqrt{4b-1}}\right)}{\sqrt{4a}\exp\left(-\frac{t}{\sqrt{4b-1}}\right)} = -\frac{1}{\sqrt{4b-1}},
\end{equation*}
we have
\begin{eqnarray*}
G_F( D_VD_XX,V) &=& k(t)\left\{-\frac{1}{\sqrt{4b-1}}f(t)^2 + l(t,\mu)f(t)^2\right\},
\end{eqnarray*}
\begin{eqnarray*}
G_F( D^*_VD^*_XX,V) &=& -k(t)\left\{ -\frac{1}{\sqrt{4b-1}}f(t)^2 - l(t,\mu)f(t)^2\right\},
\end{eqnarray*}
\begin{eqnarray*}
G_F( D_XD_VX,V) &=& XG_F( \frac{Xf}{f}V + P_XV,V) - G_F( D_XV,D^*_XV)\\
&=& \frac{\partial}{\partial t}\left\{\left(-\frac{1}{\sqrt{4b-1}}f(t)^2 + l(t,\mu)f(t)^2\right)\right\} - \left\{\left(\frac{Xf}{f}\right)^2G_F( V,V)- G_F( P_XV,P_XV)\right\}\\
&=& f(t)^2\left\{\frac{1}{4b-1} + \partial_t l - \frac{2l}{\sqrt{4b-1}} + l^2\right\}
\end{eqnarray*}
and
\begin{eqnarray*}
G_F (D^*_XD^*_VX, V ) &=& f^2\left\{\frac{1}{4b-1} - \partial_tl + \frac{2l}{\sqrt{4b-1}} + l^2\right\}.
\end{eqnarray*}
Hence, we have
\begin{eqnarray*}
R(V,X,X,V) &=& G_F( D_VD_XX,V) - G_F( D_XD_VX , V)\\
&=& f(t)^2\left\{-\frac{k(t)}{\sqrt{4b-1}} + kl - \frac{1}{4b-1} - \partial_t l + \frac{2l}{\sqrt{4b-1}} - l^2\right\}
\end{eqnarray*}
and
\begin{eqnarray*}
R^*(V,X,X,V) &=& G_F( D^*_VD^*_XX,V)- G_F( D^*_XD^*_VX, V )\\
&=& f(t)^2\left\{ \frac{k(t)}{\sqrt{4b-1}} + kl - \frac{1}{4b-1} + \partial_t l - \frac{2l}{\sqrt{4b-1}} - l^2\right\}.
\end{eqnarray*}
Since we now consider the case $R = R^* = 0$, we have
\begin{eqnarray*}
&&f(t)^2\left\{kl - \frac{1}{4b-1} - l^2\right\} = 0,\\
&&f(t)^2\left\{-\frac{k(t)}{\sqrt{4b-1}} - \partial_t l + \frac{2l}{\sqrt{4b-1}}\right\} = 0.
\end{eqnarray*}
Summarizing the above arguments, we have the following theorem.
\begin{theorem}\label{constant_connection}
For any $\gamma\in\mathbb{R}$, we define affine connections $D, D^*$ on $M$ by
\begin{eqnarray*}
&&D_{\frac{\partial}{\partial t}}\frac{\partial}{\partial t} = k\frac{\partial}{\partial t},\\
&&D_{\frac{\partial}{\partial \mu}}\frac{\partial}{\partial\mu} = \nabla_{\frac{\partial}{\partial t}}\frac{\partial}{\partial\mu} + l\frac{\partial}{\partial\mu},\\
&&D_{\frac{\partial}{\partial\mu}}\frac{\partial}{\partial\mu} = \nabla_{\frac{\partial}{\partial\mu}}\frac{\partial}{\partial\mu} + \gamma\frac{\partial}{\partial\mu} + \frac{l}{2\sigma^2}\frac{\partial}{\partial t},
\end{eqnarray*}
where $\nabla$ is the Levi-Civita connection and $k,l$ are arbitrary functions satisfying
\begin{eqnarray}\label{difeq}
\begin{cases}
kl - \frac{1}{4b-1} - l^2 = 0,\\
-\frac{k(t)}{\sqrt{4b-1}} - \partial_t l + \frac{2l}{\sqrt{4b-1}} = 0.
\end{cases}
\end{eqnarray}
Then $(M,D,D^*)$ satisfies Assumption \ref{assume_appendix}.
\end{theorem}
\begin{remark}
In Section 5, we consider Takano Gaussian space $(L^{n + 1},G_T,\nabla^{(\alpha)})$.
It was shown in Lemma \ref{prop5_4} and Theorem \ref{takano_2} that the dually flat connections $D,D^*$ on $(L^{n + 1},G_T)$ compatible with warped product structure satisfy following two equations:
\begin{equation}\label{takano_connection_1}
\mbox{Hor} P_{\frac{\partial}{\partial m_i}}\frac{\partial}{\partial m_i} = \frac{1}{2n\sigma}\frac{\partial}{\partial\sigma} \quad \mbox{or} \quad -\frac{1}{2n\sigma}\frac{\partial}{\partial\sigma},
\end{equation}
\begin{equation}\label{takano_connection_2}
D_{\frac{\partial}{\partial\sigma}}\frac{\partial}{\partial\sigma} = \frac{1}{\sigma}\frac{\partial}{\partial \sigma} \quad \mbox{or}\quad -\frac{3}{\sigma}\frac{\partial}{\partial\sigma}.
\end{equation}
Since the Fisher metric $G_T$ of Takano Gaussian space is expressed as
\begin{equation*}
G_T = \frac{dm_1^2 + \cdots + dm_n^2 + 2nd\sigma^2}{\sigma^2},
\end{equation*}
if we set the parameter $t = \sqrt{2n}\log \sigma $, we have
\begin{equation*}
G_T\left(\frac{\partial}{\partial t},\frac{\partial}{\partial t}\right) = G_T\left(\frac{\partial\sigma}{\partial t}\frac{\partial}{\partial\sigma},\frac{\partial\sigma}{\partial t}\frac{\partial}{\partial \sigma}\right) = \frac{\sigma^2}{2n}\frac{2n}{\sigma^2} = 1.
\end{equation*}
Hence, the metric is
\begin{equation*}
G_T = dt^2 + f_T(t)^2 \{dm_1^2 + \cdots + dm_n^2\},
\end{equation*}
where
\begin{equation*}
f_T(t) = \exp\left(-\frac{t}{\sqrt{2n}}\right).
\end{equation*}
From (\ref{takano_connection_1}) and (\ref{takano_connection_2}), we have
\begin{equation*}
k = \sqrt{\frac{2}{n}},\quad l = \frac{1}{\sqrt{2n}}.
\end{equation*}
This is the only solution of (\ref{difeq}) if $l$ is constant when $n = 1, a = \frac{1}{4}$ and $ b = \frac{3}{4}$.
\end{remark}
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0.132.9
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\subsection{$\alpha$-connections and dually flat connections}
Using calculations for $\alpha$-connections on elliptic distributions $\nabla^{(\alpha)}$ in \cite{elliptic}, we have
\begin{eqnarray*}
\nabla^{(\alpha)}_{\frac{\partial}{\partial\sigma}}\frac{\partial}{\partial\sigma} &=& \frac{1-4b + \alpha(6b + 4d - 1)}{(4b-1)\sigma}\frac{\partial}{\partial\sigma}\\
&=& \frac{1-4b + \alpha(6b + 4d - 1)}{(4b-1)\sigma}\frac{\partial t}{\partial\sigma}\frac{\partial}{\partial t}.
\end{eqnarray*}
On the other hand, since $\frac{\partial t}{\partial\sigma} = \frac{\sqrt{4b-1}}{\sigma}$, we have
\begin{eqnarray*}
\nabla^{(\alpha)}_{\frac{\partial}{\partial\sigma}}\frac{\partial}{\partial\sigma} &=& \nabla^{(\alpha)}_{\frac{\partial t}{\partial\sigma}\frac{\partial}{\partial t}}\frac{\partial t}{\partial \sigma}\frac{\partial}{\partial t}\\
&=& \frac{\partial t}{\partial \sigma}\sqrt{4b-1}\frac{(-1)}{\sigma^2}\frac{\partial\sigma}{\partial t}\frac{\partial }{\partial t} + \left(\frac{\partial t}{\partial \sigma}\right)^2 \nabla^{(\alpha)}_{\frac{\partial}{\partial t}}\frac{\partial}{\partial t} .
\end{eqnarray*}
Comparing two equations above, we have
\begin{equation*}
\nabla^{(\alpha)}_{\frac{\partial}{\partial t}}\frac{\partial}{\partial t} = \frac{\alpha(6b + 4d - 1)}{(4b-1)^{\frac{3}{2}}}\frac{\partial}{\partial t}.
\end{equation*}
The only solution of $(\ref{difeq})$ when $l$ is constant is
\begin{equation}\label{only_constant_sol}
k = \frac{2}{\sqrt{4b-1}},\quad l= \frac{1}{\sqrt{4b-1}}.
\end{equation}
If we set $k,l$ as (\ref{only_constant_sol}), dually flat connections which are compatible with the structure of warped products are constructed.
For dually flat connections $D,D^*$ constructed by Theorem \ref{constant_connection} using (\ref{only_constant_sol}), we have
\begin{equation*}
\frac{1}{2}(D_{\partial_t}\partial_t - D^*_{\partial_t}\partial_t) = \frac{2}{\sqrt{4b-1}}\frac{\partial}{\partial t}.
\end{equation*}
On the other hand, for $\alpha$-connections of elliptic distributions $\nabla^{(\alpha)}$, we have
\begin{equation*}
\frac{1}{2}(\nabla^{(-\alpha)}_{\partial_t}\partial_t - \nabla^{(\alpha)}_{\partial_t}\partial_t) = \frac{-\alpha(6b + 4d - 1)}{(4b-1)^{\frac{3}{2}}}\frac{\partial}{\partial t}.
\end{equation*}
\begin{example}
We compare dually flat connections on the line $B = \mathbb{R}_{>0}$ constructed by Theorem \ref{constant_connection} with dually flat connections among $\alpha$-connections in Table \ref{table:compare}.
Note that $\alpha$-connections of Cauchy distribution are not dually flat connections \cite{elliptic} and
Student's $t$ distributions are dually flat when $\alpha = \pm{\frac{k + 5}{k-1}}$.
\begin{table}[hbtp]
\caption{Comparing dually flat connections}
\label{table:compare}
\centering
\begin{tabular}{lllll}
\hline
Connections on $B = \mathbb{R}_{>0}$ & Gauss & Cauchy & Student's t \\
\hline \hline
$\frac{2}{\sqrt{4b-1}}\partial_t$ & $\sqrt{2}\partial_t$ & $2\sqrt{2}\partial_t$ & $\sqrt{\frac{2(k+3)}{k}}\partial_t$\\
$\frac{-\alpha(6b + 4d - 1)}{(4b-1)^{\frac{3}{2}}}\partial_t$ &$\sqrt{2}\partial_t (\alpha = 1)$ & none & $\frac{k\sqrt{2}}{k + 3}\partial_t (\alpha = \frac{k + 5}{k-1})$\\
\hline
\end{tabular}
\end{table}
\end{example}
Although every dually flat connections which is compatible with structure of warped product appeared before this appendix were realized as one of $\alpha$-connections, it is observed in this appendix that it does not always happen.
\begin{remark}
In \cite{furuhata}(4.2), they defined $\alpha$-connections with respect to a metric $g = \frac{dx^2 + \lambda^2 dy^2}{y^2}$ ($\lambda > 0$) on the upper half plane $\{(x,y) | x\in\mathbb{R}, y > 0\}$.
By direct calculations, we see that their $\alpha$-connections are also compatible with the structure of warped product and their $\alpha$-connections are dually flat when $\alpha = \pm{1}$.
\end{remark}
\end{document}
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\begin{document}
\title[A lower bound for $\chi (\mathcal O_S)$]
{A lower bound for $\chi (\mathcal O_S)$}
\author{Vincenzo Di Gennaro }
\address{Universit\`a di Roma \lq\lq Tor Vergata\rq\rq, Dipartimento di Matematica,
Via della Ricerca Scientifica, 00133 Roma, Italy.}
\email{[email protected]}
\abstract Let $(S,\mathcal L)$ be a smooth, irreducible, projective,
complex surface, polarized by a very ample line bundle $\mathcal L$
of degree $d > 25$. In this paper we prove that $\chi (\mathcal
O_S)\geq -\frac{1}{8}d(d-6)$. The bound is sharp, and $\chi
(\mathcal O_S)=-\frac{1}{8}d(d-6)$ if and only if $d$ is even, the
linear system $|H^0(S,\mathcal L)|$ embeds $S$ in a smooth rational
normal scroll $T\subset \mathbb P^5$ of dimension $3$, and here, as
a divisor, $S$ is linearly equivalent to $\frac{d}{2}Q$, where $Q$
is a quadric on $T$. Moreover, this is equivalent to the fact that
the general hyperplane section $H\in |H^0(S,\mathcal L)|$ of $S$ is
the projection of a curve $C$ contained in the Veronese surface
$V\subseteq \mathbb P^5$, from a point $x\in V\backslash C$.
\noindent {\it{Keywords}}: Projective surface, Castelnuovo-Halphen's
Theory, Rational normal scroll, Veronese surface.
\noindent {\it{MSC2010}}\,: Primary 14J99; Secondary 14M20, 14N15,
51N35.
\endabstract
\maketitle
\section{Introduction}
In \cite{DGF}, one proves a sharp lower bound for the
self-intersection $K^2_S$ of the canonical bundle of a smooth,
projective, complex surface $S$, polarized by a very ample line
bundle $\mathcal L$, in terms of its degree
$d={\text{deg}}\,\mathcal L$, assuming $d>35$. Refining the line of
the proof in \cite{DGF}, in the present paper we deduce a similar
result for the Euler characteristic $\chi(\mathcal O_S)$ of $S$
\cite[p. 2]{BV}, in the range $d>25$. More precisely, we prove the
following:
\begin{theorem}\label{lbound} Let $(S,\mathcal L)$ be a smooth,
irreducible, projective, complex surface, polarized by a very ample
line bundle $\mathcal L$ of degree $d > 25$. Then:
$$
\chi (\mathcal O_S)\geq -\frac{1}{8}d(d-6).
$$
The bound is sharp, and the following properties are equivalent.
(i) $\chi (\mathcal O_S)= -\frac{1}{8}d(d-6)$;
(ii) $h^0(S,\mathcal L)=6$, and the linear system $|H^0(S,\mathcal
L)|$ embeds $S$ in $\mathbb P^5$ as a scroll with sectional genus
$g=\frac{1}{8}d(d-6)+1$;
(iii) $h^0(S,\mathcal L)=6$, $d$ is even, and the linear system
$|H^0(S,\mathcal L)|$ embeds $S$ in a smooth rational normal scroll
$T\subset \mathbb P^5$ of dimension $3$, and here $S$ is linearly
equivalent to $\frac{d}{2}(H_T-W_T)$, where $H_T$ is the hyperplane
class of $T$, and $W_T$ the ruling (i.e. $S$ is linearly equivalent
to an integer multiple of a smooth quadric $Q\subset T$).
\end{theorem}
By Enriques' classification, one knows that if $S$ is unruled or
rational, then $\chi (\mathcal O_S)\geq 0$. Hence, Theorem
\ref{lbound} essentially concerns irrational ruled surfaces.
In the range $d>35$, the family of extremal surfaces for $\chi
(\mathcal O_S)$ is exactly the same for $K^2_S$. We point out there
is a relationship between this family and the Veronese surface. In
fact one has the following:
\begin{corollary}\label{Veronese}
Let $S\subseteq \mathbb P^r$ be a nondegenerate, smooth,
irreducible, projective, complex surface, of degree $d > 25$. Let
$L\subseteq \mathbb P^r$ be a general hyperplane. Then
$\chi(\mathcal O_S)=-\frac{1}{8}d(d-6)$ if and only if $r=5$, and
there is a curve $C$ in the Veronese surface $V\subseteq \mathbb
P^5$ and a point $x\in V\backslash C$ such that the general
hyperplane section $S\cap L$ of $S$ is the projection
$p_x(C)\subseteq L$ of $C$ in $L\cong\mathbb P^4$, from the point
$x$.
\end{corollary}
In particular, $S\cap L$ is not linearly normal, instead $S$ is.
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0.133.1
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\section{Proof of Theorem \ref{lbound}}
\begin{remark}\label{k8} $(i)$ We say that $S\subset \mathbb P^r$ is a {\it
scroll} if $S$ is a $\mathbb P^1$-bundle over a smooth curve, and
the restriction of $\mathcal O_S(1)$ to a fibre is $\mathcal
O_{\mathbb P^1}(1)$. In particular, $S$ is a geometrically ruled
surface, and therefore $\chi (\mathcal O_S)= \frac{1}{8}K^2_S$
\cite[Proposition III.21]{BV}.
$(ii)$ By Enriques' classification \cite[Theorem X.4 and Proposition
III.21]{BV}, one knows that if $S$ is unruled or rational, then
$\chi (\mathcal O_S)\geq 0$, and if $S$ is ruled with irregularity
$>0$, then $\chi (\mathcal O_S)\geq \frac{1}{8}K^2_S$. Therefore,
taking into account previous remark, when $d>35$, Theorem
\ref{lbound} follows from \cite[Theorem 1.1]{DGF}. In order to
examine the range $25<d\leq 35$, we are going to refine the line of
the argument in the proof of \cite[Theorem 1.1]{DGF}.
$(iii)$ When $d=2\delta$ is even, then $\frac{1}{8}d(d-6)+1$ is the genus of
a plane curve of degree $\delta$, and the genus of a curve of degree $d$ lying on
the Veronese surface.
\end{remark}
Put $r+1:=h^0(S,\mathcal L)$. Therefore, $|H^0(S,\mathcal L)|$
embeds $S$ in $\mathbb P^r$. Let $H\subseteq \mathbb P^{r-1}$ be the
general hyperplane section of $S$, so that $\mathcal L\cong \mathcal
O_S(H)$. We denote by $g$ the genus of $H$. If $2\leq r\leq 3$, then
$\chi (\mathcal O_S)\geq 1$. Therefore, we may assume $r\geq 4$.
{\bf The case $r=4$}.
We first examine the case $r=4$. In this case we only have to prove
that, for $d>25$, one has $\chi (\mathcal O_S)> -\frac{1}{8}d(d-6)$.
We may assume that $S$ is an irrational ruled surface, so
$K^2_S\leq 8\chi (\mathcal O_S)$ (compare with previous Remark
\ref{k8}, $(ii)$). We argue by contradiction, and assume also that
\begin{equation}\label{sest}
\chi (\mathcal O_S)\leq -\frac{1}{8}d(d-6).
\end{equation}
We are going to prove that this assumption implies $d\leq 25$, in contrast with our
hypothesis $d>25$.
By the double point formula:
$$
d(d-5)-10(g-1)+12\chi(\mathcal O_S)=2K^2_S,
$$
and $K^2_S\leq 8\chi (\mathcal O_S)$, we get:
$$
d(d-5)-10(g-1)\leq 4\chi(\mathcal O_S).
$$
And from $\chi (\mathcal O_S)\leq -\frac{1}{8}d(d-6)$ we obtain
\begin{equation}\label{fest}
10g\geq \frac{3}{2}d^2-8d+10.
\end{equation}
Now we distinguish two cases, according that $S$ is not
contained in a hypersurface of degree $<5$ or not.
First suppose that $S$ is not contained in a hypersurface of
$\mathbb P^4$ of degree $<5$. Since $d > 16$, by Roth's Theorem
(\cite[p. 152]{R}, \cite[p. 2, (C)]{EP}), $H$ is not contained in a
surface of $\mathbb P^3$ of degree $<5$. Using Halphen's bound
\cite{GP}, we deduce that
$$
g\leq \frac{d^2}{10} +
\frac{d}{2}+1-\frac{2}{5}(\epsilon+1)(4-\epsilon),
$$
where $d-1=5m+\epsilon$, $0\leq \epsilon<5$. It follows that
$$
\frac{3}{2}d^2-8d+10\leq \,10 g\,\leq
d^2+5d+10\left(1-\frac{2}{5}(\epsilon+1)(4-\epsilon)\right).
$$
This implies that $d\leq 25$, in contrast with our hypothesis $d>25$.
In the second case, assume that $S$ is contained in an irreducible
and reduced hypersurface of degree $s\leq 4$. When $s\in\{2,3\}$,
one knows that, for $d>12$, $S$ is of general type \cite[p.
213]{BF}. Therefore, we only have to examine the case $s=4$. In this
case $H$ is contained in a surface of $\mathbb P^3$ of degree $4$.
Since $d>12$, by Bezout's Theorem, $H$ is not contained in a surface
of $\mathbb P^3$ of degree $<4$. Using Halphen's bound \cite{GP},
and \cite[Lemme 1]{EP}, we get:
$$
\frac{d^2}{8}-\frac{9d}{8}+1\leq \, g\,\leq \frac{d^2}{8}+1.
$$
Hence, there exists a rational number $0\leq x\leq 9$ such that
$$
g=\frac{d^2}{8}+d\left(\frac{x-9}{8}\right)+1.
$$
If $0\leq x\leq \frac{15}{2}$, then $g\leq
\frac{d^2}{8}-\frac{3}{16}d+1$, and from (\ref{fest}) we get
$$
\frac{3}{20}d^2-\frac{4}{5}d+1\,\leq g\,\leq
\frac{d^2}{8}-\frac{3}{16}d+1.
$$
It follows $d\leq 24$, in contrast with our hypothesis $d>25$.
Assume $\frac{15}{2}< x\leq 9$. Hence,
$$
\left(\frac{d^2}{8}+1\right)-g=
-d\left(\frac{x-9}{8}\right)<\frac{3}{16}d.
$$
By \cite[proof of Proposition 2, and formula (2.2)]{D}, we have
$$
\chi(\mathcal O_S)\geq 1+
\frac{d^3}{96}-\frac{d^2}{16}-\frac{5d}{3}-\frac{349}{16}-(d-3)\left[\left(\frac{d^2}{8}+1\right)-g\right]
$$
$$
>1+
\frac{d^3}{96}-\frac{d^2}{16}-\frac{5d}{3}-\frac{349}{16}-(d-3)\frac{3}{16}d
= \frac{d^3}{96}-\frac{d^2}{4}-\frac{53}{48}d-\frac{333}{16}.
$$
Combining with (\ref{sest}), we get
$$
\frac{d^3}{96}-\frac{d^2}{4}-\frac{53}{48}d-\frac{333}{16}+\frac{1}{8}d(d-6)<0,
$$
i.e.
$$
d^3-12d^2-178d-1998<0.
$$
It follows $d\leq 23$, in contrast with our hypothesis $d>25$.
This concludes the analysis of the case $r=4$.
{\bf The case $r\geq 5$}.
When $r\geq 5$, by \cite[Remark 2.1]{DGF}, we know that, for $d>5$,
one has $K^2_S>-d(d-6)$, except when $r=5$, and the surface $S$ is a
scroll, $K^2_S=8\chi (\mathcal O_S)=8(1-g)$, and
\begin{equation}\label{bound}
g=\frac{1}{8}d^2-\frac{3}{4}d+\frac{(5-\epsilon)(\epsilon+1)}{8},
\end{equation}
with $d-1=4m+\epsilon$, $0<\epsilon\leq 3$. In this case, by
\cite[pp. 73-76]{DGF}, we know that, for $d>30$, $S$ is contained in
a smooth rational normal scroll of $\mathbb P^5$ of dimension $3$.
Taking into account that we may assume $K^2_S\leq 8\chi (\mathcal
O_S)$ (compare with Remark \ref{k8}, $(i)$ and $(ii)$), at this
point Theorem \ref{lbound} follows from \cite[Proposition 2.2]{DGF},
when $d>30$.
In order to examine the remaining cases $26\leq d \leq 30$, we
refine the analysis appearing in \cite{DGF}. In fact, we are going
to prove that, assuming $r=5$, $S$ is a scroll, and (\ref{bound}),
it follows that $S$ is contained in a smooth rational normal scroll
of $\mathbb P^5$ of dimension $3$ also when $26\leq d \leq 30$. Then
we may conclude as before, because \cite[Proposition 2.2]{DGF} holds
true for $d\geq 18$.
First, observe that if $S$ is contained in a threefold $T\subset
\mathbb P^5$ of dimension $3$ and minimal degree $3$, then $T$ is
necessarily a {\it smooth} rational normal scroll \cite[p. 76]{DGF}.
Moreover, observe that we may apply the same argument as in \cite[p.
75-76]{DGF} in order to exclude the case $S$ is contained in a
threefold of degree $4$. In fact the argument works for $d>24$
\cite[p. 76, first line after formula (13)]{DGF}.
In conclusion, assuming $r=5$, $S$ is a scroll, and (\ref{bound}),
it remains to exclude that $S$ is not contained in a threefold of
degree $<5$, when $26\leq d \leq 30$.
Assume $S$ is not contained in a threefold of degree $<5$. Denote by
$\Gamma\subset \mathbb P^3$ the general hyperplane section of $H$.
Recall that $26\leq d \leq 30$.
$\bullet$ Case I: $h^0(\mathbb P^3,\mathcal
I_{\Gamma}(2))\geq 2$.
It is impossible. In fact, if $d>4$, by monodromy \cite[Proposition
2.1]{CCD}, $\Gamma$ should be contained in a reduced and irreducible
space curve of degree $\leq 4$, and so, for $d>20$, $S$ should be
contained in a threefold of degree $\leq 4$ \cite[Theorem
(0.2)]{CC}.
$\bullet$ Case II: $h^0(\mathbb P^3,\mathcal
I_{\Gamma}(2))=1$ and $h^0(\mathbb P^3,\mathcal I_{\Gamma}(3))>4$.
As before, if $d>6$, by monodromy, $\Gamma$ is contained in a
reduced and irreducible space curve $X$ of degree $\deg(X)\leq 6$.
Again as before, if $\deg(X)\leq 4$, then $S$ is contained in a
threefold of degree $\leq 4$. So we may assume $5\leq \deg(X)\leq
6$.
Since $d\geq 26$, by Bezout's Theorem we have $h_{\Gamma}(i)=h_X(i)$
for all $i\leq 4$. Let $X'$ be the general plane section of $X$.
Since $h_X(i)\geq \sum_{j=0}^{i}h_{X'}(j)$, we have $h_X(3)\geq 14$
and $h_X(4)\geq 19$ \cite[pp. 81-87]{EH}. Therefore, when $d\geq
26$, taking into account \cite[Corollary (3.5)]{EH}, we get:
$$
h_{\Gamma}(1)=4,\, h_{\Gamma}(2)=9,\, h_{\Gamma}(3)\geq 14,\,
h_{\Gamma}(4)\geq 19,
$$
$$
h_{\Gamma}(5)\geq 22, \,
h_{\Gamma}(6)\geq \min\{d,\, 27\},\, h_{\Gamma}(7)=d.
$$
It follows that:
$$
p_a(C)\leq \sum_{i=1}^{+\infty}d-h_{\Gamma}(i)\leq
(d-4)+(d-9)+(d-14)+(d-19)+(d-22)+3=5d-65,
$$
which is $<\frac{1}{8}d(d-6)+1$ for $d \geq 26$. This is in contrast
with (\ref{bound}).
$\bullet$ Case III: $h^0(\mathbb P^3,\mathcal
I_{\Gamma}(2))=1$ and $h^0(\mathbb P^3,\mathcal I_{\Gamma}(3))=4$.
We have:
$$
h_{\Gamma}(1)=4,\, h_{\Gamma}(2)=9,\, h_{\Gamma}(3)=16, \,
h_{\Gamma}(4)\geq 19,
h_{\Gamma}(5)\geq 24, \, h_{\Gamma}(6)=d.
$$
It follows that:
$$
p_a(C)\leq \sum_{i=1}^{+\infty}d-h_{\Gamma}(i)\leq
(d-4)+(d-9)+(d-16)+(d-19)+(d-24)=5d-72,
$$
which is $< \frac{1}{8}d(d-6)+1$ for $d \geq 26$. This is in
contrast with (\ref{bound}).
$\bullet$ Case IV: $h^0(\mathbb P^3,\mathcal
I_{\Gamma}(2))=0$.
We have:
$$
h_{\Gamma}(1)=4,\, h_{\Gamma}(2)=10,\, h_{\Gamma}(3)\geq 13, \,
h_{\Gamma}(4)\geq 19,
$$
$$
h_{\Gamma}(5)\geq 22,\, h_{\Gamma}(6)\geq \min\{d,\, 28\}, \,
h_{\Gamma}(7)=d.
$$
It follows that:
$$
p_a(C)\leq \sum_{i=1}^{+\infty}d-h_{\Gamma}(i)\leq
(d-4)+(d-10)+(d-13)+(d-19)+(d-22)+2=5d-66,
$$
which is $< \frac{1}{8}d(d-6)+1$ for $d \geq 26$. This is in
contrast with (\ref{bound}).
This concludes the proof of Theorem \ref{lbound}.
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