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\subsection{Proof of Theorem \ref{prop1}}
Consider the map $\mu \colon \mathbb{G}_m \times \mathbb{G}_m \to \mathcal{K}_2$ obtained using the identification $\mathbb{G}_m\simeq \mathcal{K}_1$ and the multiplication $\mathcal{K}_1 \times \mathcal{K}_1 \to \mathcal{K}_2$. The functor $\mathcal{u}p$ defined as the composite
\[
\tors_Y(\mathbb{G}_m) \times \tors_Y(\mathbb{G}_m)
\xrightarrow{c_{\mathbb{G}_m, \mathbb{G}_m}}
\gerb_Y(\mathbb{G}_m \otimes \mathbb{G}_m) \xrightarrow{\mu_*} \gerb_Y(\mathcal{K}_2)
\]
is the required bi-additive functor. \qed
The functor $\mathcal{u}p$ is so-named as it categorifies the cup-product (which can be identified with the intersection product
\[
\mathrm{H}^1(Y, \mathbb{G}_m) \times \mathrm{H}^1(Y, \mathbb{G}_m) \longrightarrow \mathrm{H}^2(Y,\mathcal{K}_2) \simeq \mathsf{C}H^2(Y) \longleftarrow \mathsf{C}H^1(Y) \times \mathsf{C}H^1(Y)\,.
\]
\begin{remark}
\label{rem:biext}
The bi-additivity property of the map $c_{A,B}$ of Proposition~\ref{prop:biadditive} has the following conjectural formal interpretation. The maps $+_A$ and $+_B$, plus the commutative diagram in Remark~\ref{rem:biadditivity} and the proof of Proposition~\ref{prop:biadditive} comprise a structure that can be described as the categorification of a biextension, namely a $\stTors(A\otimes B)$-torsor (hence an $A\otimes B$-gerbe)
\begin{equation*}
\cH \longrightarrow \stTors(A) \times \stTors(B)
\end{equation*}
equipped with partial addition laws $+_A$ (resp.\ $+_B$) giving it the structure of an extension of $\stTors (A)$ (resp.\ $\stTors (B)$) by $\stTors(A \otimes B)$.
\end{remark}
\subsection{Proof of Theorem \ref{prop2}}Our proof will use the results of \S \ref{Leray} on (\ref{lowterm}) for $\pi:X \to S$ with $A=\mathcal{K}_2$ on $X$. Proposition \ref{beilinson-lemma} shows that all $\mathcal{K}_2$-gerbes are horizontal (Definition \ref{horizontal}). The functor
$\int_{\pi}$ is then defined as the composition of
\[
\cGerb_X(\mathcal{K}_2) \xrightarrow{\mathsf{T}heta} \stTors_S(\mathrm{R}^1\pi_*\mathcal{K}_2)
\xrightarrow{\mathrm{Norm}} \stTors_S(\mathcal{K}_1)\,.
\]
Our first step is to show that $\cGerb_X(\mathcal{K}_2)'$ is all of $\cGerb_X(\mathcal{K}_2)$, in other words, every $\mathcal{K}_2$-gerbe on $X$ is horizontal. This is proved by showing $\mathrm{R}^2\pi_*\mathcal{K}_2=0$ which provides the isomorphism
\[
E^2_1 \congto \mathrm{H}^2(X, \mathcal{K}_2)\, .
\]
We start with the following result, implicit in \cite[A5.1 (iv)]{MR962493}, essentially due to Beilinson-Schechtman.
\begin{proposition}[Beilinson-Schechtman]\label{beilinson-lemma}
The sheaf $\mathrm{R}^2\pi_*\mathcal{K}_2$ is zero.
\end{proposition}
This gives a map
\begin{equation}\label{map-bson}
\theta \colon \mathrm{H}^2(X, \mathcal{K}_2) \longrightarrow \mathrm{H}^1(S, \mathrm{R}^1\pi_*\mathcal{K}_2)
\end{equation}
using
\[
\mathrm{H}^2(X, \mathcal{K}_2)\leftiso E^2_1 \longrightarrow \mathrm{H}^1(S, \mathrm{R}^1\pi_*\mathcal{K}_2)\,.
\]
\begin{proof} Let $N$ be the dimension of $S$, so that $X$ has dimension $N+1$.
For any $s\in S$, we have to show that the stalk of $\mathrm{R}^2\pi_*\mathcal{K}_2$ at $s$ is zero. By definition, this is the direct limit
\[
\varinjlim_{s\in U}\mathrm{R}^2\pi_*\mathcal{K}_2(U) =
\varinjlim_{s\in U}\mathrm{H}^2(\pi^{-1}(U), \mathcal{K}_2) =
\varinjlim_{s\in U}\mathsf{C}H^2(\pi^{-1}(U))\,,
\]
where the last equality comes from the Bloch-Quillen isomorphism (valid for any smooth variety $V$)
\[
\mathrm{H}^2(V, \mathcal{K}_2) \congto \mathsf{C}H^2(V)\,.
\]
So, we have to show that for any $s \in S$, any open set $U$ containing $s$, and any codimension two cycle $Z$ in $\pi^{-1}(U) \subset X$, there exists an open subset $U' \subset U$ such that the class of $Z$ goes to zero under the map
\[
\mathsf{C}H^2(\pi^{-1}(U)) \longrightarrow \mathsf{C}H^2(\pi^{-1}(U'))\,.
\]
This is clear when $s$ is the generic point $\Spec F(S)$ of $S$: in this case, we take $U'$ to be the complement of $\pi(\abs{Z})$ in $U$. Here we have written $\abs{Z}$ for the support of $Z$.
The next the case is when $s$ is a point of codimension $i>0$, corresponding to a codimension $i$ subvariety $V$ of $S$. Let us write $Y\subset X$ for $\pi^{-1}(V)$; then $Y$ is a subset of $X$ with codimension $i$.
For any open $U \subset S$, the condition $s\in U$ means $U \cap V$ is non-empty. Let $U$ be such an open set. There are two cases to consider:
\begin{description}
\item[Case 1] If $\abs{Z}$ is disjoint from $Y$, then we can proceed as before as $\pi(Z)$ is disjoint from $V$, so we take $U'$ to be the complement of $\pi(\abs{Z})$ in $U$. Since $U' \cap V = U \cap V$, we see that $U'\cap V$ is non-empty.
Since $Z$ is in the kernel of the localization sequence for Chow groups
\[
\mathsf{C}H^2(\pi^{-1}(U)) \to \mathsf{C}H^2(\pi^{-1}(U) - \abs{Z}) \to 0\,,
\]
it is also in the kernel of the composite map
\[
\mathsf{C}H^2(\pi^{-1}(U)) \to \mathsf{C}H^2(\pi^{-1}(U) - \abs{Z}) \to \mathsf{C}H^2(\pi^{-1}(U'))\,.
\]
This finishes the proof in this case.
\item[Case 2] If $\abs{Z}$ is not disjoint from $Y$, we can find a codimension two cycle $Z'$ in $\pi^{-1}(U)$ with $[Z] = [Z']\in \mathsf{C}H^2(\pi^{-1}(U))$ which intersects $Y$ transversally. The codimension of the cycle $Z'.Y$ is $i+2$, because its dimension (= maximum of the dimensions of the irreducible components) is $N+1 - i-2 = N-1-i$. Hence the dimension of the image $\pi(Z'.Y)$ is at most $N-1-i$, and so its support $\abs{\pi(Z'.Y)}$ is a proper closed subset of $V = \pi(Y)$. If $U''$ is the complement of $\abs{\pi(Z'.Y)}$ in $U$, then the intersection of $U''$ and $V$ is empty. By definition, the cycle $Z' \cap \pi^{-1}(U'')$ is disjoint from $Y$. This means that the image of $Z'$ (= image of $Z$) under the map
\[
\mathsf{C}H^2(\pi^{-1}(U)) \longrightarrow \mathsf{C}H^2(\pi^{-1}(U''))
\]
is a cycle disjoint from $Y$. By Case 1, we can shrink $U''$ further to $U'$ such that $Z'$ (and hence $Z$ also) is in the kernel of the map
\[
\mathsf{C}H^2(\pi^{-1}(U'')) \longrightarrow \mathsf{C}H^2(\pi^{-1}(U'))\,,
\]
as required.\end{description}
\end{proof}
This gives the functor $\mathsf{T}heta$ appearing in the definition of
\[
\int_{\pi} \colon \cGerb_X(\mathcal{K}_2) \xrightarrow{\mathsf{T}heta} \stTors_S(\mathrm{R}^1\pi_*\mathcal{K}_2)
\xrightarrow{\mathrm{Norm}} \stTors_S(\mathcal{K}_1).
\]
Our next step is the definition of the map $\mathrm{R}^1\pi_*\mathcal{K}_2 \longrightarrow \mathcal{O_S^*}$.
\begin{remark}
The same proof shows that if $f\colon Y \to T$ is a smooth proper map of dimension $n$ with $Y$ and $T$ smooth, then $\mathrm{R}^jf_*\mathcal{K}_j =0$ for all $j>n$. This says that the relative Chow sheaves $\mathsf{C}H^j(Y/T)$ vanish for all $j>n$.
\end{remark}
\subsection{The norm map $\mathrm{R}^1\pi_*\mathcal{K}_2 \longrightarrow \mathcal{O_S^*}$}\label{Norm} This well known map \cite[3.4]{Rost}, \cite[pp.~262-264]{Gillet} arises from the covariant functoriality for proper maps of Rost's cycle modules (Chow groups in our case). We provide the details for the convenience of the reader. Our description proceeds via the Gersten sequence (a flasque resolution of the Zariski sheaf $\mathcal{K}_2$ on $X$)
\begin{equation}\label{gersten}
0\longrightarrow \mathcal{K}_2 \longrightarrow \mathit{\acute{e}t}a_*\mathcal{K}_{2,\mathit{\acute{e}t}a} \longrightarrow \bigoplus_{x \in X^{(1)}} i_*K_1(k(x)) \longrightarrow \bigoplus_{y\in X^{(2)}} i_*K_0(k(y)) \to 0\,;
\end{equation}
here $\mathit{\acute{e}t}a: \Spec F(X) \to X$ is the generic point of $X$ and $X^{(i)}$ denotes the set of points of codimension $i$ of $X$.
For any $U$ open in $S$, the norm map
\begin{equation*}
\mathrm{H}^1(\pi^{-1}(U), \mathcal{K}_2) \longrightarrow \mathcal O^*_S(U)
\end{equation*}
is obtained as follows. Since the first group is the homology at degree one of (\ref{gersten}), we proceed by constructing a map
\[
\bigoplus_{x \in \pi^{-1}(U)^{(1)}} i_*K_1(k(x)) \to \mathcal O^*_S(U)\,.
\]
For each such $x \in \pi^{-1}(U)$ of codimension one, the map $x \to \pi(x)$ is either finite or not, and it is zero in the second case. In the first case, there is a norm map
\[
k(x)^* \to k(\pi(x))^*\,;
\]
since $x$ has codimension one in $X$, its image $\pi(x)$ is the generic point of $S$ and hence the above norm map is a map
\[
k(x)^* \to F(S)^*\,.
\]
An element of $\mathrm{H}^1(\pi^{-1}(U), \mathcal{K}_2)$ arises from a finite collection of functions $f_x\in k(x)^*$ (for $x\in \pi^{-1}(U)$ of codimension one which is finite onto its image) which is in the kernel of the map
\[
\bigoplus_{x \in \pi^{-1}(U)^{(1)}} i_*K_1(k(x)) \to \bigoplus_{y \in \pi^{-1}(U)^{(2)}} i_*K_0(k(y))\,.
\]
On each component, this is the ord or valuation map. One checks that this means that the (finite) product of the norms of $f_x$ is an element of $F(S)^*$ with no poles on $U$ and hence defines an element of $\mathcal O_S^*(U)$. This gives the required functor
\[
\stTors_S(\mathrm{R}^1\pi_*\mathcal{K}_2) \xrightarrow{\mathrm{Norm}} \stTors_S(\mathcal{K}_1)\,,
\]
completing the definition of the functor $\int_{\pi}$ of Theorem \ref{prop2}.
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\section{Comparison with Deligne's construction}
\label{comp-deligne}
Given line bundles $L$ and $M$ (viewed as $\mathbb{G}_m$-torsors) on $X$, consider the $\mathcal{K}_2$-gerbe $G_{L,M}$ on $X$. By Proposition \ref{beilinson-lemma}, the element $[G_{L,M}]$ of $\mathrm{H}^2(X, \mathcal{K}_2)$ actually lives in $E_1^2$ and hence $G_{L,M}$ is horizontal. By Lemma \ref{lem:exercise}, $\mathsf{T}heta(G_{L,M})$ is a $\mathrm{R}^1\pi_*\mathcal{K}_2$-torsor. By definition, $\int_{\pi}G_{L,M}$ is its pushforward along the norm map of \S \ref{Norm},
\[
\mathrm{Norm} \colon \mathrm{R}^1\pi_*\mathcal{K}_2 \longrightarrow \mathcal{O_S^*}\,,
\]
which gives a line bundle $(L,M)$ on $S$. In this section, we show that this gives Deligne's line bundle $\< {L,M}\>$. Since $(L,M)$ is bi-additive and its construction is functorial, this reduces to showing the identity in Theorem \ref{prop3}:
\[
\< {\mathcal{O}(D), \mathcal{O}(E)} \cong ( {\mathcal{O}(D), \mathcal{O}(E)})\,,
\]
for any relative Cartier divisors $D$ and $E$ on $X$ with $D$ effective.
\subsection{Comparison}
To show that $\< {\mathcal{O}(D), \mathcal{O}(E)}$ is isomorphic to $( {\mathcal{O}(D), \mathcal{O}(E)})$, one just has to show that they are equal in $\mathrm{H}^1(S, \mathcal{O}^*)$. This amounts to showing that the diagram below is commutative:
\begin{equation}\label{diagram}
\begin{tikzcd}
& \mathrm{H}^1(X, \mathcal{O}^*) \ar[dl,"\mathit{\acute{e}t}a"'] \ar[dr,"\mathcal{u}p"] \\
\mathrm{H}^1(D, \mathcal{O}^*) \ar[rr,"\lambda"] \ar[d,"N_{D/S}"'] && \mathrm{H}^2(X, \mathcal{K}_2) \ar[d,"\theta"] \\
\mathrm{H}^1(S, \mathcal{O}^*) && \mathrm{H}^1(S, \mathrm{R}^1\pi_*\mathcal{K}_2) \ar[ll,"\mathrm{Norm}"']
\end{tikzcd}
\end{equation}
The map $\mathit{\acute{e}t}a$ is the restriction to $D$ of a line bundle $\mathcal{O}(E)$. The map $\mathcal{u}p$ sends $\mathcal{O}(E)$ to its cup-product with $\mathcal{O}(D)$. The boundary map $\lambda$ in the localization sequence
\[0
\to \mathcal{K}_{2,X} \longrightarrow j_*\mathcal{K}_{2,U} \longrightarrow i_*\mathcal{K}_{1,D} \to 0
\]
for $X$, $U= X - D$, and $D$. The map $\theta$ is the map (\ref{map-bson})
\[
\mathrm{H}^2(X, \mathcal{K}_2) \longrightarrow \mathrm{H}^1(S, \mathrm{R}^1\pi_*\mathcal{K}_2)\,.
\]
The commutativity of (\ref{diagram}) is an implicit consequence of the axiomatics of Rost \cite{Rost}, but we provide a direct proof.
\subsubsection{The top triangle of (\ref{diagram})} We first prove the commutativity of the top triangle of (\ref{diagram}). Let $\{U_i\}$ be a Zariski open cover of $X$ such that $D$ and $E$ are principal divisors on $U_i$. Let $\{f_i\}$ be defining equations for $D$ and $\{g_i\}$ be defining equations for $E$. Then,
\[
\{a_{ij}:= \frac{f_i}{f_j} \in \mathcal O^*(U_i \times U_j)\}, \quad
\{b_{ij}:= \frac{g_i}{g_j} \in \mathcal O^*(U_i \times U_j)\}
\]
are cocycle representatives for $\mathcal O(D)$ and $\mathcal O(E)$. By the explicit description \cite[(1-18)]{MR2362847} of the cup-product map in \v{C}ech cohomology, the map $\mathcal{u}p$ sends $\{b_{ij}\}$ to the $2$-cocycle
\begin{equation}
\label{cup-product}
\{(a_{ij}, b_{jk})\}\in K_2(U_i \times U_j \times U_k)\,.
\end{equation}
Given a cocycle $s_{ij}\in \mathcal O^*(U_i \times U_j \times D)$ relative to the cover $\{U_i \times D\}$ of $D$, one computes its image under $\lambda$ as follows. Pick $\tilde{s}_{ij} \in K_2 (U_i \times U_j \times U)$ whose tame symbol along $D$ is $s_{ij}$; then check that its \v{C}ech boundary $\partial( \tilde{s}_{ij})$ (a $2$-cochain with values in $\mathcal{K}_{2,U}$) is zero when viewed as a cochain with values in $i_*\mathcal{K}_{1,D}$. This means that $\partial (\tilde{s}_{ij})$ is a $2$-cocycle with values in $\mathcal{K}_{2,X}$; this is defined to be the image of $s_{ij}$ under $\lambda$. Let us apply this to compute the image of $\mathcal O(E)\vert_D$ under $\lambda$.
Let $\bar{b}_{ij}$ be the image of $b_{ij}$ under the map
\[
\mathcal O^*(U_i \times U_j) \to \mathcal O^*(U_i \times U_j \times D)\,.
\]
The cocycle $\{\bar{b}_{ij}\}$ represents $\mathcal O(E)\vert_D$. To compute its image under $\lambda$, consider the element (symbol)
\[
t_{ij} = (f_i,b_{ij}) \in K_2(U \times U_i \times U_j).
\]
We know that $b_{ij}$ is a unit in $U_i \times U_j$ and so defines an element of $K_1(U_i \times U_j)$; we know $f_i$ is the defining equation of $D$ on $U_i$ and so it is a unit on $U \times U_i$ and thus $f_i$ defines an element of $K_1(U \times U_i)$. So $t_{ij}$ is a well-defined element of $K_2(U \times U_i \times U_j)$. If $v$ denotes the valuation
\[
F(X)^* \longrightarrow \mathbb{Z}
\]
defined by the divisor $D$, the tame symbol map is the map
\[
K_2(U) \to K_1(D), \quad (a,b) \mapsto (-1)^{v(a)v(b)}.~ \overline{\bigg(\frac{a^{v(b)}}{b^{v(a)}}\bigg)}\,.
\]
Since $v(f_i) =1$ and $v(b_{ij}) =0$, we see that $t_{ij}$ maps to the element
\[
(-1)^{1 \times 0}.~\overline{\bigg(\frac{f_i^0}{b_{ij}^1}\bigg)} = \bar{b}_{ij}^{-1}\,.
\]
So the cochain $\{t_{ij}\}$ lifts the inverse of the cocycle $\{\bar{b}_{ij}\}$. Its \v{C}ech boundary (which represents the image under $\lambda$ of the inverse of $\{\bar{b}_{ij}\}$)
\[
t_{ij} - t_{ik} + t_{jk} = (f_i,b_{ij}) - (f_i,b_{ik}) + (f_j,b_{jk})
\]
is a $2$-cocycle with values in $\mathcal{K}_2$. Since
\[
\biggl\{b_{ij} = \frac{g_i}{g_j}\biggr\}
\]
is a cocycle, the relation
\[
b_{ik} = b_{ij}+ b_{jk}
\]
holds. Using this, the image of the inverse of $\{\bar{b}_{ij}\}$ under $\lambda$ is given by the negative of the element in (\ref{cup-product}):
\[
(f_i, b_{ij}) - (f_i, b_{ij}) -(f_i, b_{jk}) +(f_j, b_{jk})
= (\frac{f_j}{f_i}, b_{jk}) = -(a_{ij}, b_{jk})\,.
\]
This says that $\lambda$ maps $\{\bar{b}_{ij}\}$ to the class of the cup product of $\mathcal O(D)$ and $\mathcal O(E)$ in $\mathrm{H}^2(X ,\mathcal{K}_2)$ thus completing the proof of the commutativity of the top triangle in (\ref{diagram}).
\subsubsection{The bottom square of (\ref{diagram})}
We begin with an explicit description of the map
\[
\theta \colon \mathrm{H}^2(X, \mathcal{K}_2) \to \mathrm{H}^1(S, \mathrm{R}^1 \pi_*\mathcal{K}_2)
\]
in (\ref{map-bson}).
Let $G$ be a $\mathcal{K}_2$-gerbe on $X$. As $\mathsf{C}H^2(X)= \mathrm{H}^2(X,\mathcal{K}_2)$ (Bloch-Quillen), we can pick a codimension-two cycle $c$ representing $[G]$ on $X$. As $G$ is horizontal, there exists an open cover $\{V_{\alpha}\}$ of $S$ such that $[G] =0\in \mathrm{H}^2(W_{\alpha}, \mathcal{K}_2)$, with $W_{\alpha} = \pi^{-1}(V_{\alpha})$; note $\{W_{\alpha}\}$ is an open cover of $X$. In terms of the Gersten complex
\[
0\longrightarrow \mathcal{K}_2 \longrightarrow \mathit{\acute{e}t}a_*\mathcal{K}_{2,\mathit{\acute{e}t}a} \longrightarrow
\bigoplus_{x \in W_{\alpha}^{(1)}} K_1(k(x)) \xrightarrow{\mathrm{ord}}
\bigoplus_{y\in W_{\alpha}^{(2)}} K_0(k(y)) \to 0\,,
\]
which computes the cohomology of $\mathcal{K}_2$ on $W_{\alpha}$, we have that the vanishing in $H^2(W_{\alpha}, \mathcal{K}_2)$ of the restriction $c_{\alpha}$ of the codimension-two cycle $c$ representing $[G]$ on $W_{\alpha}$. Then, there exists an element $h_{\alpha} \in \bigoplus_{x \in W_{\alpha}^{(1)}}~K_1(k(x))$ such that $\textrm{ord}(h_{\alpha}) =c_{\alpha}$ in the sequence on $W_{\alpha}$. So $h_{\alpha}$ is a collection of divisors in $W_{\alpha}$ whose associated functions cut out together the codimension-two cycle $c$. Since $\textrm{ord}(h_{\alpha}) = \textrm{ord}(h_{\alpha'})$ on $W_{\alpha} \cap W_{\alpha'}$, we see that the element
$r_{\alpha,\alpha'}\coloneq h_{\alpha} -h_{\alpha'}$ on $W_{\alpha} \cap W_{\alpha'}$ defines an element of $\mathrm{H}^1(W_{\alpha} \cap W_{\alpha'}, \mathcal{K}_2)$. The cocycle condition is a formal consequence:
\[
r_{\alpha,\alpha'} + r_{\alpha',\alpha''} + r_{\alpha'',\alpha} =0\,.
\]
Namely, $\{r_{\alpha,\alpha'}\}$ defines a Čech $1$-cocycle on $S$ with values in $\mathrm{R}^1 \pi_*\mathcal{K}_2$; this is the element $\theta(G)$. Taking norms down to $S$ gives a Čech $1$-cocycle
\[
\tilde{r}_{\alpha,\alpha'} \coloneq N_{D/S}\biggl( \frac{h_{\alpha}}{h_{\alpha'}} \biggr)
\]
with values in $\mathbb G_m$ on $S$. This completes the description of the maps in the bottom square of (\ref{diagram}).
With all this in place, it is now easy to show that the bottom square of (\ref{diagram}) commutes. Recall the defining equations $g_i$ of $E$ relative to the open cover $\{U_i\}$ of $X$. Restricting $\mathcal O(E)$ to $D$ and applying $\lambda$ gives the gerbe $G= G_{\mathcal O(D), \mathcal O(E)}$, by the commutativity of the top triangle of (\ref{diagram}). We use the above description to compute the image of the gerbe under $\theta$; we see that $h_{\alpha}$ can be taken to be the collection of functions $\bar{g}_{i, \alpha} = \bar{g}_i\lvert_{D_{\alpha, i}}$ on $D_{\alpha, i}= D \cap W_{\alpha}\cap U_i$ which cuts out the codimension-two cycle corresponding to the intersection of $D$ and $E$. The norm down to $S$ of the corresponding $r_{\alpha, \alpha'}$ gives the Čech-cocycle with values in $\mathcal{K}_1$ of $S$; this is the image of $\mathcal O(E)$ along one part of the bottom square in (\ref{diagram}).
On the other hand, consider the image of $\mathcal O(E)$ under the left vertical map of (\ref{diagram}). Let
\[
\bar{g}_{i, \alpha} = \bar{g}_i\lvert_{D_{\alpha, i}}, \quad
e_{\alpha}= \prod_{i}N_{D_{\alpha, i}/{S}} \bigl(\bar{g}_{i, \alpha}\bigr) \,.
\]
The image of $\mathcal O(E)$ under the map $N_{D/S}$ is given by the cocycle
\[
c_{\alpha, \alpha'}\coloneq \frac{e_{\alpha}}{e_{\alpha'}}\in
\mathcal{O}^*(V_{\alpha}\cap V_{\alpha'})\,.
\]
It is clear that $c_{\alpha, \alpha'}$ is equal to $\mathrm{Norm}(r_{\alpha, \alpha'})$. This shows the commutativity of the diagram (\ref{diagram}), since the image of $\mathcal{O}(E)$ along the vertical left map of (\ref{diagram}) gives Deligne's line bundle $\<{\mathcal{O}(D), \mathcal{O}(E)}\>$, and the image along the other side of (\ref{diagram}) is
\[(
\mathcal O(D), \mathcal O(E)) =
\mathrm{Norm}\circ \mathsf{T}heta\circ\mathcal{u}p (\mathcal{O}(D), \mathcal{O}(E)) =
\mathrm{Norm}\circ \mathsf{T}heta\circ (G_{\mathcal{O}(D), \mathcal{O}(E)}) =
\int_{\pi} G_{\mathcal{O}(D), \mathcal{O}(E)}.
\]
This proves Theorem \ref{prop3} and therefore Theorem \ref{Main}.
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\section{Picard stacks and their endomorphisms}
\label{sec:picard-stacks-endom}
Here and elsewhere in this paper ``Picard stack,'' or ``Picard category,'' means ``strictly commutative Picard'' in the sense of Deligne \cite{10.1007/BFb0070724}. Namely, if we denote the monoidal operation of $\cP$ simply by $\mathnormal{+} \colon \cP \times \cP \to \cP$, then the symmetry condition given by the natural isomorphisms $\sigma_{x,y} \colon x + y \to y + x$ must satisfy the additional condition that $\sigma_{x,x} = \id_{x+x}$. Such stacks have the pleasant property that there exists a two-term complex of abelian sheaves $d\colon A^{-1} \to A^0$ such that $\cF$ is equivalent, as a Picard stack, to the one associated to the action groupoid formed from the complex. We denote this situation by
\begin{equation*}
\cP \simeq \bigl[ A^{-1} \stackrel{d}{\longrightarrow} A^0 \bigr]\sptilde\,.
\end{equation*}
A classical example arises from the well known divisor exact sequence of Zariski sheaves
\begin{equation}\label{divisorsequence}
0 \to \mathbb{G}_m \to \mathit{\acute{e}t}a_*F(Y)^{\times} \to \bigoplus_{y\in Y^1} (i_y)_*\mathbb Z \to 0\,,
\end{equation}
where $Y$ is a smooth scheme over a field $F$ and the sum is over the set $Y^1$ of points of codimension one in $Y$. We get the equivalence:
\begin{equation*}
\stTors_Y(\mathbb{G}_m) \simeq
\bigl[ \mathit{\acute{e}t}a_*F(Y)^{\times} \to \bigoplus_{y\in Y^1} (i_y)_*\mathbb Z \bigr]\sptilde
\end{equation*}
In the sequel we shall denote by $\mathcal{CH}^1_Y$ the Picard stack on the right hand side of the above relation and by $\mathbf{CH}^1(Y)$ the Picard category of its global sections. Therefore we have $\tors_Y(\mathbb{G}_m) \simeq \mathbf{CH}^1(Y)$.
Still from \cite{10.1007/BFb0070724}, we have that morphisms and natural transformations form a Picard stack $\cHom(\cP,\cQ)$, where the additive structure is defined pointwise: if $F,G$ are two objects, then $(F+G)(x) \coloneq F(x) +_{\cQ} G(x)$, for any object $x$ of $\cP$. It is immediate to verify that this is symmetric and in fact strictly commutative if $+_{\cQ}$ is.
\subsection{Ring structures}
\label{sec:ring-structures}
We set $\stEnd (\cP) = \cHom (\cP,\cP)$. By the above considerations, it is a Picard stack, but the composition of morphisms gives it an additional unital monoidal structure, with respect to which $\stEnd (\cP)$ acquires the structure of a stack of ring groupoids—also known as categorical rings—of the sort described in \cite{rings-tac2015} (see also \cite{drinfeld2021notion} for a résumé). Note that the ``multiplication'' monoidal structure, being given by composition of functors, is strictly associative.
\begin{remark}
\label{rem:butterflies}
If $\cP \simeq [A^{-1}\to A^0]\sptilde$, then by \cite{ButterfliesI} $\stEnd(\cP)$ is equivalent, as a stack of ring groupoids, to $\stCorr(A^\bullet,A^\bullet)$, the stack whose objects are butterfly diagrams: an object is given by an extension $0 \to A^\bullet \to E \to A^0\to 0$ such that its pullback to $A^{-1}$ via $d \colon A^{-1}\to A^0$ is trivial. Morphisms are morphisms of extensions. $\stCorr(A^\bullet,A^\bullet)$ is a stack of ring groupoids: the ``$+$'' is given by the Baer sum of extensions; the ``$\times$'' is given by concatenation of butterflies described in {loc.\,cit.}\xspace This structure is associative, but not strictly so.
\end{remark}
\subsection{Quotients and colimits}
\label{sec:quotients-colimits}
Let $F\colon \cP\to \cQ$ be a morphism of Picard stacks. Its cokernel $\mathsf{C}oker F$ is the stack associated to the following construction, the details of which can be found in the literature (see, e.g.\xspace \cite{VITALE2002383,kv2000}).\footnote{This is valid for Picard categories and stacks that not necessarily strictly commutative.} Let assume $F \colon \mathbf{P}\to \mathbf{Q}$ is a morphism of Picard \emph{categories.} The cokernel $\mathsf{C}oker F$ is a Picard category defined as follows:
\begin{enumerate}
\item its class of objects is the same as that of $\mathbf{Q}$;
\item a morphism $[f , a] \colon x \to y$ is an equivalence class of pairs $(f, a)$, where $f$ is morphism $f \colon x \to x + F(a)$ in $\mathbf{Q}$, $a \in \mathrm{Ob}j\mathbf{P}$, and two pairs $(f,a)$ and $(g,b)$ are equivalent if there exists an arrow $u \colon a\to b$ in $\mathbf{P}$ and the diagram
\begin{equation*}
\begin{tikzcd}[sep=small,cramped]
& x \ar[dl,"f"'] \ar[dr,"g"] \\
y + Fa) \ar[rr,"y+F(u)"'] && y + F(b)
\end{tikzcd}
\end{equation*}
commutes.
\item The monoidal structure is defined to be that of $\mathsf{Q}$ on objects and by the class of the composite arrow
\begin{equation*}
\begin{split}
x + y \to (x + F(a)) + (y + F(b)) &\congto (x + F(a)) + (y + F(b)) \\
&\congto (x + y) + (F(a) + F(b)) \congto (x + y) + F(a + b)
\end{split}
\end{equation*}
if $[f,a]\colon x\to x'$ and $[g,b]\colon y \to y'$.
\end{enumerate}
There is a canonical functor $p_F\colon \mathbf{Q}\to \mathsf{C}oker F$ which is the identity on objects and sends an arrow $f\colon x\to y$ in $\mathbf{Q}$ to the class of the composite:
\begin{equation*}
\begin{tikzcd}[sep=small,cramped]
x \ar[r,"f"] & y \ar[r,"\simeq"] &
y + 0_{\mathsf{Q}} \ar[r,"\simeq"] &
y + F (0_{\mathsf{P}})
\end{tikzcd}\,.
\end{equation*}
There is an isomorphism $\pi_F\colon p_F\circ F \mathrm{R}ightarrow 0\colon \mathbf{P} \to \mathbf{Q}$ given by
\begin{math}
\pi_{F,a} \colon
\begin{tikzcd}[cramped,sep=small]
F(a) \ar[r,"\simeq"] & 0_{\mathbf{Q}} + F(a)
\end{tikzcd}\,.
\end{math}
It follows that we have an abelian group isomorphism
\begin{equation*}
\pi_0(\mathsf{C}oker F) \cong
\mathsf{C}oker \bigl(\pi_0(F) \colon \pi_0(\mathbf{P}) \to \pi_0(\mathbf{Q})\bigr)\,.
\end{equation*}
As mentioned, if $F\colon \cP\to \cQ$ is a morphism of Picard \emph{stacks,} we define $\mathsf{C}oker F$ to be the Picard stack associated to the pseudo-functor
\begin{equation*}
U \rightsquigarrow \mathsf{C}oker \bigl(F_U \colon \cP (U) \to \cQ(U) \bigr)
\end{equation*}
where $U$ is in the base site.
In the following section, we apply this construction to the diagram
\begin{equation*}
\begin{tikzcd}[cramped]
\cP_1 \ar[r,"F_1"] & \cQ \ar[r,leftarrow,"F_2"] & \cP_2
\end{tikzcd}
\end{equation*}
and the resulting morphism $F_1+F_2\colon \cP_1\times \cP_2\to \cQ$, defined on objects by $(F_1 + F_2) (x_1 , x_2) = F_1(x_1) + F_2(x_2)$. We shorten or notation and simply write $\mathsf{C}oker (F_1+F_2)$ as $\cQ/(\cP_1+\cP_2)$. By the above recollection we have
\begin{equation*}
\pi_0\bigl( \cQ/(\cP_1+\cP_2) \bigr) \cong
\pi_0(\cQ) / (\pi_0(\cP_1) + \pi_0(\cP_2))\,.
\end{equation*}
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\section{Categorification of correspondences}
\label{sec:categ-corr}
In this section, $S=\Spec~F$ and a curve $C$ is a smooth projective connected one-dimensional scheme over $S$. For simplicity, we assume (just in this section) that $F$ is algebraically closed.
The main result (Theorem \ref{nuovo}) of this section is an application of Theorem \ref{Main} using \S \ref{sec:picard-stacks-endom} to the categorification of well known identities (\ref{ek}) and (\ref{caar}) about correspondences on the self-product of a curve. We will work with the category $\mathcal{V}$ whose objects are curves and the maps are (correspondences) $\mathrm{H}om_{\mathcal{V}}(D, C) = \mathsf{D}iv(C \times D)$ with composition defined by product of correspondences.
In the following we are stating our results for Picard categories, but there are parallel statements for the corresponding Picard stacks.
\subsection{Categorifying $\mathsf{C}H^1(Y)$} For any smooth scheme $Y$ over $S$, it follows from (\ref{divisorsequence}) that the Chow group $\mathsf{C}H^1(Y)$ is isomorphic to the Picard group $\Pic(Y) = \mathrm{H}^1(Y, \mathbb{G}_m)$ of $Y$; the Picard category $\mathbf{CH}^1(Y)$
is canonically equivalent to the Picard category of $\mathbb G_m$-torsors or line bundles on $Y$.
The Picard category $\mathbf{CH}^1(Y)$ categorifies the Chow group $\mathsf{C}H^1(Y)$ of divisors:
\begin{enumerate}
\item $\mathsf{C}H^1(Y) = \Pic(Y) =\mathrm{H}^1(Y, \mathbb{G}_m) = \pi_0(\mathbf{CH}^1(Y))$;
\item Any map $f \colon Y \to Y'$ of smooth schemes defines an additive functor of Picard categories
\begin{equation*}
f^*: \mathbf{CH}^1(Y') \to \mathbf{CH}^1(Y), \qquad L \mapsto f^*L\,;
\end{equation*}
the induced map on $\pi_0$ is the pullback of divisors $f^*: \mathsf{C}H^1(Y') \to \mathsf{C}H^1(Y)$.
\end{enumerate}
For a curve $C$, let $\Pic^0(C)$ be the kernel of the degree map $\Pic(C) \to \mathbb Z$. If $\mathbf{CH}^1(C)^0$ is the sub-Picard category of $\mathbf{CH}^1(C)$ consisting of line bundles of degree zero, then $\pi_0(\mathbf{CH}^1(C)^0) = \Pic^0(C)$.
\subsection{Correspondences} We refer to \cite[Chapter 16]{MR1644323} for details. Let $C$ and $D$ be curves and let $\pi_C$ and $\pi_D$ be the two projections on $C\times D$. A correspondence $\alpha: D \vdash C$ from $D$ to $C$ is a divisor $\alpha$ on $C \times D$. It defines a line bundle $\mathcal{O}(\alpha)$ on $C \times D$. The correspondence $\alpha$ acts on divisors: it induces a map
\begin{equation*}
\alpha^* \colon \Pic(D) \to \Pic(C) \quad m \mapsto (\pi_C)_*(\pi_D^*m.\alpha)
\end{equation*}
which sends a divisor $m$ on $D$ to the pushforward along $\pi_C$ of the intersection of $\alpha$ and $\pi_D^*m$ on $C\times D$. It restricts to a map $\textrm{Pic}^0(D) \to \textrm{Pic}^0(C)$: if $m$ has degree zero, then so does $\alpha^*(m)$. We get a homomorphism
\begin{equation}
\label{ek}
T\colon \mathsf{C}H^1(C \times D) \to \mathrm{H}om(\Pic(D), \Pic(C)) \to \mathrm{H}om(\Pic^0(D), \Pic^0(C))\,,
\quad \alpha \mapsto \alpha^*
\end{equation}
as $\alpha^*$ depends only on the class of $\alpha$ in $\mathsf{C}H^1(C \times D)$.
Degenerate correspondences \cite[Example 16.1.2]{MR1644323} constitute the subgroup $I(D,C) =\pi_C^*(CH^1(C)) + \pi_D^*(CH^1(D))$ of $\mathsf{C}H^1(C \times D)$. The map $T$ induces an isomorphism \cite[Proposition 3.3, Theorem 3.9]{MR1265529}
\begin{equation}
\label{doh}
T\colon \frac{\mathsf{C}H^1(C \times D)}{I(D,C)} \to \mathrm{H}om(\Pic^0(D), \Pic^0(C));
\end{equation}
see \cite[Chapter 11, Theorem 5.1]{MR2062673} for another proof when $F=\mathbb C$. Over a non-algebraically closed field, the isomorphism holds if $C$ and $D$ have rational points.
Composition of correspondences induces a ring structure on $\mathsf{C}H^1(C \times C)$ with $I(C,C)$ as an ideal \cite[Example 16.1.2]{MR1644323}. It is known that
\begin{itemize}
\item \cite[Corollary 16.1.2]{MR1644323} the map
\begin{equation}\label{teen}
T: \mathsf{C}H^1(C \times C) \to \End(\mathsf{C}H^1(C))\,, \quad \alpha \mapsto \alpha^*
\end{equation}
is a homomorphism of rings.
\item $T$ induces a ring isomorphism \cite[Example 16.1.2(c)]{MR1644323}
\begin{equation}
\label{caar}
T\colon \frac{CH^1(C \times C)}{I(C,C)} \to \End(\Pic^0(C)),
\end{equation}
as $F$ is algebraically closed; see \cite[Chapter 11, Theorem 5.1]{MR2062673} for a proof when $F=\mathbb C$.
\end{itemize}
The following result provides a categorification of the above statements.
\begin{theorem}
\label{nuovo}
There is an additive functor of Picard categories
\begin{equation}
\tilde{T}\colon \mathbf{CH}^1(C \times D) \to
\cHom(\mathbf{CH}^1(D), \mathbf{CH}^1(C)) \to
\cHom(\mathbf{CH}^1(D)^0, \mathbf{CH}^1(C)^0)
\end{equation}
which induces (\ref{ek}) on $\pi_0$. $\mathsf{T}ilde{T}$ has the following properties:
\begin{enumerate}[label=(\roman*)]
\item Let $M(D,C)= \mathsf{C}oker(\pi_C^*+ \pi_D^*)$ be the cokernel of the additive functors
\begin{equation}
\pi_C^*\colon \mathbf{CH}^1(C) \to \mathbf{CH}^1(C \times D)
\leftarrow \mathbf{CH}^1(D) \reflectbox{$\colon$} \pi_D^*.
\end{equation}
Then $\tilde{T}$ induces an additive functor
\begin{equation}
\tilde{T}_{D,C} \colon M(D,C) \to \cHom(\mathbf{CH}^1(D)^0, \mathbf{CH}^1(C)^0)
\end{equation}
which, on $\pi_0$, is (\ref{doh}).
\item $\mathbf{CH}^1(C \times C)$ comes naturally equipped with the structure of a categorical ring (\S \ref{sec:ring-structures}) which, on $\pi_0$, is the composition of correspondences.
\item the functors
\begin{equation}
\mathsf{T}ilde{T} \colon \mathbf{CH}^1(C \times C)\to \stEnd(\mathbf{CH}^1(C)), \quad
\mathsf{T}ilde{T}_{C,C} \colon M(C,C) \to \stEnd(\mathbf{CH}^1(C)^0)
\end{equation}
are functors of categorical rings (\S \ref{sec:ring-structures}). These induce (\ref{teen}), (\ref{caar}) on $\pi_0$.
\end{enumerate}
\end{theorem}
\begin{remark}
The assignment $C \mapsto \mathbf{CH}^1(C)$ and $g\in \mathsf{D}iv (C\times D) \mapsto \mathsf{T}ilde{T}(\mathcal{O} (g))$ comprise a pseudo-functor from $\mathcal{V}$ (the category of curves and correspondences) to the 2-category of Picard categories (or stacks).
\end{remark}
\subsection{Proof of Theorem \ref{nuovo}} The existence of $\tilde{T}$ is provided by the following lemma.
\begin{lemma} For any correspondence $\alpha:D \vdash C$, the map $\alpha^*: \mathsf{C}H^1(D) \to \mathsf{C}H^1(C)$ is induced by an additive functor $\tilde{\alpha}^* \colon \mathbf{CH}^1(D) \to \mathbf{CH}^1(C)$. This functor restricts to a functor $\mathbf{CH}^1(D)^0 \to \mathbf{CH}^1(C)^0$. Further, if $\beta$ is another correspondence, then $\widetilde{(\alpha+\beta)}^* = \tilde{\alpha}^*+ \tilde{\beta}^*$ as additive functors.
\end{lemma}
\begin{proof} This is a simple application of Theorem \ref{Main}. Given a line bundle $M$ on $D$, consider the pair $\pi_D^*M$ and $\mathcal{O} (\alpha)$ of line bundles on $C \times D$; \cite{ER} constructs a $\mathcal{K}_2$-gerbe $G_{(\mathcal{O} (\alpha), \pi_D^*M)}$on $C \times D$. As this gerbe is horizontal by Proposition \ref{beilinson-lemma}, one can integrate it along $\pi_C:C \times D \to C$ to get a line bundle on $C$:
\begin{equation*}
\tilde{\alpha}^*M = \int_{\pi_C}~G_{(\mathcal O(\alpha), \pi_D^*M)} \,.
\end{equation*}
Both the additivity of $\tilde{\alpha}^*$ and the property $\widetilde{(\alpha+\beta)}^* = \tilde{\alpha}^*+ \tilde{\beta}^*$ follow from the bi-additivity (Theorem \ref{prop2}) of $G$. If the line bundle $M$ on $D$ has degree zero, then so does the line bundle $ \tilde{\alpha}^*M$ on $C$ as its class in $CH^1(C)$ is $\alpha^*(M)$ which has degree zero.
\end{proof}
This gives us a bi-additive functor of Picard categories
\[\mathbf{CH}^1(C \times D) \times \mathbf{CH}^1(D) \to \mathbf{CH}^1(C), \qquad (\alpha,M) \mapsto \tilde{\alpha}^*M,\] and an additive functor
\[\tilde{T}: \mathbf{CH}^1(C \times D) \to \cHom(\mathbf{CH}^1(D), \mathbf{CH}^1(C)) \to \cHom(\mathbf{CH}^1(D)^0, \mathbf{CH}^1(C)^0)\]
where, for any pair $P$, $P'$ of Picard categories, $\cHom(P,P')$ is the Picard category of additive functors from $P$ to $P'$.
Statement (i) of Theorem \ref{nuovo} concerns the factorization of $\tilde{T}$ as
\begin{equation}\label{eq2}
\tilde{T}:M(D,C) \to \cHom(\mathbf{CH}^1(D)^0, \mathbf{CH}^1(C)^0)
\end{equation}
This, in turn, follows from the triviality of $\tilde{T}$ on $\pi_C^*\mathbf{CH}^1(C)$ and $\pi_D^*\mathbf{CH}^1(D)$:
\begin{itemize}
\item {\bf $\tilde{T}$ restricted to $\pi_D^*\mathbf{CH}^1(D)$.}
If $g:D \vdash C$ is the pullback $\pi_D^*x$ of a divisor $x$ on $D$, then $\tilde{T}(g)$ applied to a line bundle $L$ on $D$ is defined as $\int_{\pi_C}G_{(\mathcal O(g), \pi_D^*L)}$. As the construction of the $\mathcal{K}_2$-gerbe is functorial, we have
\[G_{(\pi_D^*x, \pi_D^*L)}= \pi_D^*G_{(x,L)};\] as $H^2(D, \mathcal{K}_2) =0$, the $\mathcal{K}_2$-gerbe $G_{(x,L)}$ on $D$ is trivializable. Since $\int_{\pi_C}$ is an additive functor, $\tilde{T}(g)(L)$ is trivializable. It follows that
\[\tilde{T}(g): \mathbf{CH}^1(D)^0 \to \mathbf{CH}^1(C)^0\] is the trivial functor.
\item {\bf $\tilde{T}$ restricted to $\pi_C^*\mathbf{CH}^1(C)$.}
If $g:D \vdash C$ is $\pi^*_Cx$ of a divisor $x$ on $C$ and $m=\sum m
_j y_j$ is a divisor on $D$, then $\tilde{T}(g)(m)$ corresponds to the ${\mathrm deg}~m$-th power of the line bundle $\mathcal O(x)$ and hence is trivial when $m$ has degree zero. This can be seen as follows: $\tilde{T}(g)(m)$ is the object corresponding to the line bundle
\[\<{\pi_D^*m,\pi^*_C x} = \otimes_j \<{\pi^*_D y_j, \pi_C^* x} ^{\otimes m_j}.\]
Since $\pi_C: C \times y_j \hookrightarrow C \times D \to C$ is an isomorphism for any closed point $y_j$ of $D$, one has $\<{\pi^*_D y_j, \pi_C^* x} = \mathcal O(x)$ by (\ref{norm-finite}). By bi-additivity, $$\<{\pi_D^*m,\pi^*_C x}= (\mathcal O(x))^{\mathrm{deg}~m}.$$
If $m$ has degree zero, then $\tilde{T}(g)(m)$ is trivializable.
So the functor
\[\tilde{g}^*~=~\tilde{T}(g): \mathbf{CH}^1(D)^0 \to \mathbf{CH}^1(C)^0\]
is trivial.
\end{itemize}
This completes the proof of (i) of Theorem \ref{nuovo}.
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\subsection{Composition}We show that $\tilde{T}$ is compatible with composition of correspondences. Let $X =C_1 \times C_2 \times C_3$ be the product of three curves $C_1, C_2, C_3$ and let $\pi_{ij}:X\to C_i \times C_j$ be the projections. If $g:C_2\vdash C_1$ is a correspondence on $C_1 \times C_2$ and $h:C_3\vdash C_2$ on $C_2 \times C_3$, one can compose $g$ and $h$ to get a correspondence $g\circ h:C_3\vdash C_1$ on $C_1 \times C_3$:
\[g\circ h = (\pi_{13})
_* (\pi_{23}^*h~.~\pi_{12}^*g),\]
by pulling back $g$ and $h$ to $X$ and intersecting them and pushing forward via $\pi_{13}$ to $C_1 \times C_3$. This gives a bi-additive map
\[\circ: \mathsf{C}H^1(C_1 \times C_2) \times \mathsf{C}H^1(C_2 \times C_3) \to \mathsf{C}H^1(C_1 \times C_3).\]
\begin{lemma}\label{l1}
The above bi-additive map is induced by a bi-additive functor
\[\tilde{\circ}: \mathbf{CH}^1(C_1 \times C_2) \times \mathbf{CH}^1(C_2 \times C_3) \to \mathbf{CH}^1(C_1 \times C_3).\]
\end{lemma}
\begin{proof}
The functor is defined as follows: The pair $\pi_{12}^*\mathcal{O}(g)$ and $\pi_{23}^* \mathcal{O}(h)$ of line bundles on $X$ give rise to a $\mathcal{K}_2$-gerbe $G_{(\pi_{12}^*\mathcal{O}(g),\pi_{23}^* \mathcal{O}(h))}$ on $X$. Since it is horizontal (Proposition \ref{beilinson-lemma}) for the map (a relative curve) $\pi_{13}:X \to C_1 \times C_3$, we can integrate it along $\pi_{13}$ to obtain a line bundle $\<{\pi_{12}^*\mathcal{O}(g),\pi_{23}^* \mathcal{O}(h)}$ on $C_1 \times C_3$. The functor $\tilde{\circ}$, in the notation of Theorem \ref{Main}, is
\[\tilde{\circ}:(g,h) \mapsto \int_{\pi_{13}}G_{(\pi_{12}^*\mathcal{O}(g),\pi_{23}^* \mathcal{O}(h))} = \<{\pi_{12}^*\mathcal{O}(g),\pi_{23}^* \mathcal{O}(h)}.\]
It follows from Theorem \ref{Main} that $\tilde{\circ}$ induces $\circ$ on $\pi_0$.
\end{proof}
Taking $C_1=C_2=C_3=C$ proves (ii) of Theorem \ref{nuovo}.
\begin{lemma}\label{compos-eh?}The functor $\tilde{T}$ is compatible with composition: namely, the diagram
\begin{equation}
\label{eq:100}
\begin{tikzcd}
\mathbf{CH}^1(C_1 \times C_2) \times \mathbf{CH}^1(C_2 \times C_3) \ar[r,"\tilde{\circ}"] \ar[d,"\tilde{T}\times \tilde{T}"'] &
\mathbf{CH}^1(C_1 \times C_3) \ar[d,"\tilde{T}"] \\
\cHom(\mathbf{CH}^1(C_2), \mathbf{CH}^1(C_1)) \times \cHom(\mathbf{CH}^1(C_3), \mathbf{CH}^1(C_2)) \ar[r,""'] & \cHom(\mathbf{CH}^1(C_3), \mathbf{CH}^1(C_1))
\end{tikzcd},
\end{equation}
commutes up to natural isomorphisms $\tilde{T}(g\tilde{\circ}{h}) \cong \tilde{T}(g)\circ\tilde{T}(h)$.
\end{lemma}
\begin{proof} For any smooth projective morphism $f: Y\to B$ of relative dimension one and line bundles $L_1, L_2$ on $Y$, let $\<{L
_1, L_2}_{f} = \int_fG_{(L_1, L_2)}$ denote the Deligne line bundle on $B$. Our task is to prove the existence of a natural isomorphism for any $L \in \mathbf{CH}^1(C_3)$:
\begin{equation}\label{eq:101}
\<{\<{\pi_{12}^*\mathcal{O}(g), \pi_{23}^*\mathcal{O}(h)}_{\pi_{13}}, \alpha^*_3L}_{\alpha_1} \cong \<{\mathcal{O}(g), \gamma^*_{2}\<{\mathcal{O}(h), \beta_3^*L}_{\beta_2}}_{\gamma_1}
\end{equation}
where the maps are
\[\begin{tikzcd}
C_1\times C_3 \ar[r, "\alpha_3"] \ar[d, "\alpha_1"] & C_3, & C_2\times C_3 \ar[r, "\beta_3"] \ar[d, "\beta_2"] &C_3, & C_1\times C_2 \ar[r, "\gamma_2"] \ar[d, "\gamma_1"] & C_2.\\
C_1 &&C_2 &&C_1&
\end{tikzcd} \]
By additivity in $L$, it suffices to consider the case $L = \mathcal{O}(x)$ for a closed point $x = \Spec~F$ of $C_3$. We put
\begin{gather*}
\iota_1\colon C_1 \cong D =C_1 \times x \hookrightarrow C_1 \times C_3, \quad \iota_2\colon C_2 \cong E = C_2 \times x \hookrightarrow C_2 \times C_3, \\
\iota_{12} \colon C_1 \times C_2 \cong C_1 \times C_2 \times x \hookrightarrow
C_1 \times C_2 \times C_2 = X.
\end{gather*}
By (\ref{norm-finite}), the left-hand-side of (\ref{eq:101}) is
\[ \<{\<{\pi_{12}^*\mathcal{O}(g), \pi_{23}^*\mathcal{O}(h)}_{\pi_{13}}, \alpha^*_3L}_{\alpha_1} \cong N_{D/C_1}(\<{\pi_{12}^*\mathcal{O}(g), \pi_{23}^*\mathcal{O}(h)}_{\pi_{13}}\big |_D) \cong \iota_1^*\bigg( \<{\pi_{12}^*\mathcal{O}(g), \pi_{23}^*\mathcal{O}(h)}_{\pi_{13}}\bigg).\]
On the other hand, by (\ref{norm-finite}), the right-hand-side of (\ref{eq:101}) is
\[ \<{\mathcal{O}(g), \gamma^*_{2}\<{\mathcal{O}(h), \beta_3^*L}_{\beta_2}}_{\gamma_1} \cong \<{\mathcal{O}(g), \gamma^*_{2}N_{{E}/C_2}(\mathcal{O}(h)\big |_E)}_{\gamma_1} \cong \<{\mathcal{O}(g), \gamma^*_{2}\iota_2^*\mathcal{O}(h) }_{\gamma_1} \cong\<{\mathcal{O}(g), \iota_{12}^*\pi_{23}^*\mathcal{O}(h)}_{\gamma_1},\]
using $C_2 \cong E$ for the second isomorphism and the following diagram for the last isomorphism:
\[\begin{tikzcd}
C_1\times C_2 \ar[d, "\gamma_2"] \ar[r, "\cong"] &C_1 \times C_2 \times x \ar[r] &C_1 \times C_2 \times C_3 =X \ar[d, "\pi_{23}"]\\
C_2 \ar[r, "\cong"] & E \ar[r]& C_2 \times C_3\,,
\end{tikzcd}
\]
where the top row is $\iota_{12}$ and the bottom one is $\iota_2$.
As $\pi_{12}\circ \iota_{12}$ is the identity map on $C_1 \times C_2$, we have
\[
\<{\mathcal{O}(g), \iota_{12}^*\pi_{23}^*\mathcal{O}(h)}_{\gamma_1} \cong \<{\iota_{12}^* \pi_{12}^*\mathcal{O}(g), \iota_{12}^*\pi_{23}^*\mathcal{O}(h)}_{\gamma_1}.
\]
The required natural isomorphism in (\ref{eq:101}), namely,
\[\<{\iota_{12}^*\pi_{12}^*\mathcal{O}(g), \iota_{12}^*\pi_{23}^*\mathcal{O}(h)}_{\gamma_1} \cong \iota_1^*\bigg( \<{\pi_{12}^*\mathcal{O}(g), \pi_{23}^*\mathcal{O}(h)}_{\pi_{13}}\bigg)
\]
follows from functoriality: use the map of relative curves
\[\begin{tikzcd}
C_1\times C_2 \ar[d, "\gamma_1"] \ar[r, "\cong"] &C_1 \times C_2 \times x \ar[r] &C_1 \times C_2 \times C_3 =X \ar[d, "\pi_{13}"]\\
C_1 \ar[r, "\cong"] & D \ar[r]& C_1 \times C_3\,,
\end{tikzcd}
\]
where the top row is still $\iota_{12}$ and the bottom one is now $\iota_1$.
This proves Lemma \ref{compos-eh?}.
\end{proof}
Taking $C=C_1 = C_2=C_3$ in the above lemma, we obtain that
\[ \tilde{\circ}: \mathbf{CH}^1(C\times C) \times \mathbf{CH}^1(C \times C) \to \mathbf{CH}^1(C \times C)\]
is a monoidal functor of Picard categories and that
\[\tilde{T}: \mathbf{CH}^1(C \times C) \to \stEnd(\mathbf{CH}^1(C))\] is a functor of ring categories proving (iii).
This finishes the proof of Theorem~\ref{nuovo}.
\end{document}
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\begin{document}
\title{Reply to ``Comment on `Witnessed entanglement and the geometric measure of
quantum discord' "}
\author{Tiago Debarba}
\email{[email protected]}
\author{Thiago O. Maciel}
\author{Reinaldo O. Vianna}
\affiliation{Departamento de F\'{\i}sica - ICEx - Universidade Federal de Minas Gerais,
Av. Pres. Ant\^onio Carlos 6627 - Belo Horizonte - MG - Brazil - 31270-901.}
\date{\today}
\begin{abstract}
We show that the mistakes pointed out by Rana and Parashar
[Phys. Rev. A {\bf 87}, 016301 (2013)] do not invalidate the main conclusion
of our work [Phys. Rev. A {\bf 86}, 024302 (2012)].
We show that the errors affected only a particular
application of our general results, and present the correction.
\end{abstract}
\pacs{03.67.Mn, 03.65.Aa}
\maketitle
Rana and Parashar \cite{comment} claim that our
bounds between geometrical discord and entanglement
\cite{debarbapra} are incorrect. They give examples of violations of our
bounds and suggest it has to do with non-monotonicity
of geometrical discord in the Hilbert-Schmidt norm.
The authors started their comment revising our definition of geometrical discord and
pointing a typographical error in the definition of negativity. We defined negativity
as the sum of the negative eigenvalues of the partial transpose of
the state, Eq.16 of our work, while some authors further normalize this quantity.
Their critique about the normalization of
the geometrical discord in the Hilbert-Schmidt norm is also irrelevant,
for the normalized geometrical discord is greater than ours.
The first counterexample which would violate our results is the
maximally entangled state for two qubits ($\phi_{+}$). They
consider the negativity
as $1$, while the 2-norm geometrical
discord is $1/2$.
But it is not correct. Consider Eq.20 ,
\begin{equation}
D_{(2)}(\phi_{+})\geq \frac{E_{w}^2}{Tr(W_{\phi_{+}}^2)}.
\end{equation}
We have $D_{(2)}(\phi_{+})=1/2$, and
$E_{w}=Tr(W_{\phi_{+}}\phi_{+})=Tr(P_{-}\phi_{+}^{T_1})=1/2$, where $P_{-}$ is the
projector associated to the negative eigenvalue of the partial transpose of $\phi_{+}$.
$Tr(W_{\phi_{+}}^2)$ is the number of negative eigenvalues of the partial transpose, which
is $1$.
Thus $D_{(2)} = 1/2 \geq E_{w}^2=1/4$.
The next counterexample is the $2\otimes 32$-dimension state. For this example we have
quantum discord $D_{(2)}(\rho)=0.01$ and $E_{w}^2/Tr[W_{\rho}^2]=0.0032$, where $E_{w}$
is the negativity, and Eq.20 is satisfied. However, in the
comment the equation taken was Eq.21, and via that relation we get
$\mathcal{N}^2/(d-1)^2=0.0316$,
which violates the bound. The point is we mistakenly had written that
$Tr[W_{\rho}^2]\leq d-1$, for a system with dimension $d\otimes d'$ and $d\leq d'$.
In the counterexample we have $Tr[W_{\rho}^2]=10$, i.e. the partial transpose of the
state has $10$ negative eigenvalues and
not $d-1=1$, and this is the reason of the wrong violation in Eq.21.
In the comment, the authors conclude that the violation comes from the fact
that $D_{(2)}(\rho)$ is not a monotonic distance, but monotonicity does not play
any role in our bounds.
Finally, the authors claim that Eq.27 is not valid.
Equation 27 is a particular case of
Eq.22, where we get a linear relation between geometrical discord calculated via trace norm
and witnessed entanglement. This bound is valid only for entanglement measures whose optimal
entanglement witnesses live in the domain $-\mathbb{I} \leq W \leq \mathbb{I}$,
and the entanglement witness for the negativity is not in this domain, which explains the
problem with
the bound in Eq.27. An example of entanglement measure for which
this bound is valid is the random robustness of entanglement, Eq.28.
Equation 27 can be easily corrected by means of an inequality more general than Eq.22, namely:
\begin{equation}
D_{(1)}\geq \frac{E_w}{|| W_{\rho}||_{\infty}},
\end{equation}
where $|| W_{\rho}||_{\infty}$ is the greatest eigenvalue of the optimal entanglement witness
of the state $\rho$
\footnote{Take the well known inequality
for operators $A$ and $B$,
$||A||_{q}||B||_{p}\geq |Tr[AB^{\dagger}]|$, for $1/q+1/p = 1$. Set
$A=\rho-{\xi}$, where $\xi$ is $\rho$'s nearest non-discordant state, and set $B = W_{\rho}$,
where $W_{\rho}$ is the optimal entanglement witness of $\rho$, then follows straightforwardly
$D_{(p)} \geq E_{w}/||W_{\rho}||_{q} $. For $p=1$, we have $q=\infty$.}.
Note that this bound is valid for every witnessed entanglement.
In conclusion, the main results of our work are Eq.20 and Eq.22, which are rigorously correct.
They were calculated from
{\it first principles},
via well known inequalities for operators and properties of entanglement witnesses.
We made two mistakes when specializing for the
negativity, as discussed and clarified above.
The conjecture proposed by D. Girolami
and G. Adesso \cite{girolamiadesso} about the interplay between geometrical
quantum discord and entanglement is implicit
in Eq.20 and Eq.22.
\end{document}
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\begin{document}
\title[Generalized skew Pieri rules]{Quasisymmetric and noncommutative skew Pieri rules}
\author{V. Tewari}
\address{Department of Mathematics, University of Washington, Seattle, WA 98105, USA}
\email{\href{mailto:[email protected]}{[email protected]}}
\author{S. van Willigenburg}
\address{Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada}
\email{\href{mailto:[email protected]}{[email protected]}}
\thanks{
The authors were supported in part by the National Sciences and Engineering Research Council of Canada.}
\subjclass[2010]{Primary 05E05, 16T05, 16W55; Secondary 05A05, 05E10}
\keywords{composition, composition poset, composition tableau, noncommutative symmetric function, quasisymmetric function, skew Pieri rule}
\begin{abstract}
In this note we derive skew Pieri rules in the spirit of Assaf-McNamara for skew quasisymmetric Schur functions using the Hopf algebraic techniques of Lam-Lauve-Sottile, and recover the original rules of Assaf-McNamara as a special case. We then apply these techniques a second time to obtain skew Pieri rules for skew noncommutative Schur functions.
\end{abstract}
\maketitle
\section{Introduction}\label{sec:intro}
The Hopf algebra of quasisymmetric functions, $\ensuremath{\operatorname{QSym}}$, was first defined explicitly in \cite{gessel}. It is a nonsymmetric generalization of the Hopf algebra of symmetric functions, and arises in many areas such as the representation theory of the 0-Hecke algebra \cite{BBSSZ, DKLT, konig, 0-Hecke}, probability \cite{hersh-hsiao, stanley-riffle}, and is the terminal object in the category of combinatorial Hopf algebras \cite{aguiar-bergeron-sottile}. Recently a basis of $\ensuremath{\operatorname{QSym}}$, known as the basis of quasisymmetric Schur functions, was discovered \cite{QS}, which is a nonsymmetric generalization of the symmetric function basis of Schur functions. These quasisymmetric Schur functions arose from the combinatorics of Macdonald polynomials \cite{HHL}, have been used to resolve the conjecture that $\ensuremath{\operatorname{QSym}}$ over the symmetric functions has a stable basis \cite{lauve-mason}, and have initiated the dynamic research area of discovering other quasisymmetric Schur-like bases such as row-strict quasisymmetric Schur functions \cite{ferreira, mason-remmel}, Young quasisymmetric Schur functions \cite{LMvW}, dual immaculate quasisymmetric functions \cite{BBSSZ}, type $B$ quasisymmetric Schur functions \cite{jingli, oguz}, quasi-key polynomials \cite{assafsearles, searles} and quasisymmetric Grothendieck polynomials \cite{monical}. Their name was aptly chosen since these functions not only naturally refine Schur functions, but also generalize many classical Schur function properties, such as the Littlewood-Richardson rule from the classical \cite{littlewood-richardson} to the generalized \cite[Theorem 3.5]{BLvW}, the Pieri rules from the classical \cite{pieri} to the generalized \cite[Theorem 6.3]{QS} and the RSK algorithm from the classical \cite{knuth, robinson, schensted} to the generalized \cite[Procedure 3.3]{mason}.
Dual to $\ensuremath{\operatorname{QSym}}$ is the Hopf algebra of noncommutative symmetric functions, $\ensuremath{\operatorname{NSym}}$ \cite{GKLLRT}, whose basis dual to that of quasisymmetric Schur functions is the basis of noncommutative Schur functions \cite{BLvW}. By duality this basis again has a Littlewood-Richardson rule and RSK algorithm, and, due to noncommutativity, two sets of Pieri rules, one arising from multiplication on the right \cite[Theorem 9.3]{tewari} and one arising from multiplication on the left \cite[Corollary 3.8]{BLvW}. Therefore in both $\ensuremath{\operatorname{QSym}}$ and $\ensuremath{\operatorname{NSym}}$ a key question in this realm remains: Are there \emph{skew} Pieri rules for quasisymmetric and noncommutative Schur functions? In this note we give such rules that are analogous to that of their namesake Schur functions.
More precisely, the note is structured as follows. In Section~\ref{sec:comps} we review necessary notions on compositions and define operators on them. In Section~\ref{sec:QSYMNSYM} we recall $\ensuremath{\operatorname{QSym}}$ and $\ensuremath{\operatorname{NSym}}$, the bases of quasisymmetric Schur functions and noncommutative Schur functions, and their respective Pieri rules. In Section~\ref{sec:skew} we give skew Pieri rules for quasisymmetric Schur functions in Theorem~\ref{the:QSskewPieri} and recover the Pieri rules for skew shapes of Assaf and McNamara in Corollary~\ref{cor:AM}. We close with skew Pieri rules for noncommutative Schur functions in Theorem~\ref{the:NCskewPieri}.
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\section{Compositions and diagrams}\label{sec:comps} A finite list of integers $\alpha = (\alpha _1, \ldots , \alpha _\ell)$ is called a \emph{weak composition} if $\alpha _1, \ldots , \alpha _\ell$ are nonnegative, is called a \emph{composition} if $\alpha _1, \ldots , \alpha _\ell$ are positive, and is called a \emph{partition} if $\alpha _1\geq \cdots \geq\alpha _\ell >0$. Note that every weak composition has an underlying composition, obtained by removing every zero, and in turn every composition has an underlying partition, obtained by reordering the list of integers into weakly decreasing order. Given $\alpha = (\alpha _1, \ldots , \alpha _\ell)$ we call the $\alpha _i$ the \emph{parts} of $\alpha$, also $\ell$ the \emph{length} of $\alpha$ denoted by $\ell(\alpha)$, and the sum of the parts of $\alpha$ the \emph{size} of $\alpha$ denoted by $|\alpha |$. The empty composition of length and size zero is denoted by $\emptyset$. If there exists $\alpha _{k+1} = \cdots = \alpha _{k+j} = i$ then we often abbreviate this to $i^j$. Also, given weak compositions $\alpha= (\alpha_1,\ldots ,\alpha_{\ell})$ and $\beta=(\beta_1,\ldots,\beta_m)$, we define the \emph{concatenation} of $\alpha$ and $\beta$, denoted by $\alpha \beta$, to be the weak composition $(\alpha_1,\ldots,\alpha_\ell,\beta_1,\ldots,\beta_m)$. We define the \emph{near-concatenation} of $\alpha$ and $\beta$, denoted by $\alpha \odot \beta$, to be the weak composition $(\alpha_1,\ldots,\alpha_\ell + \beta_1,\ldots,\beta_m)$.
For example, if $\alpha=(2,1,0,3)$ and $\beta=(1,4,1)$, then $\alpha\beta=(2,1,0,3,1,4,1)$ and $\alpha\odot \beta=(2,1,0,4,4,1)$.
The \emph{composition diagram} of a weak composition $\alpha$, also denoted by $\alpha$, is the array of left-justified boxes with $\alpha _i$ boxes in row $i$ from the \emph{top}, that is, following English notation for Young diagrams of partitions. We will often think of $\alpha$ as both a weak composition and as a composition diagram simultaneously, and hence future computations such as adding/subtracting 1 from the rightmost/leftmost part equalling $i$ (as a weak composition) are synonymous with adding/removing a box from the bottommost/topmost row of length $i$ (as a composition diagram).
\begin{example}\label{ex:comps}
The composition diagram of the weak composition of length 5, $\alpha=(2,0,4,3,6)$, is shown below.
$$\tableau{\ &\ \\ \\ \ &\ &\ &\ \\ \ &\ &\ \\\ &\ &\ &\ &\ &\ }$$
The composition of length 4 underlying $\alpha$ is $(2,4,3,6)$, and the partition of length 4 underlying it is $(6,4,3,2)$. They all have size 15.
\end{example}
\subsection{Operators on compositions}\label{sec:ops} In this subsection we will recall four families of operators, each of which are dependent on a positive integer parameter. These families have already contributed to the theory of quasisymmetric and noncommutative Schur functions, and will continue to cement their central role as we shall see later. Although originally defined on compositions, we will define them in the natural way on weak compositions to facilitate easier proofs. The first of these operators is the box removing operator $\mathfrak{d}$, which first appeared in the Pieri rules for quasisymmetric Schur functions \cite{QS}. The second of these is the appending operator $a$. These combine to define our third operator, the jeu de taquin or jdt operator $\mathfrak{u}$. This operator is pivotal in describing jeu de taquin slides on tableaux known as semistandard reverse composition tableaux and in describing the right Pieri rules for noncommutative Schur functions \cite{tewari}. Our fourth and final operator is the box adding operator $\mathfrak{t}$ \cite{BLvW, MNtewari}, which plays the same role in the left Pieri rules for noncommutative Schur functions \cite{BLvW} as $\mathfrak{u}$ does in the aforementioned right Pieri rules. Each of these operators is defined on weak compositions for every integer $i\geq 0$. We note that
$$\mathfrak{d} _0 = a_0 = \mathfrak{u} _0 = \mathfrak{t} _ 0 = {Id}$$namely the identity map, which fixes the weak composition it is acting on. With this in mind we now define the remaining operators for $i\geq 1$.
The first \emph{box removing operator} on weak compositions, $\mathfrak{d} _i$ for $i\geq 1$, is defined as follows. Let $\alpha$ be a weak composition. Then
$$\mathfrak{d} _i (\alpha) = \alpha '$$where $\alpha '$ is the weak composition obtained by subtracting 1 from the rightmost part equalling $i$ in $\alpha$. If there is no such part then we define $\mathfrak{d} _i(\alpha) = 0$.
\begin{example}\label{ex:down}
Let $\alpha=(2,1,2)$. Then $\mathfrak{d}_1(\alpha)=(2,0,2)$ and $\mathfrak{d}_2(\alpha)=(2,1,1)$.
\end{example}
Now we will discuss two notions that will help us state our theorems in a concise way later, as well as connect our results to those in the classical theory of symmetric functions. Let $i_1 < \cdots < i_k$ be a sequence of positive integers, and let $\alpha$ be a weak composition. Consider the operator $\mathfrak{d}_{i_1}\cdots \mathfrak{d}_{i_k}$ acting on the weak composition $\alpha$, and assume that the result is a valid weak composition. Then the boxes that are removed from $\alpha$ are said to form a \emph{$k$-horizontal strip}, and we can think of the operator $\mathfrak{d}_{i_1}\cdots \mathfrak{d}_{i_k}$ as removing a $k$-horizontal strip. Similarly, given a sequence of positive integers $i_1\geq \cdots \geq i_k$, consider the operator $\mathfrak{d}_{i_1}\cdots \mathfrak{d}row_{i_k}$ acting on $\alpha$ and suppose that the result is a valid weak composition. Then the boxes that are removed from $\alpha$ are said to form a \emph{$k$-vertical strip}. As before, we can think of the operator $\mathfrak{d}_{i_1}\cdots \mathfrak{d}row_{i_k}$ as removing a $k$-vertical strip.
\begin{example}\label{ex:horizontal and vertical strip}
Consider $\alpha=(2,5,1,3,1)$. When we compute $\mathfrak{d}_{1}\mathfrak{d}_2\mathfrak{d}_4\mathfrak{d}_5(\alpha)$, the operator $\mathfrak{d}_{1}\mathfrak{d}_2\mathfrak{d}_4\mathfrak{d}_5$ removes the $4$-horizontal strip shaded in red from $\alpha$.
$$
\ytableausetup{smalltableaux,boxsize=0.5em}
\begin{ytableau}
*(white) & *(red!80)\\
*(white) &*(white) &*(white) &*(red!80) &*(red!80) \\
*(white) \\
*(white) &*(white) &*(white) \\
*(red!80)\\
\end{ytableau}
$$
When we compute $\mathfrak{d}_3\mathfrak{d}_2\mathfrak{d}_1\mathfrak{d}_1(\alpha)$, the operator $\mathfrak{d}_3\mathfrak{d}_2\mathfrak{d}_1\mathfrak{d}_1$ removes the $4$-vertical strip shaded in red from $\alpha$.
$$
\ytableausetup{smalltableaux,boxsize=0.5em}
\begin{ytableau}
*(white) & *(red!80)\\
*(white) &*(white) &*(white) &*(white) &*(white) \\
*(red!80) \\
*(white) &*(white) &*(red!80) \\
*(red!80)\\
\end{ytableau}
$$
\end{example}
\begin{remark}\label{rem:horizontal and vertical strip}
If we consider partitions as Young diagrams in English notation, then the above notions of horizontal and vertical strips coincide with their classical counterparts. For example, consider the operator $\mathfrak{d}_1\mathfrak{d}_2\mathfrak{d}_4\mathfrak{d}_5$ acting on the partition $(5,3,2,1,1)$, in contrast to acting on the composition $(2,5,1,3,1)$ as in Example \ref{ex:horizontal and vertical strip}. Then the $4$-horizontal strip shaded in red is removed.
$$
\ytableausetup{smalltableaux,boxsize=0.5em}
\begin{ytableau}
*(white) &*(white) &*(white) &*(red!80) &*(red!80) \\
*(white) &*(white) &*(white) \\
*(white) & *(red!80)\\
*(white) \\
*(red!80)
\end{ytableau}
$$
\end{remark}
We now define the second \emph{appending operator} on weak compositions, $a_i$ for $i\geq 1$, as follows. Let $\alpha = (\alpha _1, \ldots , \alpha _{\ell(\alpha)})$ be a weak composition. Then
$$a _i (\alpha) = (\alpha _1, \ldots , \alpha _{\ell(\alpha)}, i)$$namely, the weak composition obtained by appending a part $i$ to the end of $\alpha$.
\begin{example}\label{ex:append} Let $\alpha = (2,1,3)$. Then $a_2 ((2,1,3))= (2,1,3,2)$. Meanwhile, $a_j \mathfrak{d} _2 ((3,5,1)) = 0$ for all $j\geq 0$ since $\mathfrak{d} _2 ((3,5,1)) = 0$.
\end{example}
With the definitions of $a_i$ and $\mathfrak{d} _i$ we define the third \emph{jeu de taquin} or \emph{jdt operator} on weak compositions, $\mathfrak{u} _i$ for $i\geq 1$, as
$$\mathfrak{u} _i = a_i \mathfrak{d}_1\mathfrak{d}_2\mathfrak{d}_3 \cdots \mathfrak{d} _{i-1}.$$
\begin{example}\label{ex:jdt}
We will compute $\mathfrak{u}_4(\alpha)$ where $\alpha = (3,5,2,4,1,2)$. This corresponds to computing $a_4\mathfrak{d}_1\mathfrak{d}_2\mathfrak{d}_3(\alpha)$. Now
\begin{eqnarray*}
\mathfrak{d}_1\mathfrak{d}_2\mathfrak{d}_3(\alpha)&=&\mathfrak{d}_1\mathfrak{d}_2\mathfrak{d}_3((3,5,2,4,1,2))\\&=&\mathfrak{d}_1\mathfrak{d}_2((2,5,2,4,1,2))\\&=& \mathfrak{d}_1((2,5,2,4,1,1))\\&=& (2,5,2,4,1,0).
\end{eqnarray*} Hence $\mathfrak{u}_4(\alpha)=(2,5,2,4,1,0,4)$.
\end{example}
Let $i_1 < \cdots < i_k$ be a sequence of positive integers, and let $\alpha$ be a weak composition. Consider the operator $\mathfrak{u}_{i_k}\cdots \mathfrak{u}_{i_1}$ acting on the weak composition $\alpha$, and assume that the result is a valid weak composition. Then the boxes that are added to $\alpha$ are said to form a \emph{$k$-right horizontal strip}, and we can think of the operator $\mathfrak{u}_{i_k}\cdots \mathfrak{u}_{i_1}$ as adding a $k$-right horizontal strip. Similarly, given a sequence of positive integers $i_1\geq \cdots \geq i_k$, consider the operator $\mathfrak{u}_{i_k}\cdots \mathfrak{u}_{i_1}$ acting on $\alpha$ and suppose that the result is a valid weak composition. Then the boxes that are added to $\alpha$ are said to form a \emph{$k$-right vertical strip}. As before, we can think of the operator $\mathfrak{u}_{i_k}\cdots \mathfrak{u}_{i_1}$ as adding a $k$-right vertical strip.
Lastly, we define the fourth \emph{box adding operator} on weak compositions, $\mathfrak{t} _i$ for $i\geq 1$, as follows. Let $\alpha = (\alpha _1, \ldots , \alpha _{\ell(\alpha)})$ be a weak composition. Then
$$\mathfrak{t} _1 (\alpha) = (1, \alpha _1, \ldots , \alpha _{\ell(\alpha)})$$and for $i\geq 2$
$$\mathfrak{t} _i (\alpha) = (\alpha _1, \ldots , \alpha _j + 1, \ldots ,\alpha _{\ell(\alpha)})$$where $\alpha _j$ is the leftmost part equalling $i-1$ in $\alpha$. If there is no such part, then we define $\mathfrak{t} _i (\alpha) = 0$.
\begin{example}\label{ex:boxadd}
Consider the composition $\alpha=(3,2,3,1,2)$. Then $\mathfrak{t}_1(\alpha)=(1,3,2,3,1,2)$, $\mathfrak{t}_2(\alpha)=(3,2,3,2,2)$, $\mathfrak{t}_3(\alpha)=(3,3,3,1,2)$, $\mathfrak{t}_4(\alpha)=(4,2,3,1,2)$ and $\mathfrak{t}_i(\alpha)=0$ for all $i\geq 5$.
\end{example}
As with the jdt operators let $i_1 < \cdots < i_k$ be a sequence of positive integers, and let $\alpha$ be a weak composition. Consider the operator $\mathfrak{t}_{i_k}\cdots \mathfrak{t}_{i_1}$ acting on the weak composition $\alpha$, and assume that the result is a valid weak composition. Then the boxes that are added to $\alpha$ are said to form a \emph{$k$-left horizontal strip}, and we can think of the operator $\mathfrak{t}_{i_k}\cdots \mathfrak{t}_{i_1}$ as adding a $k$-left horizontal strip. Likewise, given a sequence of positive integers $i_1\geq \cdots \geq i_k$, consider the operator $\mathfrak{t}_{i_k}\cdots \mathfrak{t}_{i_1}$ acting on $\alpha$ and suppose that the result is a valid weak composition. Then the boxes that are added to $\alpha$ are said to form a \emph{$k$-left vertical strip}, and we can think of the operator $\mathfrak{t}_{i_k}\cdots \mathfrak{t}_{i_1}$ as adding a $k$-left vertical strip.
The box adding operator is also needed to define the composition poset \cite[Definition 2.3]{BLvW}, which in turn will be needed to define skew quasisymmetric Schur functions in the next section.
\begin{definition}\label{def:RcLc} The \emph{composition poset}, denoted by $\mathcal{L}_{c}$, is the poset consisting of the set of all compositions equipped with cover relation $\lessdot _Yc$ such that for compositions $\alpha, \beta$
$$\beta \lessdot _Yc \alpha \mbox{ if and only if } \alpha = \mathfrak{t} _i (\beta)$$for some $i\geq1$.
\end{definition}
The order relation $< _{c}$ in $\mathcal{L}_{c}$ is obtained by taking the transitive closure of the cover relation $\lessdot _Yc$.
\begin{example}\label{ex:boxaddLc}
We have that $(3,2,3,1,2) \lessdot _Yc (4,2,3,1,2)$ by Example~\ref{ex:boxadd}.
\end{example}
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\section{Quasisymmetric and noncommutative symmetric functions}\label{sec:QSYMNSYM}
We now recall the basics of graded Hopf algebras before focussing on the graded Hopf algebra of quasisymmetric functions \cite{gessel} and its dual, the graded Hopf algbera of noncommutative symmetric functions \cite{GKLLRT}. We say that $\mathcal{H}$ and $\mathcal{H}^*$ form a pair of dual graded Hopf algebras each over a field $K$ if there exists a duality pairing $\langle \ ,\ \rangle : \mathcal{H}\otimes \mathcal{H}^{*} \longrightarrow K$, for which the structure of $\mathcal{H}^*$ is dual to $\mathcal{H}$ that respects the grading, and vice versa. More precisely, the duality pairing pairs the elements of any basis $\{B_i\}_{i\in I}$ of the graded piece $\mathcal{H}^N$ for some index set $I$, and the elements of its dual basis $\{D_i\}_{i\in I}$ of the graded piece $(\mathcal{H}^N)^*$,
given by
$\langle B_i, D_j\rangle = \delta_{ij}$, where the \emph{Kronecker delta}\index{Kronecker delta} $\delta_{ij} = 1$ if $i=j$ and 0 otherwise.
Duality is exhibited in that the product coefficients of one basis are the coproduct coefficients of its dual basis and vice versa, that is, \begin{eqnarray*}
B_i \cdot B_j = \sum_k b^k_{i,j} B_k &\qquad \Longleftrightarrow\qquad &
\Delta D_k = \sum_{i,j} b^k_{i,j} D_i \otimes D_j \\
D_i \cdot D_j = \sum_k d^k_{i,j} D_k &\qquad \Longleftrightarrow\qquad &
\Delta B_k = \sum_{i,j} d^k_{i,j} B_i \otimes B_j
\end{eqnarray*}where $\cdot$ denotes \emph{product} and $\Delta$ denotes \emph{coproduct}. Graded Hopf algebras also have an \emph{antipode} $S: \mathcal{H}\longrightarrow\mathcal{H}$, whose general definition we will not need. Instead we will state the specific antipodes, as needed, later. Lastly, before we define our specific graded Hopf algebras, we recall one Hopf algebraic lemma, which will play a key role later. For $h\in \mathcal{H}$ and $a\in \mathcal{H}^{*}$, let the following be the respective coproducts in Sweedler notation.
\begin{eqnarray}\label{eq:coproductH}
\Delta (h)&=& \displaystyle\sum_{h} h_{(1)}\otimes h_{(2)}
\end{eqnarray}
\begin{eqnarray}\label{eq:coproductHdual}
\Delta (a)&=&\displaystyle\sum_{a} a_{(1)}\otimes a_{(2)}
\end{eqnarray}
Now define left actions of $\mathcal{H}^{*}$ on $\mathcal{H}$ and $\mathcal{H}$ on $\mathcal{H}^{*}$, both denoted by $\rightharpoonup$, as
\begin{eqnarray}\label{eq:HdualactingonH}
a \rightharpoonup h &=& \displaystyle\sum_{h}\langle h_{(2)},a\rangle h_{(1)},
\end{eqnarray}
\begin{eqnarray}\label{eq:HactingonHdual}
h\rightharpoonup a &=& \displaystyle\sum_{a} \langle h,a_{(2)}\rangle a_{(1)},
\end{eqnarray}
where $a\in \mathcal{H}^{*}$, $h\in \mathcal{H}$. Then we have the following.
\begin{lemma}\cite{lam-lauve-sottile}\label{lem:magiclemma}
For all $g,h \in \mathcal{H}$ and $a\in \mathcal{H}^{*}$, we have that
\begin{eqnarray*}
(a\rightharpoonup g)\cdot h &= & \displaystyle\sum_{h} \left( S(h_{(2)})\rightharpoonup a \right)\rightharpoonup \left(g\cdot h_{(1)}\right)
\end{eqnarray*}
where $S:\mathcal{H}\longrightarrow \mathcal{H}$ is the antipode.
\end{lemma}
The graded Hopf algebra of quasisymmetric functions, $\ensuremath{\operatorname{QSym}}$ \cite{gessel}, is a subalgebra of $\mathbb{C} [[x_1, x_2, \ldots]]$ with a basis given by the following functions, which in turn are reliant on the natural bijection between compositions and sets, for which we first need to recall that $[i]$ for $i\geq 1$ denotes the set $\{1,2,\ldots , i\}$. Now we can state the bijection. Given a composition $\alpha = ( \alpha _1 , \ldots , \alpha _{\ell(\alpha)})$, there is a natural subset of $[|\alpha|-1]$ corresponding to it, namely,
$$\mathrm{set} (\alpha) = \{ \alpha _1 , \alpha _1 + \alpha _2, \ldots , \alpha _1+\alpha _2 + \cdots + \alpha _{\ell(\alpha)-1}\} \mbox{ and } \mathrm{set}((|\alpha|))=\emptyset.$$Conversely, given a subset $S = \{ s_1< \cdots < s_{|S|}\}\subseteq [N-1]$, there is a natural composition of size $N$ corresponding to it, namely, $$\mathrm{comp} (S) = (s_1, s_2 - s_1, \ldots , N-s_{|S|}) \mbox{ and } \mathrm{comp}(\emptyset)=(N).$$
\begin{definition}\label{def:Fbasis}
Let $\alpha = (\alpha _1, \ldots , \alpha _{\ell(\alpha)})$ be a composition. Then the \emph{fundamental quasisymmetric function} $F_\alpha$ is defined to be $F_\emptyset = 1$ and
$$F_\alpha = \sum x_{i_1} \cdots x_{i_{|\alpha|}}$$where the sum is over all $|\alpha|$-tuples $(i_1, \ldots , i_{|\alpha|})$ of indices satisfying
$$i_1\leq \cdots \leq i_{|\alpha|} \mbox{ and } i_j<i_{j+1} \mbox{ if } j \in \mathrm{set}(\alpha).$$
\end{definition}
\begin{example}\label{ex:Fbasis}
$F_{(1,2)} = x_1x_2^2 + x_1x_3^2 + \cdots + x_1x_2x_3 + x_1x_2x_4 + \cdots.$
\end{example}
Then $\ensuremath{\operatorname{QSym}}$ is a graded Hopf algebra
$$\ensuremath{\operatorname{QSym}} = \bigoplus _{N\geq 0} \ensuremath{\operatorname{QSym}} ^N$$where
$$\ensuremath{\operatorname{QSym}} ^N = \operatorname{span} \{ F_\alpha \;|\; |\alpha| = N \}.$$The product for this basis is inherited from the product of monomials and Definition~\ref{def:Fbasis}. The coproduct \cite{gessel} is given by $\Delta(1)=1\otimes1$ and
\begin{equation}\label{eq:Fcoproduct}
\Delta(F_\alpha)= \sum _{\beta\gamma = \alpha \atop \mbox{ or }\beta\odot\gamma = \alpha}F_\beta \otimes F_\gamma
\end{equation}and the antipode, which was discovered independently in \cite{ehrenborg-1, malvenuto-reutenauer}, is given by $S(1)=1$ and
\begin{equation}\label{eq:antipode}
S(F_\alpha)= (-1)^{|\alpha|}F_{\mathrm{comp}(\mathrm{set}(\alpha)^c)}
\end{equation}where $\mathrm{set}(\alpha)^c$ is the complement of $\mathrm{set}(\alpha)$ in the set $[|\alpha| -1]$.
\begin{example}\label{ex:Fcoprodantipode}
$$\Delta (F_{(1,2)})=F_{(1,2)}\otimes 1 + F_{(1,1)}\otimes F_{(1)} + F_{(1)}\otimes F_{(2)} + 1\otimes F_{(1,2)}$$and $S(F_{(1,2)})=(-1)^3 F_{(2,1)}$.
\end{example}
However, this is not the only basis of $\ensuremath{\operatorname{QSym}}$ that will be useful to us. For the second basis we will need to define skew composition diagrams and then standard skew composition tableaux.
For the first of these, let $\alpha, \beta$ be two compositions such that $\beta < _{c} \alpha$. Then we define the \emph{skew composition diagram} $\alpha {/\!\!/} \beta $ to be the array of boxes that are contained in $\alpha$ but not in $\beta$. That is, the boxes that arise in the saturated chain $\beta \lessdot _Yc \cdots \lessdot _Yc \alpha$. We say the \emph{size} of $\alpha {/\!\!/} \beta$ is $|\alpha {/\!\!/} \beta | = |\alpha| - |\beta|$. Note that if $\beta=\emptyset$, then we recover the composition diagram $\alpha$.
\begin{example}\label{ex:skewshape} The skew composition diagram $(2,1,3){/\!\!/} (1)$ is drawn below with $\beta$ denoted by $\bullet$.
$$\tableau{\ &\ \\
\ \\
\bullet&\ &\ \\}$$
\end{example}
We can now define standard skew composition tableaux. Given a saturated chain, $C$, in $\mathcal{L}_{c}$
$$\beta = \alpha ^0 \lessdot _Yc \alpha ^1 \lessdot _Yc \cdots \lessdot _Yc \alpha ^{|\alpha {/\!\!/} \beta|} = \alpha$$we define the \emph{standard skew composition tableau} ${\tau} _C$ of \emph{shape} $\alpha{/\!\!/} \beta$ to be the skew composition diagram $\alpha {/\!\!/} \beta$ whose boxes are filled with integers such that the number $|\alpha {/\!\!/} \beta| -i +1$ appears in the box in ${\tau} _C$ that exists in $\alpha ^i$ but not $\alpha ^{i-1}$ for $1\leq i\leq |\alpha {/\!\!/} \beta|$. If $\beta = \emptyset$, then we say that we have a \emph{standard composition tableau}. Given a standard skew composition tableau, ${\tau}$, whose shape has size $N$ we say that the \emph{descent set} of ${\tau}$ is
$$\mathrm{Des} ({\tau}) = \{ i \;|\; i+1 \mbox{ appears weakly right of } i \} \subseteq [N-1]$$and the corresponding \emph{descent composition} of ${\tau}$ is $\mathrm{comp}({\tau})= \mathrm{comp}(\mathrm{Des}({\tau}))$.
\begin{example}\label{ex:skewCT} The saturated chain
$$(1)\lessdot _Yc (2) \lessdot _Yc (1,2) \lessdot _Yc (1,1,2) \lessdot _Yc (1,1,3) \lessdot _Yc (2,1,3)$$gives rise to the standard skew composition tableau ${\tau}$ of shape $(2,1,3){/\!\!/} (1)$ below.
$$\tableau{3 &1\\
4 \\
\bullet&5 &2 \\}$$Note that $\mathrm{Des}({\tau}) = \{1,3, 4\}$ and hence $\mathrm{comp}({\tau}) = (1,2,1,1)$.
\end{example}
With this is mind we can now define skew quasisymmetric Schur functions \cite[Proposition 3.1]{BLvW}.
\begin{definition}\label{def:QSbasis}
Let $\alpha {/\!\!/} \beta$ be a skew composition diagram. Then the \emph{skew quasisymmetric Schur function} ${\mathcal{S}} _{\alpha{/\!\!/} \beta}$ is defined to be
$${\mathcal{S}} _{\alpha {/\!\!/} \beta} = \sum F_{\mathrm{comp} ({\tau})}$$where the sum is over all standard skew composition tableaux ${\tau}$ of shape $\alpha{/\!\!/} \beta$. When $\beta = \emptyset$ we call ${\mathcal{S}} _\alpha$ a \emph{quasisymmetric Schur function}.
\end{definition}
\begin{example}\label{ex:QSbasis} We can see that ${\mathcal{S}} _{(n)} = F_{(n)}$ and ${\mathcal{S}} _{(1^n)} = F_{(1^n)}$ and
$${\mathcal{S}} _{(2,1,3){/\!\!/} (1)} = F_{(2,1,2)}+ F_{(2,2,1)} + F_{(1,2,1,1)}$$from the standard skew composition tableaux below.
$$\tableau{2 &1\\
3 \\
\bullet&5 &4 \\}\qquad
\tableau{2 &1\\
4 \\
\bullet&5 &3 \\}\qquad
\tableau{3 &1\\
4 \\
\bullet&5 &2 \\}$$
\end{example}
Moreover, the set of all quasisymmetric Schur functions forms another basis for $\ensuremath{\operatorname{QSym}}$ such that
$$\ensuremath{\operatorname{QSym}} ^N = \operatorname{span} \{ {\mathcal{S}} _\alpha \;|\; |\alpha| = N\}$$and while explicit formulas for their product and antipode are still unknown, their coproduct \cite[Definition 2.19]{BLvW} is given by
\begin{eqnarray}\label{eq:coproductquasischur}
\Delta({\mathcal{S}}_{\alpha})=\displaystyle\sum_{\gamma} {\mathcal{S}}_{\alpha{/\!\!/}\gamma}\otimes {\mathcal{S}}_{\gamma}
\end{eqnarray}where the sum is over all compositions $\gamma$. As discussed in the introduction, quasisymmetric Schur functions have many interesting algebraic and combinatorial properties, one of the first of which to be discovered was the exhibition of Pieri rules that utilise our box removing operators \cite[Theorem 6.3]{QS}.
\begin{theorem}\emph{(Pieri rules for quasisymmetric Schur functions)}\label{the:QSPieri}
Let $\alpha $ be a composition and $n$ be a positive integer. Then
\begin{align*}
{\mathcal{S}}_{\alpha}\cdot {\mathcal{S}}_{(n)}=\sum {\mathcal{S}}_{\alpha^+}
\end{align*}
where $\alpha^+$ is a composition such that $\alpha$ can be obtained by removing an $n$-horizontal strip from it.
Similarly,
\begin{align*}
{\mathcal{S}}_{\alpha}\cdot {\mathcal{S}}_{(1^n)}=\sum {\mathcal{S}}_{\alpha^+ }
\end{align*}
where $\alpha^+$ is a composition such that $\alpha$ can be obtained by removing an $n$-vertical strip from it.
\end{theorem}
Dual to $\ensuremath{\operatorname{QSym}}$ is the graded Hopf algebra of noncommutative symmetric functions, $\ensuremath{\operatorname{NSym}}$, itself a subalgebra of $\mathbb{C} << x_1, x_2, \ldots >>$ with many interesting bases \cite{GKLLRT}. The one of particular interest to us is the following \cite[Section 2]{BLvW}.
\begin{definition}\label{def:NCbasis} Let $\alpha$ be a composition. Then the \emph{noncommutative Schur function} ${\mathbf{s}} _\alpha$ is the function under the duality pairing $\langle \ ,\ \rangle :\ensuremath{\operatorname{QSym}} \otimes \ensuremath{\operatorname{NSym}} \rightarrow \mathbb{C}$ that satisfies
$$\langle {\mathcal{S}} _\alpha , {\mathbf{s}} _\beta \rangle = \delta _{\alpha\beta}$$where $\delta _{\alpha\beta} = 1$ if $\alpha = \beta$ and $0$ otherwise.\end{definition}
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\begin{example}\label{ex:skewCT} The saturated chain
$$(1)\lessdot _Yc (2) \lessdot _Yc (1,2) \lessdot _Yc (1,1,2) \lessdot _Yc (1,1,3) \lessdot _Yc (2,1,3)$$gives rise to the standard skew composition tableau ${\tau}$ of shape $(2,1,3){/\!\!/} (1)$ below.
$$\tableau{3 &1\\
4 \\
\bullet&5 &2 \\}$$Note that $\mathrm{Des}({\tau}) = \{1,3, 4\}$ and hence $\mathrm{comp}({\tau}) = (1,2,1,1)$.
\end{example}
With this is mind we can now define skew quasisymmetric Schur functions \cite[Proposition 3.1]{BLvW}.
\begin{definition}\label{def:QSbasis}
Let $\alpha {/\!\!/} \beta$ be a skew composition diagram. Then the \emph{skew quasisymmetric Schur function} ${\mathcal{S}} _{\alpha{/\!\!/} \beta}$ is defined to be
$${\mathcal{S}} _{\alpha {/\!\!/} \beta} = \sum F_{\mathrm{comp} ({\tau})}$$where the sum is over all standard skew composition tableaux ${\tau}$ of shape $\alpha{/\!\!/} \beta$. When $\beta = \emptyset$ we call ${\mathcal{S}} _\alpha$ a \emph{quasisymmetric Schur function}.
\end{definition}
\begin{example}\label{ex:QSbasis} We can see that ${\mathcal{S}} _{(n)} = F_{(n)}$ and ${\mathcal{S}} _{(1^n)} = F_{(1^n)}$ and
$${\mathcal{S}} _{(2,1,3){/\!\!/} (1)} = F_{(2,1,2)}+ F_{(2,2,1)} + F_{(1,2,1,1)}$$from the standard skew composition tableaux below.
$$\tableau{2 &1\\
3 \\
\bullet&5 &4 \\}\qquad
\tableau{2 &1\\
4 \\
\bullet&5 &3 \\}\qquad
\tableau{3 &1\\
4 \\
\bullet&5 &2 \\}$$
\end{example}
Moreover, the set of all quasisymmetric Schur functions forms another basis for $\ensuremath{\operatorname{QSym}}$ such that
$$\ensuremath{\operatorname{QSym}} ^N = \operatorname{span} \{ {\mathcal{S}} _\alpha \;|\; |\alpha| = N\}$$and while explicit formulas for their product and antipode are still unknown, their coproduct \cite[Definition 2.19]{BLvW} is given by
\begin{eqnarray}\label{eq:coproductquasischur}
\Delta({\mathcal{S}}_{\alpha})=\displaystyle\sum_{\gamma} {\mathcal{S}}_{\alpha{/\!\!/}\gamma}\otimes {\mathcal{S}}_{\gamma}
\end{eqnarray}where the sum is over all compositions $\gamma$. As discussed in the introduction, quasisymmetric Schur functions have many interesting algebraic and combinatorial properties, one of the first of which to be discovered was the exhibition of Pieri rules that utilise our box removing operators \cite[Theorem 6.3]{QS}.
\begin{theorem}\emph{(Pieri rules for quasisymmetric Schur functions)}\label{the:QSPieri}
Let $\alpha $ be a composition and $n$ be a positive integer. Then
\begin{align*}
{\mathcal{S}}_{\alpha}\cdot {\mathcal{S}}_{(n)}=\sum {\mathcal{S}}_{\alpha^+}
\end{align*}
where $\alpha^+$ is a composition such that $\alpha$ can be obtained by removing an $n$-horizontal strip from it.
Similarly,
\begin{align*}
{\mathcal{S}}_{\alpha}\cdot {\mathcal{S}}_{(1^n)}=\sum {\mathcal{S}}_{\alpha^+ }
\end{align*}
where $\alpha^+$ is a composition such that $\alpha$ can be obtained by removing an $n$-vertical strip from it.
\end{theorem}
Dual to $\ensuremath{\operatorname{QSym}}$ is the graded Hopf algebra of noncommutative symmetric functions, $\ensuremath{\operatorname{NSym}}$, itself a subalgebra of $\mathbb{C} << x_1, x_2, \ldots >>$ with many interesting bases \cite{GKLLRT}. The one of particular interest to us is the following \cite[Section 2]{BLvW}.
\begin{definition}\label{def:NCbasis} Let $\alpha$ be a composition. Then the \emph{noncommutative Schur function} ${\mathbf{s}} _\alpha$ is the function under the duality pairing $\langle \ ,\ \rangle :\ensuremath{\operatorname{QSym}} \otimes \ensuremath{\operatorname{NSym}} \rightarrow \mathbb{C}$ that satisfies
$$\langle {\mathcal{S}} _\alpha , {\mathbf{s}} _\beta \rangle = \delta _{\alpha\beta}$$where $\delta _{\alpha\beta} = 1$ if $\alpha = \beta$ and $0$ otherwise.\end{definition}
Noncommutative Schur functions also have rich and varied algebraic and combinatorial properties, including Pieri rules, although due to the noncommutative nature of $\ensuremath{\operatorname{NSym}}$ there are now Pieri rules arising both from multiplication on the right \cite[Theorem 9.3]{tewari}, and from multiplication on the left \cite[Corollary 3.8]{BLvW}. We include them both here for completeness, and for use later.
\begin{theorem}\emph{(Right Pieri rules for noncommutative Schur functions)}\label{the:RightPieri}
Let $\alpha $ be a composition and $n$ be a positive integer. Then
\begin{align*}
{\mathbf{s}}_{\alpha }\cdot {\mathbf{s}}_{(n)}=\sum {\mathbf{s}}_{\alpha^+}
\end{align*}
where $\alpha^+$ is a composition such that it can be obtained by adding an $n$-right horizontal strip to $\alpha$.
Similarly,
\begin{align*}
{\mathbf{s}}_{\alpha }\cdot {\mathbf{s}}_{(1^n)}=\sum {\mathbf{s}}_{\alpha^+}
\end{align*}
where $\alpha^+$ is a composition such that it can be obtained by adding an $n$-right vertical strip to $\alpha$.
\end{theorem}
\begin{theorem}\emph{(Left Pieri rules for noncommutative Schur functions)}\label{the:LeftPieri}
Let $\alpha $ be a composition and $n$ be a positive integer. Then
\begin{align*}
{\mathbf{s}}_{(n)} \cdot {\mathbf{s}}_{\alpha } =\sum {\mathbf{s}}_{\alpha^+}
\end{align*}
where $\alpha^+$ is a composition such that it can be obtained by adding an $n$-left horizontal strip to $\alpha$.
Similarly,
\begin{align*}
{\mathbf{s}}_{(1^n)} \cdot {\mathbf{s}}_{\alpha } =\sum {\mathbf{s}}_{\alpha^+ }
\end{align*}
where $\alpha^+$ is a composition such that it can be obtained by adding an $n$-left vertical strip to $\alpha$.\end{theorem}
Note that since quasisymmetric and noncommutative Schur functions are indexed by compositions, \emph{if any parts of size 0 arise during computation, then they are ignored}.
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\section{Generalized skew Pieri rules}\label{sec:skew}
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\subsection{Quasisymmetric skew Pieri rules}\label{subsec:QSymskewPieri} We now turn our attention to proving skew Pieri rules for skew quasisymmetric Schur functions. The statement of the rules is in the spirit of the Pieri rules for skew shapes of Assaf and McNamara \cite{assaf-mcnamara}, and this is no coincidence as we recover their rules as a special case in Corollary~\ref{cor:AM}. However first we prove a crucial proposition.
\begin{proposition}\label{ob:skewingisharpooning} Let $\alpha, \beta$ be compositions. Then
${\mathbf{s}}b \rightharpoonup {\mathcal{S}}_{\alpha} = {\mathcal{S}}_{\alpha{/\!\!/}\beta}$.
\end{proposition}
\begin{proof}
Recall Equation~\eqref{eq:coproductquasischur} states that
$$
\Delta({\mathcal{S}}_{\alpha})=\displaystyle\sum_{\gamma} {\mathcal{S}}_{\alpha{/\!\!/}\gamma}\otimes {\mathcal{S}}_{\gamma}
$$
where the sum is over all compositions $\gamma$. Thus using Equations \eqref{eq:HdualactingonH} and \eqref{eq:coproductquasischur} we obtain
\begin{eqnarray}\label{eq:harpoons}
{\mathbf{s}}b \rightharpoonup {\mathcal{S}}_{\alpha}= \displaystyle\sum_{\gamma} \langle {\mathcal{S}}_{\gamma},{\mathbf{s}}b\rangle {\mathcal{S}}_{\alpha{/\!\!/}\gamma}
\end{eqnarray}
where the sum is over all compositions $\gamma$. Since by Definition~\ref{def:NCbasis}, $\langle {\mathcal{S}}_{\gamma},{\mathbf{s}}b\rangle$ equals $1$ if $\beta= \gamma$ and $0$ otherwise, the claim follows.
\end{proof}
\begin{remark}\label{rem:terms that equal 0 for poset reasons}
The proposition above does not tell us when ${\mathbf{s}}b \rightharpoonup {\mathcal{S}}_{\alpha} = {\mathcal{S}}_{\alpha{/\!\!/}\beta}$ equals $0$. However, by the definition of $\alpha {/\!\!/} \beta$ this is precisely when $\alpha$ and $\beta$ satisfy $\beta \not< _{c} \alpha$. Consequently in the theorem below the \emph{nonzero} contribution will only be from those $\alpha^+$ and $\beta^-$ that satisfy $\beta^- < _{c} \alpha^+$. As always if any parts of size 0 arise during computation, then they are ignored.
\end{remark}
\begin{theorem}\label{the:QSskewPieri}
Let $\alpha, \beta$ be compositions and $n$ be a positive integer. Then
\begin{align*}
{\mathcal{S}}_{\alpha{/\!\!/}\beta}\cdot {\mathcal{S}}_{(n)}=\sum_{i+j=n}(-1)^j{\mathcal{S}}_{\alpha^+{/\!\!/}\beta^-}
\end{align*}
where $\alpha^+$ is a composition such that $\alpha$ can be obtained by removing an $i$-horizontal strip from it, and $\beta^{-}$ is a composition such that it can be obtained by removing a $j$-vertical strip from $\beta$.
Similarly,
\begin{align*}
{\mathcal{S}}_{\alpha{/\!\!/}\beta}\cdot {\mathcal{S}}_{(1^n)}=\sum_{i+j=n}(-1)^j{\mathcal{S}}_{\alpha^+{/\!\!/}\beta^-}
\end{align*}
where $\alpha^+$ is a composition such that $\alpha$ can be obtained by removing an $i$-vertical strip from it, and $\beta^{-}$ is a composition such that it can be obtained by removing a $j$-horizontal strip from $\beta$.
\end{theorem}
\begin{proof}
For the first part of the theorem, our aim is to calculate ${\mathcal{S}}_{\alpha{/\!\!/}\beta}\cdot {\mathcal{S}}_{(n)}$, which in light of Proposition \ref{ob:skewingisharpooning}, is the same as calculating $({\mathbf{s}}b \rightharpoonup {\mathcal{S}}_{\alpha})\cdot {\mathcal{S}}_{(n)}$.
Taking $a={\mathbf{s}}b$, $g={\mathcal{S}}_{\alpha}$ and $h={\mathcal{S}}_{(n)}$ in Lemma \ref{lem:magiclemma} gives the LHS as $({\mathbf{s}}b \rightharpoonup {\mathcal{S}}_{\alpha})\cdot {\mathcal{S}}_{(n)}$. For the RHS observe that, by Definition~\ref{def:QSbasis}, ${\mathcal{S}} _{(n)} = {F} _{(n)}$ and by Equation~\eqref{eq:Fcoproduct} we have that
\begin{eqnarray}\label{eq:coproductF}
\Delta({F}_{(n)})&=& \sum_{i+j=n}{F}_{(i)}\otimes {F}_{(j)}.
\end{eqnarray}
Substituting these in yields
\begin{eqnarray}\label{eq:firststeprhs}
\displaystyle\sum_{i+j=n}(S({F}_{(j)})\rightharpoonup {\mathbf{s}}b)\rightharpoonup({\mathcal{S}}_{\alpha}\cdot {F}_{(i)}).
\end{eqnarray}
Now, by Equation~\eqref{eq:antipode}, we have that $S({F}_{(j)})=(-1)^j{F}_{(1^j)}$.
This reduces \eqref{eq:firststeprhs} to
\begin{eqnarray}\label{eq:secondsteprhs}
\displaystyle\sum_{i+j=n}((-1)^j{F}_{(1^j)}\rightharpoonup {\mathbf{s}}b)\rightharpoonup({\mathcal{S}}_{\alpha}\cdot {F}_{(i)}).
\end{eqnarray}
We will first deal with the task of evaluating ${F}_{(1^j)}\rightharpoonup {\mathbf{s}}b$. We need to invoke Equation \eqref{eq:HactingonHdual} and thus we need $\Delta({\mathbf{s}}b)$. Assume that
\begin{eqnarray}
\Delta({\mathbf{s}}b)=\sum_{\gamma,\delta}b_{\gamma,\delta}^{\beta}{\mathbf{s}}_{\gamma}\otimes {\mathbf{s}}_{\delta}
\end{eqnarray}
where the sum is over all compositions $\gamma,\delta$.
Thus Equation~\eqref{eq:HactingonHdual} yields
\begin{eqnarray}
{F}_{(1^j)}\rightharpoonup {\mathbf{s}}b&=& \displaystyle\sum_{\gamma,\delta} b_{\gamma,\delta}^{\beta}\langle {F}_{(1^j)},{\mathbf{s}}_{\delta}\rangle{\mathbf{s}}_{\gamma}.
\end{eqnarray}
Observing that, by Definition~\ref{def:QSbasis}, ${F}_{(1^j)}={\mathcal{S}}_{(1^j)}$ and that, by Definition~\ref{def:NCbasis}, $\langle {\mathcal{S}}_{(1^j)},{\mathbf{s}}_{\delta}\rangle$ equals $1$ if $\delta = (1^j)$ and equals $0$ otherwise, we obtain
\begin{eqnarray}\label{eq:eharpooningncsstep1}
{F}_{(1^j)}\rightharpoonup {\mathbf{s}}b &=& \displaystyle\sum_{\gamma} b_{\gamma,(1^j)}^{\beta}{\mathbf{s}}_{\gamma}.
\end{eqnarray}
Since by Definition~\ref{def:NCbasis} and the duality pairing we have that $\langle {\mathcal{S}}_{\gamma}\otimes {\mathcal{S}}_{\delta},\Delta({\mathbf{s}}b)\rangle=\langle {\mathcal{S}}_{\gamma}\cdot{\mathcal{S}}_{\delta},{\mathbf{s}}b\rangle=b_{\gamma,\delta}^{\beta}$, we get that
\begin{eqnarray}
\langle {\mathcal{S}}_{\gamma}\cdot{\mathcal{S}}_{(1^j)},{\mathbf{s}}b\rangle&=& b_{\gamma,(1^j)}^{\beta}.
\end{eqnarray}
The Pieri rules for quasisymmetric Schur functions in Theorem~\ref{the:QSPieri} state that $b_{\gamma,(1^j)}^{\beta}$ is $1$ if there exists a weakly decreasing sequence $\ell_1\geq \ell_2\geq\cdots \geq \ell_j$ such that $\mathfrak{d}_{\ell_1}\cdots \mathfrak{d}_{\ell_j}(\beta)=\gamma$, and is $0$ otherwise. Thus this reduces Equation~\eqref{eq:eharpooningncsstep1} to
\begin{eqnarray}\label{eq:eharpooningncsstep2}
{F}_{(1^j)}\rightharpoonup {\mathbf{s}}b &=& \displaystyle\sum_{\substack{\mathfrak{d}_{\ell_1}\cdots \mathfrak{d}_{\ell_j}(\beta)=\gamma\\\ell_1\geq\cdots \geq \ell_j}}{\mathbf{s}}_{\gamma}.
\end{eqnarray}
Since ${\mathcal{S}}_{(i)}={F}_{(i)}$, by Definition~\ref{def:QSbasis}, the Pieri rules in Theorem~\ref{the:QSPieri} also imply that
\begin{eqnarray}\label{eq:rowpieriruleqschur}
{\mathcal{S}}_{\alpha}\cdot {F}_{(i)}&=& \displaystyle \sum_{\substack{\mathfrak{d}_{r_1}\cdots \mathfrak{d}_{r_i}(\varepsilon)=\alpha\\r_1<\cdots < r_i}}{\mathcal{S}}_{\varepsilon}.
\end{eqnarray}
Using Equations~\eqref{eq:eharpooningncsstep2} and \eqref{eq:rowpieriruleqschur} in \eqref{eq:secondsteprhs}, we get
\begin{eqnarray}
\displaystyle\sum_{i+j=n}((-1)^j{F}_{(1^j)}\rightharpoonup {\mathbf{s}}b)\rightharpoonup({\mathcal{S}}_{\alpha}\cdot {F}_{(i)})&=& \displaystyle\sum_{i+j=n}\left((-1)^j\displaystyle\sum_{\substack{\mathfrak{d}_{\ell_1}\cdots \mathfrak{d}_{\ell_j}(\beta)=\gamma\\\ell_1\geq\cdots \geq \ell_j}}{\mathbf{s}}_{\gamma}\right)\rightharpoonup \left( \sum_{\substack{\mathfrak{d}_{r_1}\cdots \mathfrak{d}_{r_i}(\varepsilon)=\alpha\\r_1<\cdots < r_i}}{\mathcal{S}}_{\varepsilon}
\right).
\nonumber\\
\end{eqnarray}
Using Proposition \ref{ob:skewingisharpooning}, we obtain that
\begin{eqnarray}\label{eq:penultimateskewpieri}
\displaystyle\sum_{i+j=n}((-1)^j{F}_{(1^j)}\rightharpoonup {\mathbf{s}}b)\rightharpoonup({\mathcal{S}}_{\alpha}\cdot {F}_{(i)})&=&\displaystyle\sum_{i+j=n}\left(\sum_{\substack{\mathfrak{d}_{\ell_1}\cdots \mathfrak{d}_{\ell_j}(\beta)=\gamma\\\ell_1\geq\cdots \geq \ell_j\\\mathfrak{d}_{r_1}\cdots \mathfrak{d}_{r_i}(\varepsilon)=\alpha\\r_1<\cdots < r_i}}(-1)^j{\mathcal{S}}_{\varepsilon{/\!\!/}\gamma}\right).
\end{eqnarray}
Thus
\begin{eqnarray}\label{eq:skewpierirule-row}
{\mathcal{S}}_{\alpha{/\!\!/}\beta}\cdot {\mathcal{S}}_{(n)}&=&\displaystyle\sum_{i+j=n}\left(\sum_{\substack{\mathfrak{d}_{\ell_1}\cdots \mathfrak{d}_{\ell_j}(\beta)=\gamma\\\ell_1\geq\cdots \geq \ell_j\\\mathfrak{d}_{r_1}\cdots \mathfrak{d}_{r_i}(\varepsilon)=\alpha\\r_1<\cdots < r_i}}(-1)^j{\mathcal{S}}_{\varepsilon{/\!\!/}\gamma}\right).
\end{eqnarray} The first part of the theorem now follows from the definitions of $i$-horizontal strip and $j$-vertical strip.
For the second part of the theorem we use the same method as the first part, but this time calculate
$$({\mathbf{s}}b \rightharpoonup {\mathcal{S}}_{\alpha})\cdot {\mathcal{S}}_{(1^n)}.$$
\end{proof}
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\begin{remark}\label{rem:noomega}
Notice that as opposed to the classical case where one can apply the $\omega$ involution to obtain the corresponding Pieri rule, we can not do this here. This is because the image of the skew quasisymmetric Schur functions under the $\omega$ involution is not yet known explicitly. Notice that the $\omega$ map applied to quasisymmetric Schur functions results in the row-strict quasisymmetric Schur functions of Mason and Remmel \cite{mason-remmel}.
\end{remark}
\begin{example}\label{ex:skewQSPierirow}
Let us compute ${\mathcal{S}}_{(1,3,2){/\!\!/} (2,1)}\cdot {\mathcal{S}}_{(2)}$. We first need to compute all compositions $\gamma$ that can be obtained by removing a vertical strip of size at most 2 from $\beta=(2,1)$. These compositions correspond to the white boxes in the diagrams below, while the boxes in the darker shade of red correspond to the vertical strips that are removed from $\beta$.
$$
\ytableausetup{smalltableaux,boxsize=0.5em}
\begin{ytableau}
*(white) & *(white)\\
*(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) & *(white)\\
*(red!80)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) & *(red!80)\\
*(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) & *(red!80)\\
*(red!80)
\end{ytableau}
$$
Next we need to compute all compositions $\varepsilon$ such that a horizontal strip of size at most $2$ can be removed from it so as to obtain $\alpha$. We list these $\varepsilon$s below with the boxes in the lighter shade of green corresponding to horizontal strips that need to be removed to obtain $\alpha$.
$$
\ytableausetup{smalltableaux,boxsize=0.5em}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(white) & *(white)\\
*(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(green!70)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(green!70)\\
*(white) & *(white) & *(white)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(white) & *(white) & *(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) &*(green!70)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) \\
*(green!70)\\
*(white) & *(white) & *(green!70)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) & *(green!70)\\
*(green!70)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(green!70)\\
*(white) & *(white) & *(white)\\
*(white) & *(white) & *(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(green!70)\\
*(white) & *(white) & *(white) & *(green!70)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) \\
*(white) & *(white) \\
*(green!70) & *(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) \\
*(white) & *(white) & *(green!70)\\
*(green!70) \\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) & *(green!70) \\
*(white) & *(white) \\
*(green!70) \\
\end{ytableau}
\hspace{5mm}
$$
$$
\ytableausetup{smalltableaux,boxsize=0.5em}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) \\
*(white) & *(white) & *(green!70) & *(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) & *(green!70) \\
*(white) & *(white) & *(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) & *(green!70) & *(green!70)\\
*(white) & *(white) \\
\end{ytableau}
$$
Now to compute ${\mathcal{S}}_{(1,3,2){/\!\!/} (2,1)}\cdot {\mathcal{S}}_{(2)}$, our result tells us that for every pair of compositions in the above lists $(\varepsilon,\gamma)$ such that (the number of green boxes in $\varepsilon$)+(the number of red boxes in $\gamma$)=2, and $\gamma < _{c} \varepsilon$ we have a term ${\mathcal{S}}_{\varepsilon {/\!\!/} \gamma}$ with a sign $(-1)^{\text{number of red boxes}}$. Hence we have the following expansion, suppressing commas and parentheses in compositions for ease of comprehension.
\begin{align*}
{\mathcal{S}}_{132{/\!\!/} 21}\cdot {\mathcal{S}}_{2}=&{\mathcal{S}}_{132{/\!\!/} 1}-{\mathcal{S}}_{1321{/\!\!/} 11}-{\mathcal{S}}_{1312{/\!\!/} 2}-{\mathcal{S}}_{1132{/\!\!/}2}-{\mathcal{S}}_{1132{/\!\!/} 11}-{\mathcal{S}}_{133{/\!\!/} 2}-{\mathcal{S}}_{133{/\!\!/} 11}\\&-{\mathcal{S}}_{142{/\!\!/} 2}-{\mathcal{S}}_{142{/\!\!/} 11}+{\mathcal{S}}_{1133{/\!\!/} 21}+{\mathcal{S}}_{1142{/\!\!/} 21}+{\mathcal{S}}_{1322{/\!\!/} 21}+{\mathcal{S}}_{1331{/\!\!/} 21}\\&+{\mathcal{S}}_{1421{/\!\!/} 21}+{\mathcal{S}}_{143{/\!\!/} 21}+{\mathcal{S}}_{152{/\!\!/} 21}
\end{align*}
\end{example}
\begin{example}\label{ex:skewQSPiericol}
Let us compute ${\mathcal{S}}_{(1,3,2){/\!\!/} (2,1)}\cdot {\mathcal{S}}_{(1,1)}$. We first need to compute all compositions $\gamma$ that can be obtained by removing a horizontal strip of size at most 2 from $\beta=(2,1)$. These compositions correspond to the white boxes in the diagrams below, while the boxes in the darker shade of red correspond to the horizontal strips that are removed from $\beta$.
$$
\ytableausetup{smalltableaux,boxsize=0.5em}
\begin{ytableau}
*(white) & *(white)\\
*(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) & *(white)\\
*(red!80)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) & *(red!80)\\
*(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) & *(red!80)\\
*(red!80)
\end{ytableau}
$$
Next we need to compute all compositions $\varepsilon$ such that a vertical strip of size at most $2$ can be removed from it so as to obtain $\alpha$. We list these $\varepsilon$s below with the boxes in the lighter shade of green corresponding to vertical strips that need to be removed to obtain $\alpha$.
$$
\ytableausetup{smalltableaux,boxsize=0.5em}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(white) & *(white)\\
*(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(green!70)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(green!70)\\
*(white) & *(white) & *(white)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(white) & *(white) & *(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) &*(green!70)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(white) & *(white) \\
*(green!70)\\
*(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(green!70)\\
*(white) & *(white) \\
*(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(green!70)\\
*(white) & *(white) & *(white)\\
*(white) & *(white) \\
*(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(green!70)\\
*(green!70)\\
*(white) & *(white) \\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(green!70)\\
*(white) & *(white) & *(white)\\
*(green!70)\\
*(white) & *(white) \\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(green!70)\\
*(green!70)\\
*(white) & *(white) & *(white)\\
*(white) & *(white) \\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) \\
*(white) & *(white) & *(green!70) \\
*(green!70) \\
\end{ytableau}
\hspace{5mm}
$$
$$
\ytableausetup{smalltableaux,boxsize=0.5em}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) \\
*(green!70)\\
*(white) & *(white) & *(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(green!70)\\
*(white) & *(white) & *(white) \\
*(white) & *(white) & *(green!70) \\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) & *(green!70)\\
*(white) & *(white) & *(white) \\
*(white) & *(white) & *(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) \\
*(white) & *(white) & *(white) & *(green!70)\\
*(white) & *(white) \\
*(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) \\
*(white) & *(white) & *(white) & *(green!70)\\
*(green!70)\\
*(white) & *(white) \\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) \\
*(green!70)\\
*(white) & *(white) & *(white) & *(green!70)\\
*(white) & *(white) \\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) \\
*(white) & *(white) & *(white) & *(green!70) \\
*(white) & *(white) & *(green!70)\\
\end{ytableau}
$$
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0.189.7
|
$$
\ytableausetup{smalltableaux,boxsize=0.5em}
\begin{ytableau}
*(white) & *(white)\\
*(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) & *(white)\\
*(red!80)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) & *(red!80)\\
*(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) & *(red!80)\\
*(red!80)
\end{ytableau}
$$
Next we need to compute all compositions $\varepsilon$ such that a vertical strip of size at most $2$ can be removed from it so as to obtain $\alpha$. We list these $\varepsilon$s below with the boxes in the lighter shade of green corresponding to vertical strips that need to be removed to obtain $\alpha$.
$$
\ytableausetup{smalltableaux,boxsize=0.5em}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(white) & *(white)\\
*(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(green!70)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(green!70)\\
*(white) & *(white) & *(white)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(white) & *(white) & *(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) &*(green!70)\\
*(white) & *(white)
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(white) & *(white) \\
*(green!70)\\
*(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(green!70)\\
*(white) & *(white) \\
*(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(green!70)\\
*(white) & *(white) & *(white)\\
*(white) & *(white) \\
*(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white)\\
*(green!70)\\
*(green!70)\\
*(white) & *(white) \\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(green!70)\\
*(white) & *(white) & *(white)\\
*(green!70)\\
*(white) & *(white) \\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(green!70)\\
*(green!70)\\
*(white) & *(white) & *(white)\\
*(white) & *(white) \\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) \\
*(white) & *(white) & *(green!70) \\
*(green!70) \\
\end{ytableau}
\hspace{5mm}
$$
$$
\ytableausetup{smalltableaux,boxsize=0.5em}
\begin{ytableau}
*(white)\\
*(white) & *(white) & *(white) \\
*(green!70)\\
*(white) & *(white) & *(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white)\\
*(green!70)\\
*(white) & *(white) & *(white) \\
*(white) & *(white) & *(green!70) \\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) & *(green!70)\\
*(white) & *(white) & *(white) \\
*(white) & *(white) & *(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) \\
*(white) & *(white) & *(white) & *(green!70)\\
*(white) & *(white) \\
*(green!70)\\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) \\
*(white) & *(white) & *(white) & *(green!70)\\
*(green!70)\\
*(white) & *(white) \\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) \\
*(green!70)\\
*(white) & *(white) & *(white) & *(green!70)\\
*(white) & *(white) \\
\end{ytableau}
\hspace{5mm}
\begin{ytableau}
*(white) \\
*(white) & *(white) & *(white) & *(green!70) \\
*(white) & *(white) & *(green!70)\\
\end{ytableau}
$$
Now to compute ${\mathcal{S}}_{(1,3,2){/\!\!/} (2,1)}\cdot {\mathcal{S}}_{(1,1)}$, our result tells us that for every pair of compositions in the above lists $(\varepsilon,\gamma)$ such (that the number of green boxes in $\varepsilon$)+(the number of red boxes in $\gamma$)=2 and $\gamma < _{c} \varepsilon$, we have a term ${\mathcal{S}}_{\varepsilon{/\!\!/} \gamma}$ with a sign $(-1)^{\text{number of red boxes}}$. Hence we have the following expansion, suppressing commas and parentheses in compositions for ease of comprehension.
\begin{align*}
{\mathcal{S}}_{132{/\!\!/} 21}\cdot {\mathcal{S}}_{11}=&{\mathcal{S}}_{132{/\!\!/} 1}-{\mathcal{S}}_{1321{/\!\!/} 11}-{\mathcal{S}}_{1312{/\!\!/} 2}-{\mathcal{S}}_{1132{/\!\!/}2}-{\mathcal{S}}_{1132{/\!\!/} 11}-{\mathcal{S}}_{133{/\!\!/} 2}-{\mathcal{S}}_{133{/\!\!/} 11}\\&-{\mathcal{S}}_{142{/\!\!/} 2}-{\mathcal{S}}_{142{/\!\!/} 11}+{\mathcal{S}}_{13121{/\!\!/} 21}+{\mathcal{S}}_{11321{/\!\!/} 21}+{\mathcal{S}}_{11132{/\!\!/} 21}+{\mathcal{S}}_{1331{/\!\!/} 21}\\&+{\mathcal{S}}_{1133{/\!\!/} 21}+{\mathcal{S}}_{233{/\!\!/} 21}+{\mathcal{S}}_{1421{/\!\!/} 21}+{\mathcal{S}}_{1142{/\!\!/} 21}+{\mathcal{S}}_{143{/\!\!/} 21}
\end{align*}
\end{example}
We now turn our attention to skew Schur functions, which we will define in the next paragraph, after we first discuss some motivation for our attention.
Skew Schur functions can be written as a sum of skew quasisymmetric Schur functions \cite[Lemma 2.23]{SSQSS}, so one might ask whether we can recover the Pieri rules for skew shapes of Assaf and McNamara by expanding a skew Schur function as a sum of skew quasisymmetric Schur functions, applying our quasisymmetric skew Pieri rules and then collecting suitable terms. However, a much simpler proof exists.
A skew Schur function $s _{\lambda/\mu}$ for partitions $\lambda, \mu$ where $\ell(\lambda)\geq\ell(\mu)$, can be defined, given an $M>\ell(\lambda)$, by \cite[Section 5.1]{BLvW}
\begin{equation}\label{eq:skewSchur}s _{\lambda/\mu}={\mathcal{S}} _{\lambda + 1^{M} {/\!\!/} \mu + 1^{M}}\end{equation}where $\lambda + 1^{M} = (\lambda _1 + 1, \ldots, \lambda_{\ell(\lambda)} +1, 1^{M-\ell(\lambda)})$, and $\mu + 1^{M} = (\mu _1 + 1, \ldots, \mu_{\ell(\mu)} +1, 1^{M -\ell(\mu)})$. It follows immediately that $s_{(n)}=s_{(n)/\emptyset}={\mathcal{S}} _{(n)}$ and $s_{(1^n)}=s_{(1^n)/\emptyset}={\mathcal{S}} _{(1^n)}$ by Equation~\eqref{eq:skewSchur}. Then as a corollary of our skew Pieri rules we recover the skew Pieri rules of Assaf and McNamara as follows.
\begin{corollary}\cite[Theorem 3.2]{assaf-mcnamara}\label{cor:AM} Let $\lambda, \mu$ be partitions where $\ell(\lambda)\geq\ell(\mu)$ and $n$ be a positive integer. Then
$$s_{\lambda/\mu}\cdot s_{(n)} = \sum _{i+j = n} (-1) ^j s _{\lambda^+/\mu ^-}$$ where $\lambda^+$ is a partition such that the boxes of $\lambda ^+$ not in $\lambda$ are $i$ boxes such that no two lie in the same column, and $\mu ^-$ is a partition such that the boxes of $\mu$ not in $\mu ^-$ are $j$ boxes such that no two lie in the same row.
Similarly,
$$s_{\lambda/\mu}\cdot s_{(1^n)} = \sum _{i+j = n} (-1) ^j s _{\lambda^+/\mu ^-}$$ where $\lambda^+$ is a partition such that the boxes of $\lambda ^+$ not in $\lambda$ are $i$ boxes such that no two lie in the same row, and $\mu ^-$ is a partition such that the boxes of $\mu$ not in $\mu ^-$ are $j$ boxes such that no two lie in the same column.
\end{corollary}
\begin{proof} Let $N>\ell(\lambda)+n+1$. Then consider the product ${\mathcal{S}} _{\lambda + 1^N {/\!\!/} \mu + 1^N} \cdot {\mathcal{S}} _{(n)}$ (respectively, ${\mathcal{S}} _{\lambda + 1^N {/\!\!/} \mu + 1^N} \cdot {\mathcal{S}} _{(1^n)}$) where $\lambda, \mu$ are partitions and $\ell(\lambda)\geq\ell(\mu)$. By the paragraph preceding the corollary, this is equivalent to what we are trying to compute.
To begin, we claim that if
$$\mathfrak{d} _1\mathfrak{d} _{r_2} \cdots \mathfrak{d} _{r_i} (\alpha ')= \lambda + 1^N$$where $1<r_2<\cdots<r_i$ (respectively, $\mathfrak{d} _{r _1} \cdots \mathfrak{d} _{r _{i-q}} \mathfrak{d} _1^q (\alpha ')= \lambda + 1^N$ where $q\geq 0$ and $r _1 \geq \cdots \geq r _{i-q} >1$) and
$$\mathfrak{d} _{\ell _1} \cdots \mathfrak{d} _{\ell _{j-p}} \mathfrak{d} _1^p (\mu+ 1^N) = \beta '$$ where $p\geq 0$ and $\ell _1 \geq \cdots \geq \ell _{j-p} >1$ (respectively, $\mathfrak{d} _1\mathfrak{d} _{\ell_2} \cdots \mathfrak{d} _{\ell_j} (\mu+ 1^N) = \beta '$ where $1<\ell_2<\cdots< \ell_j$) and
$$\mathfrak{d} _{r_2} \cdots \mathfrak{d} _{r_i} (\alpha '')= \lambda + 1^N$$(respectively, $\mathfrak{d} _{r _1} \cdots \mathfrak{d} _{r _{i-q}} \mathfrak{d} _1^{q+1}(\alpha '')= \lambda + 1^N$) and
$$\mathfrak{d} _{\ell _1} \cdots \mathfrak{d} _{\ell _{j-p}} \mathfrak{d} _1^{p+1} (\mu+ 1^N) = \beta ''$$(respectively, $\mathfrak{d} _{\ell_2} \cdots \mathfrak{d} _{\ell_j}(\mu+ 1^N) = \beta ''$) then $\beta ' < _{c} \alpha ' $ if and only if $\beta '' < _{c} \alpha ''$. This follows from three facts. Firstly $\ell(\lambda) \geq \ell(\mu)$, secondly $\alpha ' $ and $\alpha ''$ only differ by $\mathfrak{d} _1$, and thirdly $\beta '$ and $\beta ''$ only differ by $\mathfrak{d} _1$. Moreover, ${\alpha ' {/\!\!/} \beta '}= {\alpha '' {/\!\!/} \beta ''}$. Furthermore, by our skew Pieri rules in Theorem~\ref{the:QSskewPieri}, the summands ${\mathcal{S}} _{\alpha ' {/\!\!/} \beta '}$ and ${\mathcal{S}}_{\alpha '' {/\!\!/} \beta ''}$ will be of opposite sign, and thus will cancel since ${\alpha ' {/\!\!/} \beta '}= {\alpha '' {/\!\!/} \beta ''}$. Consequently, any nonzero summand appearing in the product ${\mathcal{S}} _{\lambda + 1^N {/\!\!/} \mu + 1^N} \cdot {\mathcal{S}} _{(n)}$ (respectively, ${\mathcal{S}} _{\lambda + 1^N {/\!\!/} \mu + 1^N} \cdot {\mathcal{S}} _{(1^n)}$) is such that no box can be removed from the first column of $(\lambda + 1^N)^+$ to obtain $\lambda + 1^N$, nor from the first column of $\mu+ 1^N$ to obtain $(\mu+ 1^N)^-$.
Next observe that we can obtain $\lambda + 1^N$ by removing an $i$-horizontal (respectively, $i$-vertical) strip not containing a box in the first column from $(\lambda + 1^N)^+$ if and only if $(\lambda + 1^N)^+ = \lambda ^+ + 1^N$ where $\lambda ^+$ is a partition such that the boxes of $\lambda ^+$ not in $\lambda$ are $i$ boxes such that no two lie in the same column (respectively, row).
Similarly, we can obtain $(\mu+ 1^N)^-$ by removing a $j$-vertical (respectively, $j$-horizontal) strip not containing a box in the first column from $\mu+ 1^N$ if and only if $(\mu+ 1^N)^- = \mu^-+ 1^N$ where $\mu ^-$ is a partition such that the boxes of $\mu$ not in $\mu ^-$ are $j$ boxes such that no two lie in the same row (respectively, column).
\end{proof}
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\subsection{Noncommutative skew Pieri rules}\label{subsec:NSymskewPieri} It is also natural to ask whether skew Pieri rules exist for the dual counterparts to skew quasisymmetric Schur functions and whether our methods are applicable in order to prove them. To answer this we first need to define these dual counterparts, namely skew noncommutative Schur functions.
\begin{definition}\label{def:ncsskew} Given compositions $\alpha, \beta$, the \emph{skew noncommutative Schur function} ${\mathbf{s}}_{\alpha/\beta}$ is defined implicitly via the equation
\begin{eqnarray*}
\Delta({\mathbf{s}}a)&=&\displaystyle\sum_{\beta}{\mathbf{s}}_{\alpha/\beta}\otimes {\mathbf{s}}b
\end{eqnarray*}
where the sum ranges over all compositions $\beta$.
\end{definition}
With this definition and using Equation~\eqref{eq:HactingonHdual} we can deduce that
$${\mathcal{S}}_{\beta} \rightharpoonup {\mathbf{s}}a= {\mathbf{s}}_{\alpha/\beta}$$via a proof almost identical to that of Proposition~\ref{ob:skewingisharpooning}. We know from Definition~\ref{def:QSbasis} that ${\mathcal{S}} _{(n)} = F_{(n)}$ and ${\mathcal{S}} _{(1^n)} = F_{(1^n)}$. Combined with the product for fundamental quasisymmetric functions using Definition~\ref{def:Fbasis}, Definition~\ref{def:NCbasis}, and the duality pairing, it is straightforward to deduce that for $n\geq 1$ the coproduct on ${\mathbf{s}} _{(n)}$ and ${\mathbf{s}} _{(1^n)}$ is given by
$$\Delta({\mathbf{s}} _{(n)}) = \sum _{i+j=n} {\mathbf{s}} _{(i)}\otimes {\mathbf{s}} _{(j)} \qquad\Delta({\mathbf{s}} _{(1^n)}) = \sum _{i+j=n} {\mathbf{s}} _{(1^i)}\otimes {\mathbf{s}} _{(1^j)}.$$Also the action of the antipode $S$ on ${\mathbf{s}} _{(n)}$ and ${\mathbf{s}} _{(1^n)}$ is given by
$$S({\mathbf{s}} _{(j)})= (-1)^j {\mathbf{s}} _{(1^j)} \qquad S({\mathbf{s}} _{(1^j)})= (-1)^j {\mathbf{s}} _{(j)}.$$Using all the above in conjunction with the right Pieri rules for noncommutative Schur functions in Theorem~\ref{the:RightPieri} yields our concluding theorem, whose proof is analogous to the proof of Theorem~\ref{the:QSskewPieri}, and hence is omitted. As always if any parts of size 0 arise during computation, then they are ignored.
\begin{theorem}\label{the:NCskewPieri}
Let $\alpha, \beta$ be compositions and $n$ be a positive integer. Then
\begin{align*}
{\mathbf{s}}_{\alpha/\beta}\cdot {\mathbf{s}}_{(n)}=\sum_{i+j=n}(-1)^j{\mathbf{s}}_{\alpha^+/\beta^-}
\end{align*}
where $\alpha^+$ is a composition such that it can be obtained by adding an $i$-right horizontal strip to $\alpha$, and $\beta^{-}$ is a composition such that $\beta$ can be obtained by adding a $j$-right vertical strip to it.
Similarly,
\begin{align*}
{\mathbf{s}}_{\alpha/\beta}\cdot {\mathbf{s}}_{(1^n)}=\sum_{i+j=n}(-1)^j{\mathbf{s}}_{\alpha^+/\beta^-}
\end{align*}
where $\alpha^+$ is a composition such that it can be obtained by adding an $i$-right vertical strip to $\alpha$, and $\beta^{-}$ is a composition such that $\beta$ can be obtained by adding a $j$-right horizontal strip to it.
\end{theorem}
\end{document}
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\begin{document}
\title{Fields of character values for finite special unitary groups}
\begin{abstract}
Turull has described the fields of values for characters of $SL_n(q)$ in terms of the parametrization of the characters of $GL_n(q)$. In this article, we extend these results to the case of $SU_n(q)$.\\
\\
\noindent 2010 {\em AMS Subject Classification}: 20C15, 20C33
\end{abstract}
\section{Introduction}
It is a problem of general interest to understand the fields of values of the complex characters of finite groups, as these fields often reflect important or subtle properties of the group itself. Turull \cite[Section 4]{turull01} computed the fields of character values of the finite special linear groups $SL_n(q)$ by using properties of degenerate Gelfand-Graev characters of $GL_n(q)$. In this paper, we extend these methods to compute the fields of character values for the finite special unitary groups $SU_n(q)$. In particular, we use properties of generalized Gelfand-Graev characters of $SU_n(q)$ and the full unitary group $GU_n(q)$ to get this information. Further, we frame these methods so that we obtain many results for both $SL_n(q)$ and $SU_n(q)$ simultaneously.
Turull also computes the Schur indices of the characters of $SL_n(q)$. This appears to be a much more difficult problem for $SU_n(q)$. For example, it is helpful in the $SL_n(q)$ case that the Schur index for every character of $GL_n(q)$ is 1. However, the Schur indices of the characters of $GU_n(q)$ are not all explicitly known, but are known to take values other than 1.
This paper is organized as follows. In Section \ref{sec:Chars}, we establish the necessary results from character theory that are needed for the main arguments. In Sections \ref{sec:LusztigInd} and \ref{sec:Param}, we give some tools from Deligne-Lusztig theory and the parameterization of the characters of $GL^{\epsilon}_n(q)$, respectively, and we use these to describe the characters of $SL^{\epsilon}_n(q)$ in Section \ref{sec:Restriction}. We introduce generalized Gelfand-Graev characters in Section \ref{sec:GGGR}. In Section \ref{sec:Initial}, we obtain some preliminary results on fields of character values which follow quickly from the material in Section \ref{sec:Chars}. To deal with the harder cases, we need some explicit information on unipotent elements obtained in Section \ref{sec:Unipotent}, and we apply this information to generalized Gelfand-Graev characters in Section \ref{sec:MoreGGGR}. Finally, in Section \ref{sec:Main} we prove our main results in Theorem \ref{thm:turullext1} and Corollary \ref{cor:turullext}, which give explicitly the fields of values of any character of $SU_n(q)$ and a description of the real-valued characters of $SU_n(q)$, as well as recover the corresponding results for $SL_n(q)$ originally found in \cite[Section 4]{turull01}.
\subsection*{Notation}
We will often use the notations found in \cite{turull01}, for clarity of analogous statements. For example, the natural action of a Galois automorphism $\sigma\in\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on a character $\mathrm{Char}i$ of a group will be denoted $\sigma\mathrm{Char}i$. Here for a group element $g$, the value of $\sigma\mathrm{Char}i$ is given by $\sigma\mathrm{Char}i(g)=\sigma(\mathrm{Char}i(g))$. We write $\mathbb{Q}(\mathrm{Char}i)$ for the field obtained from $\mathbb{Q}$ by adjoining all values of the character $\mathrm{Char}i$.
For an integer $n$, we will write $n=n_2n_{2'}$ where $n_2$ is a $2$-power and $n_{2'}$ is odd. Further, for an element $x$ of a finite group $Y$, we write $x=x_2x_{2'}$ where $x_2$ has $2$-power order and $x_{2'}$ has odd order. We denote by $|x|$ the order of the element $x$ (we also use this notation for cardinality and size of partitions, which will be clear from context). We write $\mathrm{Irr}(Y)$ for the set of all irreducible complex characters of the group $Y$. Given two elements $g, x$ in $Y$, we write $g^x = x^{-1}gx$, and for $\mathrm{Char}i \in \mathrm{Irr}(Y)$, we define $\mathrm{Char}i^x$ by $\mathrm{Char}i^x(g) = \mathrm{Char}i(xgx^{-1})$.
For a subgroup $X\leq Y$, we write $\mathrm{Ind}_X^Y(\varphi)$ for the character of $Y$ induced from a character $\varphi$ of $X$, and we write $\res^Y_X(\mathrm{Char}i)$ for the character of $X$ restricted from a character $\mathrm{Char}i$ of $Y$. We will further use $\mathrm{Irr}(Y|\varphi)$ and $\mathrm{Irr}(X|\mathrm{Char}i)$ to denote the set of irreducible constituents of $\mathrm{Ind}_X^Y(\varphi)$ and $\res^Y_X(\mathrm{Char}i)$, respectively.
Throughout the article, let $q$ be a power of a prime $p$ and let $G=SL^\epsilon_n(q)$ and $\wt{G}=GL^\epsilon_n(q)$, where $\epsilon\in\{\pm1\}$. Here when $\epsilon = 1$, we mean $\wt{G}=GL_n(q)$ and $G=SL_n(q)$, and when $\epsilon=-1$, we mean $\wt{G}=GU_n(q)$ and $G=SU_n(q)$. We also write $\bg{G}=SL_n(\bar{\mathbb{F}}_q)$ and $\wt{\bg{G}}=GL_n(\bar{\mathbb{F}}_q)$ for the corresponding algebraic groups, so that $\wt{G}=\wt{\bg{G}}^{F_\epsilon}$ and $G=\bg{G}^{F_\epsilon}$ for an appropriate Frobenius morphism $F_{\epsilon}$.
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\section{Characters} \label{sec:Chars}
\subsection{Lusztig Induction} \label{sec:LusztigInd}
For this section, we let $\mathbf{H}$ be any connected reductive group over $\bar{\mathbb{F}}_q$ with Frobenius map $F$, and write $H = \mathbf{H}^F$. For any $F$-stable Levi subgroup $\mathbf{L}$ of $\mathbf{H}$, contained in a parabolic subgroup $\mathbf{P}$, we write $L = \mathbf{L}^F$ and denote by $R_L^H = R_{\mathbf{L} \subset \mathbf{P}}^{\mathbf{H}}$ the Lusztig (or twisted) induction functor. When $\mathbf{P}$ may be chosen to be an $F$-stable parabolic, then $R_L^H$ becomes Harish-Chandra induction. When $\mathbf{L} = \mathbf{T}$ is chosen to be a maximal torus and $\theta$ is a character of $T = \mathbf{T}^F$, then $R_T^H(\theta)$ is the corresponding Deligne-Lusztig (virtual) character. We need the following basic result regarding actions on characters of finite reductive groups obtained through twisted induction.
\begin{lemma} \label{DLlemma}
Let $\mathbf{H}$ and $H = \mathbf{H}^F$ be as above. Let $\mathbf{L}$ be an $F$-stable Levi subgroup of $\mathbf{H}$, and write $L = \mathbf{L}^F$. Let $\mathrm{Char}i$ be a character of $L$, $\sigma \in \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$, and $\alpha$ a linear character of $H$ which is trivial on unipotent elements. Then
$$ \sigma R_{L}^H(\mathrm{Char}i) = R_L^H(\sigma \mathrm{Char}i) \quad \text{ and } \quad \alpha R_L^H(\mathrm{Char}i) = R_L^H(\alpha \mathrm{Char}i).$$
In particular, when $\mathbf{L} = \mathbf{T}$ is a maximal torus and $\mathrm{Char}i = \theta$ is a character of $T = \mathbf{T}^F$, then we have
$$ \sigma R_{T}^H(\theta) = R_T^H(\sigma \theta) \quad \text{ and } \quad \alpha R_T^H(\theta) = R_T^H(\alpha \theta).$$
\end{lemma}
\begin{proof} From \cite[Proposition 11.2]{dignemichel}, for any $g \in H$ we have
$$R_L^H(\mathrm{Char}i)(g) = \frac{1}{|L|}\sum_{l \in L} \mathrm{Tr}((g,l^{-1})|X) \mathrm{Char}i(l),$$
where $\mathrm{Tr}((g,l^{-1})|X)$ is the Lefschetz number corresponding to the $H \times L$-action on the $\ell$-adic cohomology $X$ of the relevant Deligne-Lusztig variety. In particular, these numbers are rational integers (by \cite[Corollary 10.6]{dignemichel}, for example). Thus,
$$\sigma R_L^H(\mathrm{Char}i)(g) = \frac{1}{|L|}\sum_{l \in L} \mathrm{Tr}((g,l^{-1})|X) \sigma\mathrm{Char}i(l) = R_L^H(\sigma \mathrm{Char}i).$$
Now let $g \in H$ have Jordan decomposition $g=su$, so that $s \in H$ is semisimple and $u \in H$ is unipotent, and we have $\alpha(g)=\alpha(s)$. From \cite[Proposition 12.2]{dignemichel} we have
\begin{equation} \label{DLGreen}
R_L^H(\mathrm{Char}i)(g) = \frac{1}{|L||C_{\mathbf{H}}^{\circ}(s)^F|} \sum_{\{ h \in H \, \mid \, s \in h \mathbf{L} h^{-1} \}} |C_{h \mathbf{L}h^{-1}}^{\circ} (s)^F| \sum_{ v \in C_{h \mathbf{L}h^{-1}}^{\circ} (s)^F_{\mathrm{u}}} Q_{C_{h \mathbf{L}h^{-1}}^{\circ} (s)}^{C_{\mathbf{H}}^{\circ}(s)}(u, v^{-1}) \mathrm{Char}i^h(sv),
\end{equation}
where $C_{\mathbf{H}}^{\circ}(s)$ denotes the connected component of the centralizer, and $Q_{C_{h\mathbf{L}h^{-1}}^{\circ}(s)}^{C_{\mathbf{H}}^{\circ}(s)}$ denotes the Green function. Note that for any $h \in H$ from the first sum of \eqref{DLGreen}, and any unipotent $v$ from the second sum of \eqref{DLGreen}, we have
$$\alpha(g) = \alpha(s) = \alpha(sv) = \alpha(h sv h^{-1}) = \alpha^h(sv).$$
So, if we multiply \eqref{DLGreen} by $\alpha(g)$, we may pass this factor through the sums to obtain
$$ \alpha(g) \mathrm{Char}i^h(sv) = \alpha^h(sv) \mathrm{Char}i^h(sv) = (\alpha \mathrm{Char}i)^h(sv).$$
It follows that we have $\alpha(g) R_L^H(\mathrm{Char}i)(g) = R_L^H(\alpha \mathrm{Char}i)(g)$, as claimed.
\end{proof}
\subsection{Parametrization of Characters of $GL^\epsilon_n(q)$} \label{sec:Param}
We identify $GL_1(\bar{\mathbb{F}}_q)$ with $\bar{\mathbb{F}}_q^{\times}$, and so $F_{\epsilon}$ acts on $\bar{\mathbb{F}}_q^{\times}$ via $F_{\epsilon}(a)=a^{\epsilon q}$. For any integer $k \geq 1$, we define $T_k$ to be the multiplicative subgroup of $\bar{\mathbb{F}}_q^{\times}$ fixed by $F_{\epsilon}^k$, that is
$$T_k = (\bar{\mathbb{F}}_q^{\times})^{F_{\epsilon}^k}.$$
We denote by $\wh{T}_k$ the multiplicative group of complex-valued linear characters of $T_k$. Whenever $d|k$, we have the natural norm map $\mathrm{Nm}_{k,d}=\mathrm{Nm}$ from $T_k$ to $T_d$, and the transpose map $\wh{\mathrm{Nm}}$ gives a norm map from $\wh{T}_d$ to $\wh{T}_k$, where $\wh{\mathrm{Nm}}(\xi) = \xi \circ \mathrm{Nm}$. We consider the direct limit of the character groups $\wh{T}_k$ with respect to these norm maps, $\displaystyle \lim_{\longrightarrow} \wh{T_k}$,
on which $F_{\epsilon}$ acts through its natural action on the groups $T_k$. Moreover, the fixed points of $\displaystyle \lim_{\longrightarrow} \wh{T_k}$ under $F_{\epsilon}^d$ can be identified with $\wh{T_d}$. We let $\Theta$ denote the set of $F_{\epsilon}$-orbits of $\displaystyle \lim_{\longrightarrow} \wh{T_k}$. The elements of $\Theta$ are sometimes called {\em simplices} (in \cite{Green, turull01} for example). They are naturally dual objects to polynomials with roots given by an $F_{\epsilon}$-orbit of $\bar{\mathbb{F}}_q^{\times}$.
For any orbit $\phi \in \Theta$, let $|\phi|$ denote the size of the orbit. Let $\mathcal{P}$ denote the set of all partitions of non-negative integers, where we write $|\nu| = n$ if $\nu$ is a partition of $n$, and let $\mathcal{P}_n$ denote the set of all partitions of $n$. The irreducible characters of $\wt{G}=GL^{\epsilon}_n(q)$ are parameterized by partition-valued functions on $\Theta$. Specifically, given a function $\lambda: \Theta \rightarrow \mathcal{P}$, define $|\lambda|$ by
$$|\lambda| = \sum_{\phi \in \Theta} |\phi| |\lambda(\phi)|,$$
and define $\mathcal{F}_n$ by
$$ \mathcal{F}_n = \{ \lambda: \Theta \rightarrow \mathcal{P} \, \mid \, |\lambda| = n\}.$$
Then $\mathcal{F}_n$ gives a parametrization of the irreducible complex characters of $\wt{G}$. Given $\lambda \in \mathcal{F}_n$, we let $\wt{\mathrm{Char}i}_{\lambda}$ denote the irreducible character corresponding to it.
We need several details regarding the structure of the character $\wt{\mathrm{Char}i}_{\lambda}$. In the case $\epsilon =1$, these facts all follow from the original work of Green \cite{Green}, and also appear from a slightly different point of view in the book of Macdonald \cite[Chapter IV]{MacBook}. For the case $\epsilon=-1$, the facts we need appear in \cite{TV07}, which contains relevant results from \cite{dignemichelunitary, LuszSrin}.
First consider some $\lambda \in \mathcal{F}_n$ such that $\lambda(\phi)$ is a nonempty partition for exactly one $\phi \in \Theta$, and write $\wt{\mathrm{Char}i}_{\lambda}= \wt{\mathrm{Char}i}_{\lambda(\phi)}$. Suppose that $|\phi|=d$, so that $|\lambda(\phi)| = n/d$. Then let $\omega^{\lambda(\phi)}$ be the irreducible character of the symmetric group $S_{n/d}$ parameterized by $\lambda(\phi) \in \mathcal{P}_{n/d}$. We fix this parametrization so that the partition $(1, 1, \ldots, 1)$ corresponds to the trivial character. For any $\gamma =(\gamma_1, \gamma_2, \ldots, \gamma_{\ell}) \in \mathcal{P}_{n/d}$, let $\omega^{\lambda(\phi)}(\gamma)$ denote the character $\omega^{\lambda(\phi)}$ evaluated at the conjugacy class parameterized by $\gamma$ (where $(1, 1, \ldots, 1)$ corresponds to the identity), and let $z_{\gamma}$ the size of the centralizer in $S_{n/d}$ of the class corresponding to $\gamma$. Let $T_{\gamma}$ be the torus
$$ T_{\gamma} = T_{d\gamma_1} \times T_{d \gamma_2} \times \cdots \times T_{d \gamma_{\ell}},$$
and let $\theta \in \phi$. Then we have
\begin{equation} \label{DLLinComb}
\wt{\mathrm{Char}i}_{\lambda(\phi)} = \pm \sum_{\gamma \in \mathcal{P}_{n/d}} \frac{ \omega^{\lambda(\phi)}(\gamma)}{z_{\gamma}} R_{T_{\gamma}}^{\wt{G}}(\theta),
\end{equation}
where the sign can be determined explicitly (see the Remark after \cite[Theorem 4.3]{TV07}, for example), but the sign will not have any impact for us. Note that from \eqref{DLLinComb}, it follows from our parametrization of characters of the symmetric group and \cite[Proposition 12.13]{dignemichel} that the trivial character of $\wt{G}$ corresponds to $\lambda({\bf 1})=(1, 1, \ldots, 1)$.
For an arbitrary $\lambda \in \mathcal{F}_n$, let $\phi_1, \phi_2, \ldots, \phi_r$ be precisely those elements in $\Theta$ such that $\lambda(\phi_i)$ is a nonempty partition, and let $d_i = |\phi_i|$. Let $n_i = d_i |\lambda(\phi_i)|$, and define $L$ to be the Levi subgroup $L = GL_{n_1}^{\epsilon}(q) \times \cdots \times GL_{n_r}^{\epsilon}(q)$. The character $\wt{\mathrm{Char}i}_{\lambda}$ is then given by
\begin{equation} \label{Lusztigprod}
\wt{\mathrm{Char}i}_{\lambda} = \pm R_{L}^{\wt{G}}\left(\wt{\mathrm{Char}i}_{\lambda(\phi_1)} \times \cdots \times \wt{\mathrm{Char}i}_{\lambda(\phi_r)}\right).
\end{equation}
The sign only appears in the $\epsilon = -1$ case, and again can be determined explicitly. Note that \eqref{Lusztigprod} is Harish-Chandra induction in the case $\epsilon = 1$.
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\subsection{Restriction to $SL^\epsilon_n(q)$ and Actions on the Parametrization} \label{sec:Restriction}
We now turn to the parametrization of the characters of $G = SL^{\epsilon}_n(q)$ in terms of the characters of $\wt{G}$ described in the previous section. This is done for the case $\epsilon=1$ in \cite{karkargreen,Lehrer}, and we adapt the methods there to handle the more general case of $\epsilon = \pm 1$.
Consider any Galois automorphism $\sigma \in \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. Through the natural action of $\sigma$ on the character values of any $\theta \in \wh{T}_d$, we have an action of $\sigma$ on the orbits $\phi \in \Theta$. Given any $\lambda \in \mathcal{F}_n$, we define $\sigma \lambda$ by
$$ \sigma \lambda(\phi) = \lambda(\sigma \phi).$$
For $\alpha\in\wh{T}_1$, define $\alpha\wt{\mathrm{Char}i}$ and $\alpha\theta$ by usual product of characters in $\mathrm{Irr}(\wt{G})$ and $\wh{T}_d$, where we compose $\alpha$ with determinant and the norm maps, respectively. Then $\alpha$ acts on the orbits $\phi \in \Theta$ as well, and we get an action of $\alpha$ on $\mathcal{F}_n$ by defining $\alpha\lambda$ as
$$\alpha\lambda(\phi)=\lambda(\alpha\phi).$$
We will need the following statements regarding these actions on the characters of $\wt{G}$.
\begin{lemma}\label{lem:actions}
Let $\lambda\in\mathcal{F}_n$. For any $\sigma \in \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ and any $\alpha \in \wh{T}_1$, we have
$$ \sigma \wt{\mathrm{Char}i}_{\lambda} = \wt{\mathrm{Char}i}_{\sigma \lambda} \quad \text{ and } \quad \alpha \wt{\mathrm{Char}i}_{\lambda} = \wt{\mathrm{Char}i}_{\alpha \lambda}.$$
\end{lemma}
\begin{proof} We proceed in a manner similar to the proof of \cite[Proposition of Section 3]{karkargreen}, which proves this statement in the $\epsilon=1$ case with the $\wh{T}_1$ action. We begin by considering $\lambda \in \mathcal{F}_n$ such that $\lambda(\phi)$ is a nonempty partition for precisely one $\phi \in \Theta$, and so $\wt{\mathrm{Char}i}_{\lambda}=\wt{\mathrm{Char}i}_{\lambda(\phi)}$ is given by \eqref{DLLinComb}. Given $\sigma \in \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$, since each $\omega^{\lambda(\phi)}(\gamma)$ and each $z_{\gamma}$ is a rational integer, we have
$$\sigma \wt{\mathrm{Char}i}_{\lambda(\phi)} = \pm \sum_{\gamma \in \mathcal{P}_{n/d}} \frac{ \omega^{\lambda(\phi)}(\gamma)}{z_{\gamma}} \sigma R_{T_{\gamma}}^{\wt{G}}(\theta) = \pm \sum_{\gamma \in \mathcal{P}_{n/d}} \frac{ \omega^{\lambda(\phi)}(\gamma)}{z_{\gamma}} R_{T_{\gamma}}^{\wt{G}}(\sigma\theta),$$
by Lemma \ref{DLlemma}. Since $\sigma \theta \in \sigma \phi$, then we have $\sigma \wt{\mathrm{Char}i}_{\lambda(\phi)} = \wt{\mathrm{Char}i}_{\sigma \lambda(\phi)}$. Similarly, if $\alpha \in \wh{T}_1$, we have by Lemma \ref{DLlemma}
$$\alpha \wt{\mathrm{Char}i}_{\lambda(\phi)} = \pm \sum_{\gamma \in \mathcal{P}_{n/d}} \frac{ \omega^{\lambda(\phi)}(\gamma)}{z_{\gamma}} \alpha R_{T_{\gamma}}^{\wt{G}}(\theta) = \pm \sum_{\gamma \in \mathcal{P}_{n/d}} \frac{ \omega^{\lambda(\phi)}(\gamma)}{z_{\gamma}} R_{T_{\gamma}}^{\wt{G}}(\alpha\theta),$$
and since $\alpha \theta \in \alpha \phi$, we have $\alpha \wt{\mathrm{Char}i}_{\lambda(\phi)} = \wt{\mathrm{Char}i}_{\alpha \lambda(\phi)}$.
Now consider an arbitrary $\lambda \in \mathcal{F}_n$, with $\wt{\mathrm{Char}i}_{\lambda}$ given by \eqref{Lusztigprod}. By applying Lemma \ref{DLlemma}, along with the first case just proved, we have
\begin{align*}
\sigma \wt{\mathrm{Char}i}_{\lambda} & = \pm R_L^{\wt{G}}\left(\sigma(\wt{\mathrm{Char}i}_{\lambda(\phi_1)} \times \cdots \times \wt{\mathrm{Char}i}_{\lambda(\phi_r)})\right) \\
& = \pm R_L^{\wt{G}}\left(\sigma\wt{\mathrm{Char}i}_{\lambda(\phi_1)} \times \cdots \times \sigma\wt{\mathrm{Char}i}_{\lambda(\phi_r)})\right) \\
& = \pm R_L^{\wt{G}}\left(\wt{\mathrm{Char}i}_{\sigma\lambda(\phi_1)} \times \cdots \times \wt{\mathrm{Char}i}_{\sigma\lambda(\phi_r)}\right) \\
& = \wt{\mathrm{Char}i}_{\sigma \lambda}.
\end{align*}
Similarly, if we replace $\sigma$ with $\alpha \in \wh{T}_1$, we have $\alpha \wt{\mathrm{Char}i}_{\lambda} = \wt{\mathrm{Char}i}_{\alpha \lambda}$ as claimed.
\end{proof}
Note that we may identify $\wt{G}/G$ with $T_1$, and directly from Clifford theory we know every character $\mathrm{Char}i$ of $G$ appears in some multiplicity-free restriction of a character $\wt{\mathrm{Char}i}_{\lambda}$ of $\wt{G}$. The restrictions of two different irreducible characters of $\wt{G}$ are either equal, or have no irreducible constituents of $\mathrm{Irr}(G)$ in common. With this, the next result is all that is needed to parameterize $\mathrm{Irr}(G)$.
\begin{lemma}\label{lem:irrparam}
Let $\lambda, \mu \in \mathcal{F}_n$ with $\wt{\mathrm{Char}i}_{\lambda}, \wt{\mathrm{Char}i}_{\mu} \in \mathrm{Irr}(\wt{G})$ the corresponding characters. Then
$$\mathbb{R}es^{\wt{G}}_G (\wt{\mathrm{Char}i}_{\lambda}) = \mathbb{R}es^{\wt{G}}_G (\wt{\mathrm{Char}i}_{\mu})$$
if and only if there exists some $\alpha \in \wh{T}_1$ such that $\lambda = \alpha \mu$.
\end{lemma}
\begin{proof}This follows directly from \cite[Theorem 1(i)]{karkargreen} and Lemma \ref{lem:actions}.
\end{proof}
Consider any irreducible character $\mathrm{Char}i$ of $G$, so $\mathrm{Char}i$ is a constituent of $\mathbb{R}es^{\wt{G}}_G(\wt{\mathrm{Char}i}_{\lambda})$ for some $\lambda \in \mathcal{F}_n$. That is, $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\mathrm{Char}i}_\lambda)$. The other constituents of this restriction are $\wt{G}$-conjugates of $\mathrm{Char}i$. Note that the field of values of $\mathrm{Char}i$ is invariant under conjugation by $\wt{G}$, and so in studying this field of character values it is not important which constituent we choose.
Given $\lambda \in \mathcal{F}_n$, we define the group $\mathcal{I}(\lambda)$ as
$$\mathcal{I}(\lambda)=\bigcap\{\ker\alpha \, \mid \, \alpha\in\wh{T}_1 \hbox{ such that } \alpha\lambda=\lambda\}.$$
We collect some basic properties of $\mathcal{I}(\lambda)$ in the following.
\begin{proposition}\label{prop:indstabdivides}
Let $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\mathrm{Char}i}_\lambda)$. Then:
\begin{itemize}
\item The stabilizer in $\wt{G}$ of $\mathrm{Char}i$ is the set of elements with determinant in $\mathcal{I}(\lambda)$.
\item The stabilizer of $\lambda$ in $\wh{T}_1$ is the set of elements whose kernel contains $\mathcal{I}(\lambda)$.
\item The index $[T_1: \mathcal{I}(\lambda)]$ divides $\gcd(q-\epsilon,n)$.
\end{itemize}
\end{proposition}
\begin{proof}
The proof is exactly as in \cite[Propositions 4.2 and 4.3 and Corollary 4.4]{turull01}, using Clifford theory and \prettyref{lem:actions}.
\end{proof}
\subsection{Remarks on Generalized Gelfand-Graev Characters}\label{sec:GGGR}
We recall here some subgroups described in \cite[Section 2]{Geck04} used in the construction of the characters of generalized Gelfand-Graev representations (GGGRs). We introduce only the essentials for our purposes, and refer the reader to \cite{Geck04, kawanaka1985, Taylor16}, for example, for more details.
First, let $\bg{T}\leq \bg{B}=\bg{T}\bg{U}$ be an $F_\epsilon$-stable maximal torus and Borel subgroup, respectively, of $\bg{G}$, with unipotent radical $\bg{U}$. Let $\Phi$ be the root system of $\bf{G}$ with respect to $\bg{T}$ and $\Phi^+\subset \Phi$ the set of positive roots determine by $\bg{B}$.
To each unipotent class $\mathcal{C}$ in $\bg{G}$ (or, equivalently, in $\wt{\bg{G}}$), there is associated a weighted Dynkin diagram $d\colon \Phi\rightarrow\mathbb{Z} $ and $F_\epsilon$-stable groups
\[\bg{U}_{d,i}:=\langle X_{\alpha}|\alpha\in\Phi^+, d(\alpha)\geq i\rangle \leq \bg{U},\]
where $X_\alpha$ denotes the root subgroup corresponding to $\alpha$. In particular, $\bg{P}_d:=N_{\wt{\bg{G}}}(\bg{U}_{d,1})$ is an $F_\epsilon$-stable parabolic subgroup of $\wt{\bg{G}}$ and $\bg{U}_{d,i}\lhd \bg{P}_d$ for each $i=1,2,3,\ldots$. We will further write $U_{d,i}:=\bg{U}_{d,i}^{F_\epsilon}$ and $P_d:=\bg{P}_d^{F_\epsilon}$.
Given $u\in\mathcal{C}\cap U_{d,2}$, the characters of GGGRs (which we will also refer to as GGGRs) of $\wt{G}$, respectively $G$, are constructed by inducing certain linear characters $\varphi_u\colon U_{d,2}\rightarrow\mathbb{C}^\times$ to $\wt{G}$, resp. ${G}$. In particular, the values of $\varphi_u$ are all $p$th roots of unity. Strictly speaking, the GGGRs are actually rational multiples of the induced character:
\[\wt{\Gamma}_u=[U_{d,1}:U_{d,2}]^{-1/2}\mathrm{Ind}_{U_{d,2}}^{\wt{G}}(\varphi_u)\quad\hbox{ and }\quad \Gamma_u=[U_{d,1}:U_{d,2}]^{-1/2}\mathrm{Ind}_{U_{d,2}}^{G}(\varphi_u).\]
The following is \cite[Proposition 10.11]{SFTaylorTypeA}, which is a consequence of \cite[Theorem 1.8, Lemma 2.6, and Theorem 10.10]{tiepzalesski04}.
\begin{proposition}\label{prop:GGGRvalues}
Let $\Gamma_u$ be a GGGR of $G$. Then the following hold.
\begin{enumerate}
\item If $q$ is a square, $n$ is odd, or $n/(n,q-\epsilon)$ is even, then the values of $\Gamma_u$ are integers.
\item Otherwise, the values of $\Gamma_u$ lie in $\mathbb{Q}(\sqrt{\eta p})$, where $\eta\in\{\pm1\}$ is such that $p\equiv\eta\mod 4$.
\end{enumerate}
\end{proposition}
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\section{Initial Results on Fields of Values} \label{sec:Initial}
Keep the notation from above, so that $G=SL^\epsilon_n(q)$, $\wt{G}=GL^\epsilon_n(q)$, and the characters of $\wt{G}$ are denoted by $\wt{\mathrm{Char}i}_\lambda$ for $\lambda\in\mathcal{F}_n$.
For $\lambda\in\mathcal{F}_n$, let $\mathbb{Q}(\lambda)$ denote the field obtained from $\mathbb{Q}$ by adjoining the values of the characters in the orbits $\phi\in\Theta$ such that $\lambda(\phi)$ is nonempty.
We define $\mathrm{Galg}(\lambda)$ and $\mathrm{Galr}(\lambda)$ as in \cite{turull01}. That is, $\mathrm{Galg}(\lambda)$ is the stabilizer of $\lambda$ in $\mathrm{Gal}(\mathbb{Q}(\lambda)/\mathbb{Q})$ and
\[\mathrm{Galr}(\lambda)=\{\sigma\in\mathrm{Gal}(\mathbb{Q}(\lambda)/\mathbb{Q})\, \mid \, \sigma\lambda=\alpha\lambda \hbox{ for some $\alpha\in\wh{T}_1$}\}.\]
\begin{theorem}
Let $\lambda\in\mathcal{F}_n$. Then
$\mathbb{Q}(\wt{\mathrm{Char}i}_\lambda)=\mathbb{Q}(\lambda)^{\mathrm{Galg}(\lambda)}$ and $ \mathbb{Q}(\res^{\wt{G}}_G(\wt{\mathrm{Char}i}_\lambda))=\mathbb{Q}(\lambda)^{\mathrm{Galr}(\lambda)}.$ That is, the field of values for $\wt{\mathrm{Char}i}_\lambda$ and its restriction to $G$ are the fixed fields of $\mathrm{Galg}(\lambda)$ and $\mathrm{Galr}(\lambda)$, respectively.
\end{theorem}
\begin{proof}
Given \prettyref{lem:actions}, the proof is exactly the same as that of \cite[Propositions 2.8, 3.4]{turull01}.
\end{proof}
Note that since the members of $\phi\in\Theta$ are characters of $T_d$ for some $d$, it follows that $\mathbb{Q}(\lambda)=\mathbb{Q}(\zeta_m)$ is the field obtained from $\mathbb{Q}$ by adjoining some primitive $m$th root of unity $\zeta_m$, where $\gcd(m,p)=1$.
\begin{remark}\label{rem:sigmainv}
We further remark that, as in the proof of \cite[Proposition 6.2]{turull01}, the Galois automorphism $\sigma_{-1}\colon\mathbb{Q}(\zeta_m)\rightarrow\mathbb{Q}(\zeta_m)$ satisfying $\sigma_{-1}(\zeta_m)=\zeta_m^{-1}$ induces complex conjugation on $\mathbb{Q}(\lambda)$. Hence $\wt{\mathrm{Char}i}_\lambda$, respectively $\res^{\wt{G}}_G(\wt{\mathrm{Char}i}_\lambda)$, is real-valued if and only if $\sigma_{-1}\in \mathrm{Galg}(\lambda)$, respectively $\sigma_{-1}\in\mathrm{Galr}(\lambda)$.
\end{remark}
For the remainder of the article, we let $\mathbb{F}_\lambda$ denote the field of values of $\res^{\wt{G}}_G(\wt{\mathrm{Char}i}_\lambda)$. That is, $\mathbb{F}_\lambda$ is the fixed field of $\mathrm{Galr}(\lambda)$. Since $\mathbb{F}_\lambda\subseteq\mathbb{Q}(\lambda)=\mathbb{Q}(\zeta_m)$ and $\gcd(m,p)=1$, we have $\mathbb{F}_\lambda\cap \mathbb{Q}(\zeta_p)=\mathbb{Q}$ for any primitive $p$th root of unity $\zeta_p$.
\begin{proposition}\label{prop:initialcase}
Let $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\mathrm{Char}i}_\lambda)$. Keep the notation above. Then
\begin{enumerate}
\item If $q$ is square, $n$ is odd, or $n/(n,q-\epsilon)$ is even, then $\mathbb{Q}(\mathrm{Char}i)=\mathbb{F}_\lambda$.
\item Otherwise, $\mathbb{F}_\lambda\subseteq\mathbb{Q}(\mathrm{Char}i)\subseteq \mathbb{F}_\lambda(\sqrt{\eta p})$, where $\eta\in\{\pm1\}$ is such that $p\equiv\eta\mod 4$.
\end{enumerate}
In particular, $\mathrm{Char}i$ is real-valued if and only if $\res^{\wt{G}}_G(\wt{\mathrm{Char}i}_\lambda)$ is, except possibly when $q\equiv 3\mod4$ and $2\leq n_2\leq (q-\epsilon)_2$.
\end{proposition}
\begin{proof}
Write $\mathbb{F}:=\mathbb{F}_\lambda$ and $\wt{\mathrm{Char}i}:=\wt{\mathrm{Char}i}_\lambda$. First, we remark that certainly $\mathbb{F}\subseteq \mathbb{Q}(\mathrm{Char}i)$, by its definition, since $\mathbb{R}es_G^{\widetilde{G}}(\widetilde{\mathrm{Char}i})$ is the sum of $\wt{G}$-conjugates of $\mathrm{Char}i$.
Let $\wt{\Gamma}$ be a GGGR of $\wt{G}$ such that such that $\langle \widetilde{\Gamma}, \widetilde{\mathrm{Char}i}\rangle_{\widetilde{G}} = 1$, which exists by a well-known result of Kawanaka (see \cite[3.2.18]{kawanaka1985} or \cite[15.7]{Taylor16}). Further, there exists a GGGR, $\Gamma$, of $G$ such that $\widetilde{\Gamma} = \mathrm{Ind}_G^{\widetilde{G}}(\Gamma)$. Then Frobenius reciprocity yields that there is a unique irreducible constituent $\mathrm{Char}i_0\in\mathrm{Irr}(G|\wt{\mathrm{Char}i})$ satisfying $\langle \Gamma, \mathrm{Char}i_0 \rangle_G = 1$. Without loss, we may assume $\mathrm{Char}i$ is this $\mathrm{Char}i_0$, as the field of values is invariant under $\wt{G}$-conjugation.
Write $\mathbb{K}=\mathbb{F}(\sqrt{\eta p})$. Let $\sigma\in\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{F})$ in case (1), and let $\sigma\in\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{K})$ in case (2). Then by \prettyref{prop:GGGRvalues}, $\sigma\mathrm{Char}i$ is also a constituent of $\Gamma = \sigma\Gamma$ occurring with multiplicity 1. However, as $\mathbb{R}es_G^{\wt{G}}(\wt{\mathrm{Char}i})$ is invariant under $\sigma$, we have $\sigma\mathrm{Char}i$ is also a constituent of the restriction $\mathbb{R}es_G^{\wt{G}}(\wt{\mathrm{Char}i})$. Hence we see that $\sigma\mathrm{Char}i=\mathrm{Char}i$, by uniqueness, and hence $\mathbb{Q}(\mathrm{Char}i)\subseteq \mathbb{F}$ in case (1) and $\mathbb{Q}(\mathrm{Char}i)\subseteq \mathbb{K}$ in case (2).
\end{proof}
In our main results below, characters of $T_1$ of $2$-power order will play an important role. In particular, we denote by $\mathrm{sgn}$ the unique member of $\wh{T}_1$ of order $2$.
\begin{lemma}\label{lem:orbiteven}
Let $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\mathrm{Char}i}_\lambda)$ and write $I:=\mathrm{stab}_{\wt{G}}(\mathrm{Char}i)$. Then $[\wt{G}:I]$ is even if and only if $\mathrm{sgn}\lambda=\lambda$.
\end{lemma}
\begin{proof}
Note that $2$ divides $[\wt{G}:I]$ if and only if $[I:G]_2\leq \frac{1}{2}(q-\epsilon)_2$, if and only if $\mathcal{I}(\lambda)$ is contained in the unique subgroup of $\wt{G}/G$ of order $\frac{1}{2}(q-\epsilon)$. But notice that this is exactly the kernel of $\mathrm{sgn}$ as an element of $\wh{T_1}$.
\end{proof}
\begin{lemma}\label{lem:sgn}
Let $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\mathrm{Char}i}_\lambda)$. If $\mathrm{sgn}\lambda\neq\lambda$, then $\mathbb{F}_\lambda=\mathbb{Q}(\mathrm{Char}i)$.
\end{lemma}
\begin{proof}
Write $\mathbb{F}:=\mathbb{F}_\lambda$ and recall that $\mathbb{F}\subseteq\mathbb{Q}(\mathrm{Char}i)\subseteq \mathbb{F}(\sqrt{\eta p})$. Let $\mathbb{K}=\mathbb{F}(\sqrt{\eta p})$, so that $\mathbb{K}$ is a quadratic extension of $\mathbb{F}$. Let $\tau$ be the generator of $\mathrm{Gal}(\mathbb{K}/\mathbb{F})$ and write $I$ for the stabilizer of $\mathrm{Char}i$ under $\wt{G}$. Then note that $\tau^2$ necessarily fixes $\mathrm{Char}i$, and by definition $\tau$ fixes $\mathbb{R}es^{\wt{G}}_G(\wt{\mathrm{Char}i})$, which by Clifford theory is the sum of the $[\wt{G}:I]$ conjugates of $\mathrm{Char}i$ under the action of $\wt{G}$.
We prove the contrapositive. Suppose $\mathbb{F}\neq\mathbb{Q}(\mathrm{Char}i)$, so that $\tau$ does not fix $\mathrm{Char}i$. Then since the field of values is invariant under $\wt{G}$-conjugation, it follows that the orbit of $\mathrm{Char}i$ under $\wt{G}$ can be partitioned into pairs conjugate to $\{\mathrm{Char}i, \tau\mathrm{Char}i\}$. Hence the size of the orbit, $[\wt{G}:I]$, must be even, so $\mathrm{sgn}\lambda=\lambda$ by \prettyref{lem:orbiteven}.
\end{proof}
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In our main results below, characters of $T_1$ of $2$-power order will play an important role. In particular, we denote by $\mathrm{sgn}$ the unique member of $\wh{T}_1$ of order $2$.
\begin{lemma}\label{lem:orbiteven}
Let $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\mathrm{Char}i}_\lambda)$ and write $I:=\mathrm{stab}_{\wt{G}}(\mathrm{Char}i)$. Then $[\wt{G}:I]$ is even if and only if $\mathrm{sgn}\lambda=\lambda$.
\end{lemma}
\begin{proof}
Note that $2$ divides $[\wt{G}:I]$ if and only if $[I:G]_2\leq \frac{1}{2}(q-\epsilon)_2$, if and only if $\mathcal{I}(\lambda)$ is contained in the unique subgroup of $\wt{G}/G$ of order $\frac{1}{2}(q-\epsilon)$. But notice that this is exactly the kernel of $\mathrm{sgn}$ as an element of $\wh{T_1}$.
\end{proof}
\begin{lemma}\label{lem:sgn}
Let $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\mathrm{Char}i}_\lambda)$. If $\mathrm{sgn}\lambda\neq\lambda$, then $\mathbb{F}_\lambda=\mathbb{Q}(\mathrm{Char}i)$.
\end{lemma}
\begin{proof}
Write $\mathbb{F}:=\mathbb{F}_\lambda$ and recall that $\mathbb{F}\subseteq\mathbb{Q}(\mathrm{Char}i)\subseteq \mathbb{F}(\sqrt{\eta p})$. Let $\mathbb{K}=\mathbb{F}(\sqrt{\eta p})$, so that $\mathbb{K}$ is a quadratic extension of $\mathbb{F}$. Let $\tau$ be the generator of $\mathrm{Gal}(\mathbb{K}/\mathbb{F})$ and write $I$ for the stabilizer of $\mathrm{Char}i$ under $\wt{G}$. Then note that $\tau^2$ necessarily fixes $\mathrm{Char}i$, and by definition $\tau$ fixes $\mathbb{R}es^{\wt{G}}_G(\wt{\mathrm{Char}i})$, which by Clifford theory is the sum of the $[\wt{G}:I]$ conjugates of $\mathrm{Char}i$ under the action of $\wt{G}$.
We prove the contrapositive. Suppose $\mathbb{F}\neq\mathbb{Q}(\mathrm{Char}i)$, so that $\tau$ does not fix $\mathrm{Char}i$. Then since the field of values is invariant under $\wt{G}$-conjugation, it follows that the orbit of $\mathrm{Char}i$ under $\wt{G}$ can be partitioned into pairs conjugate to $\{\mathrm{Char}i, \tau\mathrm{Char}i\}$. Hence the size of the orbit, $[\wt{G}:I]$, must be even, so $\mathrm{sgn}\lambda=\lambda$ by \prettyref{lem:orbiteven}.
\end{proof}
\section{Unipotent Elements} \label{sec:Unipotent}
To deal with the remaining cases (in particular, when
$q\equiv \eta\pmod 4$ is nonsquare, $\eta\in\{\pm1\}$, $\epsilon=-1$, and $2\leq n_2\leq (q+1)_2$), we will continue to employ the use of GGGRs. For this, we will need to analyze certain aspects of conjugacy of unipotent elements. Here the authors' observations in \cite{SFVinroot} on this subject will be useful.
In particular, if a unipotent element of $\wt{G}$ has $m_k$ Jordan blocks of size $k$ (that is, $m_k$ elementary divisors of the form $(t-1)^k$), then we may find a conjugate in $\wt{G}$ of the form $\bigoplus_k \wt{J}_k^{m_k}$, where the sum is over only those $k$ such that $m_k \neq 0$ and each $\wt{J}_k\in GL^\epsilon_k(q)$. The following lemma, which is \cite[Lemma 3.2]{SFVinroot}, will be useful throughout the section.
\begin{lemma}\label{lem:SFV3.2}
Let $u$ be a unipotent element in $\wt{G}$ with $m_k$ Jordan blocks of size $k$ for each $1\leq k\leq n$. For each $k$ such that $m_k\neq 0$, let $\delta_k\in T_1$ be arbitrary. Then there exists some $g \in C_{\wt{G}}(u)$ such that $\mathrm{det}(g) = \prod_k \delta_k^k$.
\end{lemma}
Let $\zeta_p$ be a primitive $p$th root of unity in $\mathbb{C}$. In what follows, we let $b$ be a fixed integer such that $\mathrm{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$ is generated by the map $\tau\colon \zeta_p\mapsto \zeta_p^b$. Note that $(b,p)=1$ and $b$ has multiplicative order $p-1$ modulo $p$. Further, note that $\tau$ also induces the map $\sqrt{\eta p}\mapsto-\sqrt{\eta p}$ generating $\mathrm{Gal}(\mathbb{Q}(\sqrt{\eta p})/\mathbb{Q})$. Let $\bar{b}$ denote the image of $b$ under a fixed isomorphism $(\mathbb{Z}/p\mathbb{Z})^\times \rightarrow \mathbb{F}_p^\times$, so that $\bar{b}$ generates $\mathbb{F}_{p}^\times$.
Note that by \cite[Theorem 1.9]{tiepzalesski04}, every unipotent element $u$ of $GU_n(q)$ is conjugate to $u^b$ in $C_{\widetilde{G}}(s)$, where $s$ is a semisimple element in $C_{\wt{G}}(u)$. We are interested in making precise statements about such a conjugating element.
To begin, let $u$ be a regular unipotent element of $GU_n(q)$, identified as in \cite[Lemma 5.1]{SFVinroot}. Arguing as there, we see that an element conjugating $u$ to $u^b$ must have diagonal \[(\bar{b}^{n-1}\beta, \bar{b}^{n-2}\beta,...,\bar{b}\beta,\beta),\] where $\beta\in\mathbb{F}_{q^2}^\times$ and $\bar{b}^{n-1}\beta^{q+1}=1$.
Note that the determinant of such an element is $\bar{b}^{\binom{n-1}{2}}\beta^n$ and that the condition that $\bar{b}^{n-1}\beta^{q+1}=1$ yields that $\beta^{q+1}$ is a $(p-1)$-root of unity.
\begin{lemma}\label{lem:regunip}\label{lem:beta2}
Let $q\equiv \eta\mod 4$ be nonsquare with $\eta\in\{\pm1\}$ and let
$u$ be a regular unipotent element of $GU_n(q)$. Keep the notation above. Then
\begin{enumerate}[label=(\alph*)]
\item If $n$ is even, then $\beta_2$ is a primitive $(q^2-1)_2$-root of unity in $\mathbb{F}_{q^2}^\times$.
\item There is an element $x$ in $GU_n(q)$ such that $u^x=u^b$ and $|\det(x)|$ is a $2$-power.
\item If $n\not\equiv 0 \pmod 4$, there is an element $x$ in $GU_n(q)$ such that $u^x=u^b$ and $|\det(x)| = (q+1)_2$.
\end{enumerate}
\end{lemma}
\begin{proof}
For part (a) note that $|\bar{b}^{n-1}|$ has the same $2$-part as $|\bar{b}|$ since $n$ is even, so $|\beta_2^{q+1}|=(p-1)_2=(q-1)_2$ since $q$ is nonsquare. Hence the multiplicative order of $\beta_2$ is $2(q-\eta)_2=(q^2-1)_2$.
To prove the rest, we begin by showing that if $n\not\equiv 0\pmod4$, then we may find an $x$ in $GU_n(q)$ such that $u^x=u^b$ and $|\det(x)|_2=(q+1)_2$. If $n\equiv 2\pmod 4$, then $\beta_2$ is a primitive $(q^2-1)_2$-root of unity by (a), and hence $\beta_2^n$ is a $(q-\eta)_2$ root of unity. If $\eta=-1$, then since $|\bar{b}|_2=2,$ we see $\bar{b}^{n-1\mathrm{Char}oose 2}$ has odd order. Then any $x\in GU_n(q)$ satisfying $u^x=u^b$ must satisfy $|\det(x)|_2 = (q+1)_2$ in this case. If $\eta=1$, note that $(q+1)_2=2$, so we must just show that $\det(x)$ has even order. Here $|\bar{b}|_2=(q-1)_2,$ and $\bar{b_2}^{\binom{n-1}{2}}$ has order strictly smaller than $\bar{b}_2$. Hence $\bar{b}^{\binom{n-1}{2}}\cdot\beta^n$ has even order, so $|\det(x)|_2 = (q+1)_2$ again in this case.
Now assume $n$ is odd and let $\wt{x}$ be an element in $GU_n(q)$ satisfying $u^{\wt{x}}=u^b$. Then certainly $\det(\wt{x})\in {T}_1$, so we may use \prettyref{lem:SFV3.2} to replace $\wt{x}$ with some $x\in GU_n(q)$ satisfying $\det(x)=\det(\wt{x})\cdot \delta^n$ for any $\delta\in {T}_1$. In particular, note that $|\delta^n|=|\delta|$ for any $(q+1)_2$-root of unity $\delta$, since $n$ is odd. Then we may choose $\delta$ so that $\det(\wt{x})_2\delta^n$ is a primitive $(q+1)_2$-root of unity, yielding $|\det(x)|_2 = (q+1)_2$.
It remains to show that in all cases, $x$ can be chosen such that $|\det(x)|_{2'}=1$. Since $\beta^{q+1}$ is a $(p-1)$-root of unity, we may decompose the determinant of $x$ into $\beta_2^n\cdot \beta_{(q+1)_{2'}}^n\cdot y$, where $y$ is a $(p-1)$-root of unity. However, we also know that the determinant is a $(q+1)$-root and an odd prime cannot divide both $p-1$ and $q+1$. Hence $y$ must be a $2$-power root of unity, and we may replace $x$ with an element of determinant $\beta_2^n\cdot y$, using \prettyref{lem:SFV3.2}.
\end{proof}
We remark that arguing similarly, we see that in fact there is no $x$ satisfying the conclusion of \prettyref{lem:regunip}(c) if $u$ is a regular unipotent element when $4$ divides $n$. We can, however, generalize to the following statement about more general unipotent elements when $4\not|n$.
\begin{corollary}\label{cor:unipconj}
Let $q\equiv \eta\mod 4$ be nonsquare with $\eta\in\{\pm1\}$. If $u$ is a unipotent element of $GU_n(q)$ satisfying at least one of the following:
\begin{enumerate}
\item $u$ has an odd number of elementary divisors of the form $(t-1)^k$ with $k\equiv 2\mod 4$;
\item $u$ has an elementary divisor of the form $(t-1)^k$ with $k$ odd,
\end{enumerate}
then $u$ is conjugate to $u^b$ by an element $x$ satisfying $|\det(x)| = (q+1)_2$.
In particular, if $n$ is not divisible by $4$, any unipotent element is conjugate to $u^b$ by an element $x$ satisfying $|\det(x)| = (q+1)_2$.
\end{corollary}
\begin{proof}
Indeed, viewing $u$ as $\bigoplus_k \wt{J}_k^{m_k}$ as in \cite[Section 3.2]{SFVinroot}, we may find elements $x_k$ for each $1\leq k\leq n$ as in \prettyref{lem:regunip} conjugating each $\tilde{J}_k$ to $\tilde{J}_k^b$. In case (1), we see that the product $\bigoplus_k{x_k}^{m_k}$ will satisfy the statement, after possibly again using \prettyref{lem:SFV3.2} to replace $x_k$ for any odd $k$ with an element satisfying $|\det(x_k)|=1$.
If (2) holds, but (1) does not hold, $y=\bigoplus_{2|k} {x_k}^{m_k}$ will satisfy $|\det(y)|=|\det(y)|_2 < (q+1)_2$. We may use \prettyref{lem:regunip} to obtain $x_k$ for some $k$ odd such that $|\det(x_k)| = (q+1)_2$, and replace the remaining $x_k$ for odd $k$ with an element satisfying $|\det(x_k)|=1$. The resulting $\bigoplus_k{x_k}^{m_k}$ will satisfy the statement.
The last statement follows, since if $n$ is odd, we must be in case (2), and if $n\equiv2\pmod4$, we must be in case (1) or (2).
\end{proof}
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We remark that arguing similarly, we see that in fact there is no $x$ satisfying the conclusion of \prettyref{lem:regunip}(c) if $u$ is a regular unipotent element when $4$ divides $n$. We can, however, generalize to the following statement about more general unipotent elements when $4\not|n$.
\begin{corollary}\label{cor:unipconj}
Let $q\equiv \eta\mod 4$ be nonsquare with $\eta\in\{\pm1\}$. If $u$ is a unipotent element of $GU_n(q)$ satisfying at least one of the following:
\begin{enumerate}
\item $u$ has an odd number of elementary divisors of the form $(t-1)^k$ with $k\equiv 2\mod 4$;
\item $u$ has an elementary divisor of the form $(t-1)^k$ with $k$ odd,
\end{enumerate}
then $u$ is conjugate to $u^b$ by an element $x$ satisfying $|\det(x)| = (q+1)_2$.
In particular, if $n$ is not divisible by $4$, any unipotent element is conjugate to $u^b$ by an element $x$ satisfying $|\det(x)| = (q+1)_2$.
\end{corollary}
\begin{proof}
Indeed, viewing $u$ as $\bigoplus_k \wt{J}_k^{m_k}$ as in \cite[Section 3.2]{SFVinroot}, we may find elements $x_k$ for each $1\leq k\leq n$ as in \prettyref{lem:regunip} conjugating each $\tilde{J}_k$ to $\tilde{J}_k^b$. In case (1), we see that the product $\bigoplus_k{x_k}^{m_k}$ will satisfy the statement, after possibly again using \prettyref{lem:SFV3.2} to replace $x_k$ for any odd $k$ with an element satisfying $|\det(x_k)|=1$.
If (2) holds, but (1) does not hold, $y=\bigoplus_{2|k} {x_k}^{m_k}$ will satisfy $|\det(y)|=|\det(y)|_2 < (q+1)_2$. We may use \prettyref{lem:regunip} to obtain $x_k$ for some $k$ odd such that $|\det(x_k)| = (q+1)_2$, and replace the remaining $x_k$ for odd $k$ with an element satisfying $|\det(x_k)|=1$. The resulting $\bigoplus_k{x_k}^{m_k}$ will satisfy the statement.
The last statement follows, since if $n$ is odd, we must be in case (2), and if $n\equiv2\pmod4$, we must be in case (1) or (2).
\end{proof}
\begin{remark}\label{rem:condsnmod4}
We remark that at least one of conditions 1 and 2 of \prettyref{cor:unipconj} must occur if $n\equiv 2\pmod 4$, and that condition 1 implies condition 2 if $n\equiv 0\mod 4$. Further, when $\eta=1$, the condition $n_2\leq (q+1)_2$ induced from \prettyref{prop:initialcase} yields that $n\equiv2\pmod 4$.
\end{remark}
We now address the case that $4$ divides $n$, $q\equiv 3\pmod 4$, and that neither of the conditions in \prettyref{cor:unipconj} occur.
\begin{lemma}\label{lem:unipconj0mod4}
Let $q\equiv 3\mod 4$ and let $n\equiv 0\pmod 4$ such that $n_2\leq (q+1)_2$. Let $u$ be a unipotent element of $GU_n(q)$ with no elementary divisors $(t-1)^k$ with $k$ odd. Then $u$ is conjugate to $u^b$ by an element $x$ satisfying $|\det(x)| = \frac{(q^2-1)_2}{n_2}$. \end{lemma}
\begin{proof}
As in the proof of \prettyref{cor:unipconj}, let $\wt{x}=\bigoplus_k{x_k}^{m_k}$, where for each $k$ such that $m_k\neq 0$, $x_k$ is an element of $GU_k(q)$ conjugating $\tilde{J}_k$ to $\tilde{J}_k^b$ as in \prettyref{lem:regunip}. Now, each $x_k$ has determinant $\pm{(\beta_k)_2}^k $, where ${(\beta_k)_2}$ is a $(q^2-1)_2$-root of unity in $\mathbb{F}_{q^2}^\times$, by \prettyref{lem:beta2}, since the $y$ found there has multiplicative order $(p-1)_2=2$. Then taking $\delta_k\in\mathbb{F}_{q^2}^\times$ to be the primitive $(q+1)_2$ root of unity $\delta_k=(\beta_k)_2^2$, we may use
\prettyref{lem:SFV3.2} to replace $x_k$ with an element whose determinant is $\pm(\beta_k)_2^k\delta_k^{rk}=\pm(\beta_k)_2^{k(2r+1)}$ for any odd $r$, yielding that we may replace each $x_k$ with an element whose determinant is $\pm\beta_2^k$ for a fixed $(q^2-1)_2$-root of unity $\beta_2$. Hence the resulting $x$ satisfies $\det(x)=\pm\beta_2^n$, which has the stated order.
\end{proof}
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\section{Application to GGGRs} \label{sec:MoreGGGR}
Here we keep the notation of \prettyref{sec:GGGR} and return to the more general case that $\wt{G}=GL_n^\epsilon(q)$ and $G=SL_n^\epsilon(q)$ for $\epsilon\in\{\pm1\}$.
\begin{lemma}\label{lem:varphiuconj}
Let $u\in \mathcal{C}\cap U_{d,2}$ and suppose that $x$ is an element normalizing $U_{d,2}$ and conjugating $u$ to $u^b$. Then $\varphi_u^x=\varphi_u^b$.
\end{lemma}
\begin{proof}
This follows from the construction of $\varphi_u$ in \cite[Section 5]{Taylor16} or \cite[Section 2]{Geck04}. Indeed, for each $g$ in $U_{d,2}$, we have $\varphi_u^x(g)=\varphi_u(xgx^{-1})=\varphi_{x^{-1}ux}(g)=\varphi_{u^x}(g)=\varphi_{u^b}(g)=\varphi_u(g)^b$, where the second equality is noted in \cite[Remark 2.2]{Geck04}.
\end{proof}
\begin{lemma}\label{lem:conjinP}
Let $u\in \mathcal{C}\cap U_{d,2}$ and $\epsilon=-1$. Then the elements $x$ found in \prettyref{cor:unipconj} and \prettyref{lem:unipconj0mod4} are members of $P_{d}$, and hence normalize $U_{d,2}$.
\end{lemma}
\begin{proof}
First, note that $C_{\wt{\bg{G}}}(u)\leq \bg{P}_{d}$. Indeed, this is noted in \cite[Theorem 2.1.1]{Kawanaka86} for simply connected groups, and here we have $\wt{\bg{G}}=\bg{G}Z(\wt{\bg{G}})$. Further, $u$ is conjugate to $u^b$ in $\bg{P}_{d}$ by \cite[Lemma 4.6]{SFTaylorTypeA}. So $u^x=u^b=u^y$ for some $y\in \bg{P}_{d}$, which yields that $xy^{-1}\in C_{\wt{\bg{G}}}(u)$, and hence $x\in \wt{G}\cap \bg{P}_{d}$. This shows that $x$ is contained in $P_{d}$, which contains $U_{d,2}$ as a normal subgroup.
\end{proof}
\begin{lemma}\label{lem:GGGRarg}
Let $G=SL^\epsilon_n(q)$ and $\wt{G}=GL^\epsilon_n(q)$, with $\epsilon\in\{\pm1\}$. Let $\wt{\mathrm{Char}i}:=\wt{\mathrm{Char}i}_\lambda\in\mathrm{Irr}(\wt{G})$ and let $\wt{\Gamma}_u=[U_{1,d}:U_{2,d}]^{-1/2}\mathrm{Ind}_{U_{d,2}}^{\wt{G}}(\varphi_u)$ be a Generalized Gelfand-Graev character of $\wt{G}$ such that $\langle \wt{\Gamma}_u, \wt{\mathrm{Char}i}\rangle_{\wt{G}}=1$. Let $\sigma\in\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{F}_\lambda)$ and let $x\in\wt{G}$ normalizing $U_{d,2}$ such that $\sigma\varphi_u=\varphi_u^x$. Then for $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\mathrm{Char}i})$, there is some conjugate $\mathrm{Char}i_0$ of $\mathrm{Char}i$ such that $\sigma\mathrm{Char}i_0=\mathrm{Char}i_0^x$.
\end{lemma}
\begin{proof}
Let $\Gamma_u$ be such that $\wt{\Gamma}_u=\mathrm{Ind}_G^{\wt{G}}\Gamma_u$ and $\Gamma_u=r\cdot\mathrm{Ind}_{U_{d,2}}^{{G}}(\varphi_u)$ where $r=[U_{1,d}:U_{2,d}]^{-1/2}$.
Then by Clifford theory and Frobenius reciprocity, there is a unique conjugate, $\mathrm{Char}i_0$, of $\mathrm{Char}i$ such that $\mathrm{Char}i_0\in\mathrm{Irr}(G|\wt{\mathrm{Char}i})$ and $\langle \Gamma_u, \mathrm{Char}i_0\rangle_G=1$.
Since $\res_G^{\wt{G}}(\wt{\mathrm{Char}i})$ is fixed by $\sigma$, we also see $\sigma\mathrm{Char}i_0$ is the unique member of $\mathrm{Irr}({G}|\wt{\mathrm{Char}i})$ satisfying $\langle \sigma\Gamma_u, \sigma\mathrm{Char}i_0\rangle_G=1.$ But note that \[\sigma\Gamma_u=r\cdot\mathrm{Ind}_{U_{d,2}}^{G}(\sigma\varphi_u)=r\cdot\mathrm{Ind}_{U_{d,2}}^{G}(\varphi_u^x)=\Gamma_u^x.\]
Then
$\langle \sigma\Gamma_u, \mathrm{Char}i_0^x\rangle_{G}=\langle \Gamma_u^x, \mathrm{Char}i_0^x\rangle_G=1,$ forcing $\mathrm{Char}i_0^x=\sigma\mathrm{Char}i_0$ by uniqueness, since $\mathrm{Char}i_0^x\in\mathrm{Irr}(G|\wt{\mathrm{Char}i})$.
\end{proof}
\section{Main Results} \label{sec:Main}
We begin by stating our main results. The first is an extension of \cite[Theorem 4.8]{turull01} to the case of $SU_n(q)$, describing the field of values $\mathbb{Q}(\mathrm{Char}i)$ for each $\mathrm{Char}i\in\mathrm{Irr}(G)$.
\begin{theorem}\label{thm:turullext1}
Let $G=SL^\epsilon_n(q)$ and $\wt{G}=GL^\epsilon_n(q)$, with $\epsilon\in\{\pm1\}$. Let $\lambda\in \mathcal{F}_n$ and let $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\mathrm{Char}i}_\lambda)$. Then $\mathbb{Q}(\mathrm{Char}i)=\mathbb{F}_\lambda$ unless all of the following hold:
\begin{itemize}
\item $p$ is odd,
\item $q$ is not square,
\item $2\leq n_2\leq (q-\epsilon)_2$, and
\item $\alpha\lambda=\lambda$ for any element $\alpha\in\widehat{T}_1$ of order $n_2$.
\end{itemize} In the latter case, $\mathbb{Q}(\mathrm{Char}i)=\mathbb{F}_\lambda(\sqrt{\eta p})$, where $\eta\in\{\pm1\}$ and $p\equiv\eta\pmod4$.
\end{theorem}
Taking into consideration \prettyref{rem:sigmainv}, \prettyref{thm:turullext1} immediately yields the following extension of \cite[Proposition 6.2]{turull01}.
\begin{corollary}\label{cor:turullext}
Let $G=SL^\epsilon_n(q)$ and $\wt{G}=GL^\epsilon_n(q)$, with $\epsilon\in\{\pm1\}$. Let $\lambda\in \mathcal{F}_n$ and let $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\mathrm{Char}i}_\lambda)$. Then the following are equivalent:
\begin{itemize}
\item $\mathrm{Char}i$ is real-valued.
\item There exists some $\alpha' \in\widehat{T}_1$
such that $\sigma_{-1}\lambda=\alpha'\lambda$, and if $p$ is odd, $q$ is not a square, $2\leq n_2\leq (q-\epsilon)_2$, and $\alpha\lambda=\lambda$ for any element $\alpha\in\widehat{T}_1$
of order $n_2$, then $p\equiv1\pmod4$.
\end{itemize}
\end{corollary}
The remainder of this section will be devoted to proving \prettyref{thm:turullext1}. We begin with an observation restricting the situation of \prettyref{cor:unipconj}.
\begin{proposition}\label{prop:oddelemoddindex}
Let $G=SL^\epsilon_n(q)$ and $\wt{G}=GL^\epsilon_n(q)$, with $\epsilon\in\{\pm1\}$. Let $\wt{\mathrm{Char}i}\in\mathrm{Irr}(\wt{G})$ and let $\wt{\Gamma}_u$ be a GGGR of $\wt{G}$ such that $\langle \wt{\Gamma}_u, \wt{\mathrm{Char}i}\rangle_{\wt{G}}=1$. Further, assume that $u$ has an elementary divisor of the form $(t-1)^k$ with $k$ odd. Then $[\wt{G}:I]$ is odd, where $I=\mathrm{stab}_{\wt{G}}(\mathrm{Char}i)$ for any $\mathrm{Char}i\in\mathrm{Irr}({G}|\wt{\mathrm{Char}i})$.
In particular, in this case, $\mathbb{F}_\lambda=\mathbb{Q}(\mathrm{Char}i)$ by \prettyref{lem:sgn}.
\end{proposition}
\begin{proof}
Write $\wt{\Gamma}_u=[U_{1,d}:U_{2,d}]^{-1/2}\mathrm{Ind}_{U_{d,2}}^{\wt{G}}(\varphi_u)$. By \prettyref{lem:SFV3.2}, there is some $x\in C_{\wt{G}}(u)$ with determinant $\delta^k$, where $\delta$ is a $(q+1)_2$-root of unity in $\mathbb{F}_{q^2}^\times$. In particular, $|\det(x)|=(q+1)_2$ since $k$ is odd, and $\varphi_u^x=\varphi_u$ since $x$ normalizes $U_{d,2}$ as in the proof of \prettyref{lem:varphiuconj}. Then applying \prettyref{lem:GGGRarg} with $\sigma$ trivial yields that some conjugate $\mathrm{Char}i_0$ of $\mathrm{Char}i$ satisfies $\mathrm{Char}i_0^x=\mathrm{Char}i_0$. This implies $[I:G]$ is divisible by $(q+1)_2$, so that $[\wt{G}:I]$ must be odd.
\end{proof}
For the remainder of this section, we will consider the case $\epsilon=-1$, so that $\wt{G}=GU_n(q)$ and $G=SU_n(q)$. In particular, \prettyref{prop:oddelemoddindex} yields that if $[\wt{G}:I]$ is even, then neither condition in \prettyref{cor:unipconj} holds if $n$ is divisible by $4$, and condition 1 holds if $n\equiv 2\pmod 4$, taking into account \prettyref{rem:condsnmod4}.
\begin{proposition}\label{prop:n2mod4}
Let $\epsilon=-1$ and suppose that $q\equiv \eta\mod 4$ is nonsquare with $\eta\in\{\pm1\}$
and that $n\equiv 2\pmod 4$. Then the converse of \prettyref{lem:sgn} holds. That is, for $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\mathrm{Char}i}_\lambda)$, $\mathbb{F}_\lambda=\mathbb{Q}(\mathrm{Char}i)$ if and only if $\mathrm{sgn} \lambda\neq\lambda$. Alternatively, $\mathbb{F}_\lambda(\sqrt{\eta p})=\mathbb{Q}(\mathrm{Char}i)$ if and only if $\mathrm{sgn} \lambda = \lambda$.
\end{proposition}
\begin{proof}
We must show that if $\mathrm{sgn}\lambda= \lambda$, then $\mathbb{F}_\lambda\neq \mathbb{Q}(\mathrm{Char}i)$. First, recall that this condition on $\lambda$ is equivalent to the condition that $[\wt{G}:I]=[T_1:\mathcal{I}(\lambda)]$ is even, by \prettyref{lem:orbiteven}. Since $n_2=2$, \prettyref{prop:indstabdivides} yields that $[\wt{G}:I]_2=2$.
This means that no $\wt{G}$-conjugate of $\mathrm{Char}i$ can be fixed by any $\wt{g}\in \wt{G}$ whose determinant satisfies $|\det(\wt{g})|_2=(q+1)_2$.
As an abuse of notation, we let $\tau$ also denote the unique element of $\mathrm{Gal}(\mathbb{F}_\lambda(\zeta_p)/\mathbb{F}_\lambda)$ that restricts to our fixed generator $\tau$ of $\mathrm{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$.
In the notation of \prettyref{lem:GGGRarg}, we have $\varphi_u^x=\varphi_u^b=\tau\varphi_u$ for some $x\in P_{d}$ satisfying $|\det(x)|=(q+1)_2$, by \prettyref{cor:unipconj} and Lemmas \ref{lem:varphiuconj} and \ref{lem:conjinP}. Then by \prettyref{lem:GGGRarg}, there is a conjugate $\mathrm{Char}i_0$ of $\mathrm{Char}i$ such that $\mathrm{Char}i_0^x=\tau\mathrm{Char}i_0$. In particular, note that the condition on the determinant yields that $\mathrm{Char}i_0^x\neq \mathrm{Char}i_0$, so $\tau\mathrm{Char}i_0\neq\mathrm{Char}i_0$. Since $\mathrm{Char}i$ and $\mathrm{Char}i_0$ have the same field of values, we see $\tau\mathrm{Char}i\neq \mathrm{Char}i$, and we have $\mathbb{F}_\lambda\neq \mathbb{Q}(\mathrm{Char}i)$.
\end{proof}
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For the remainder of this section, we will consider the case $\epsilon=-1$, so that $\wt{G}=GU_n(q)$ and $G=SU_n(q)$. In particular, \prettyref{prop:oddelemoddindex} yields that if $[\wt{G}:I]$ is even, then neither condition in \prettyref{cor:unipconj} holds if $n$ is divisible by $4$, and condition 1 holds if $n\equiv 2\pmod 4$, taking into account \prettyref{rem:condsnmod4}.
\begin{proposition}\label{prop:n2mod4}
Let $\epsilon=-1$ and suppose that $q\equiv \eta\mod 4$ is nonsquare with $\eta\in\{\pm1\}$
and that $n\equiv 2\pmod 4$. Then the converse of \prettyref{lem:sgn} holds. That is, for $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\mathrm{Char}i}_\lambda)$, $\mathbb{F}_\lambda=\mathbb{Q}(\mathrm{Char}i)$ if and only if $\mathrm{sgn} \lambda\neq\lambda$. Alternatively, $\mathbb{F}_\lambda(\sqrt{\eta p})=\mathbb{Q}(\mathrm{Char}i)$ if and only if $\mathrm{sgn} \lambda = \lambda$.
\end{proposition}
\begin{proof}
We must show that if $\mathrm{sgn}\lambda= \lambda$, then $\mathbb{F}_\lambda\neq \mathbb{Q}(\mathrm{Char}i)$. First, recall that this condition on $\lambda$ is equivalent to the condition that $[\wt{G}:I]=[T_1:\mathcal{I}(\lambda)]$ is even, by \prettyref{lem:orbiteven}. Since $n_2=2$, \prettyref{prop:indstabdivides} yields that $[\wt{G}:I]_2=2$.
This means that no $\wt{G}$-conjugate of $\mathrm{Char}i$ can be fixed by any $\wt{g}\in \wt{G}$ whose determinant satisfies $|\det(\wt{g})|_2=(q+1)_2$.
As an abuse of notation, we let $\tau$ also denote the unique element of $\mathrm{Gal}(\mathbb{F}_\lambda(\zeta_p)/\mathbb{F}_\lambda)$ that restricts to our fixed generator $\tau$ of $\mathrm{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$.
In the notation of \prettyref{lem:GGGRarg}, we have $\varphi_u^x=\varphi_u^b=\tau\varphi_u$ for some $x\in P_{d}$ satisfying $|\det(x)|=(q+1)_2$, by \prettyref{cor:unipconj} and Lemmas \ref{lem:varphiuconj} and \ref{lem:conjinP}. Then by \prettyref{lem:GGGRarg}, there is a conjugate $\mathrm{Char}i_0$ of $\mathrm{Char}i$ such that $\mathrm{Char}i_0^x=\tau\mathrm{Char}i_0$. In particular, note that the condition on the determinant yields that $\mathrm{Char}i_0^x\neq \mathrm{Char}i_0$, so $\tau\mathrm{Char}i_0\neq\mathrm{Char}i_0$. Since $\mathrm{Char}i$ and $\mathrm{Char}i_0$ have the same field of values, we see $\tau\mathrm{Char}i\neq \mathrm{Char}i$, and we have $\mathbb{F}_\lambda\neq \mathbb{Q}(\mathrm{Char}i)$.
\end{proof}
\begin{proposition}\label{prop:n0mod4}
Let $\epsilon=-1$ and suppose that $q\equiv 3\pmod 4$ and $4\leq n_2\leq (q+1)_2$, and let $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\mathrm{Char}i}_\lambda)$ and $I:=\mathrm{stab}_{\wt{G}}(\mathrm{Char}i)$. Then $\mathbb{F}_\lambda=\mathbb{Q}(\mathrm{Char}i)$ if and only if $[\wt{G}:I]_2<n_2$.
\end{proposition}
\begin{proof}
First, note that $[\wt{G}:I]_2\leq n_2$ by \prettyref{prop:indstabdivides}. Note that by \prettyref{lem:sgn}, we may assume that $[\wt{G}:I]$ is even and therefore that $\mathrm{Char}i\in\mathrm{Irr}(G|\wt{\Gamma}_u)$ where $u$ has no odd-power elementary divisor, by \prettyref{prop:oddelemoddindex}.
By Lemmas \ref{lem:unipconj0mod4}, \ref{lem:varphiuconj}, and \ref{lem:conjinP}, there is some $x\in \wt{G}$ such that $\varphi_u^x=\varphi_u^b=\tau\varphi_u$ and $|\det(x)|=\frac{2(q+1)_2}{n_2}$, which is divisible by $2$ since $n_2\leq(q+1)_2$. By \prettyref{lem:GGGRarg}, there is a conjugate $\mathrm{Char}i_0$ of $\mathrm{Char}i$ such that $\mathrm{Char}i_0^x=\tau\mathrm{Char}i_0$.
Suppose first that $[\wt{G}:I]_2=n_2$, so that $x$ cannot stabilize $\mathrm{Char}i_0$, since $[I:G]_2=\frac{(q+1)_2}{n_2}$ (and the same is true for the stabilizer of $\mathrm{Char}i_0$). This yields that $\mathrm{Char}i_0\neq \tau\mathrm{Char}i_0$, so the same holds for $\mathrm{Char}i$. Hence if $[\wt{G}:I]_2=n_2$, then $\mathbb{F}_\lambda\neq\mathbb{Q}(\mathrm{Char}i)$.
Now suppose $[\wt{G}:I]_2<n_2$. That is, $[\wt{G}:I]_2\leq \frac{n_2}{2}$. Then the stabilizer of $\mathrm{Char}i_0$ must contain $x$, since $I/G\cong \mathcal{I}(\lambda)$ is cyclic and contains the unique subgroup of $\wt{G}/G$ of size $\frac{2(q+1)_2}{n_2}$. Then $\mathrm{Char}i_0=\tau\mathrm{Char}i_0$, and the same is true for $\mathrm{Char}i$, so $\mathbb{F}_\lambda=\mathbb{Q}(\mathrm{Char}i)$.
\end{proof}
\begin{proof}[Proof of \prettyref{thm:turullext1}]
Note that the case $[\wt{G}:I]_2=n_2$, for any $n$ even, is equivalent to having $\alpha\lambda=\lambda$ for any $\alpha\in \wh{T}_1$ of order $n_2$. In case $n\equiv 2\pmod 4$, we remark that this $\alpha$ is $\mathrm{sgn}$. Hence Propositions \ref{prop:initialcase}, \ref{prop:n2mod4}, and \ref{prop:n0mod4} combine to yield the statement.
\end{proof}
\end{document}
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\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{document}
\hbox{\goth h}box{\goth f}irstpage
\hbox{\goth h}box{\goth n}ewtheorem{odstavec}[Theorem]{\hbox{\goth h}skip -2mm}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{abstract}This paper is concerned with energy properties of the wave equation
associated to the Dunkl-Cherednik Laplacian.
We establish the conservation of the total energy,
the strict equipartition of energy under suitable assumptions
and the asymptotic equipartition in the general case.\\
{\hbox{\goth s}l Mathematics Subject Index 2000:} {Primary 35L05\,; Secondary 22E30, 33C67, 35L65, 58J45}\\
{\hbox{\goth s}l Keywords and phrases:} {Dunkl-Cherednik operator, wave equation, energy, \hbox{\goth h}box{\goth l}inebreak equipartition}
\hbox{\goth e}nd{abstract}
\hbox{\goth v}badness=100000
\hbox{\goth s}ection{Introduction}
\hbox{\goth h}box{\goth n}oindent
We use \cite{O2} as a reference for the Dunkl-Cherednik theory.
Let $\hbox{\goth m}athfrak{a}$ be a Euclidean vector space of dimension $d$ equipped
with an inner product $\hbox{\goth h}box{\goth l}angle \cdot,\cdot\rangle.$ Let
$\hbox{\goth m}athcal{R}$ be a crystallographic root system in $\hbox{\goth m}athfrak{a},$
$\hbox{\goth m}athcal{R}^+$ a positive subsystem and
$W$ the Weyl group generated by the reflections
\,$r_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha(x)\!=\!x\!
-\!2\hbox{\goth h}box{\goth f}rac{\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha,x\rangle}{\|\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\|^2}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha$
\,along the roots $\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\!\in\!\hbox{\goth m}athcal{R}.$ We let $k:\hbox{\goth m}athcal{R}\hbox{\goth t}o[\,0,+\infty)$
denote a multiplicity function on the root system $\hbox{\goth m}athcal R,$ and
$\rho=\hbox{\goth h}box{\goth f}rac12\hbox{\goth s}um_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\in\hbox{\goth m}athcal{R}^+}k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha$.
We note that $k$ is $W$-invariant. The Dunkl-Cherednik operators are
the following differential-difference operators,
which are deformations of partial derivatives
and still commute pairwise\,:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
T_\xi f(x)
=\hbox{\goth h}box{\goth p}artial_{\xi}f(x)-\hbox{\goth h}box{\goth l}angle\rho,\xi\rangle f(x)
+\hbox{\goth s}um_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\in\hbox{\goth m}athcal{R}^{+}}k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\,
\hbox{\goth h}box{\goth f}rac{\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha,\xi\rangle}{1\,-\,e^{-\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha,x\rangle}}\,
\{f(x)\!-\!f(r_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}x)\}\,.
\hbox{\goth e}nd{equation*}
Given an orthonormal basis $\{\xi_1,\dots,\xi_d\}$ of $\hbox{\goth m}athfrak{a}$,
the Dunkl-Cherednik Laplacian is defined by
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
Lf(x)=\hbox{\goth s}um_{j=1}^{d}T_{\xi_j}^{2}f(x)\,.
\hbox{\goth e}nd{equation*}
More explicit formulas for $L$ exist but they will not be used in
this paper. The Laplacian $L$ is selfadjoint with respect to the
measure $\hbox{\goth m}u(x) dx$ where
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth m}u(x)=\hbox{\goth h}box{\goth p}rod_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\in\hbox{\goth m}athcal{R}^+}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl|\,2\hbox{\goth s}inh\hbox{\goth h}box{\goth f}rac{\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha,x\rangle}2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr|^{2k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}\,.
\hbox{\goth e}nd{equation*}
Consider the wave equation
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}
\hbox{\goth h}box{\goth l}abel{WaveEquation}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{cases}
\,\hbox{\goth h}box{\goth p}artial_t^2u(t,x)=L_xu(t,x),\\
\,u(0,x)=f(x),
\,\hbox{\goth h}box{\goth p}artial_t\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|_{t=0}u(t,x)=g(x),
\hbox{\goth e}nd{cases}
\hbox{\goth e}nd{equation}
with smooth and compactly supported initial data $(f,g).$
Let us introduce:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{itemize}
\item
the \hbox{\goth t}extit{kinetic energy}
\,$\hbox{\goth m}athcal{K}[u](t)
=\hbox{\goth h}box{\goth f}rac12{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|\hbox{\goth h}box{\goth p}artial_tu(x,t)\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|^{2}\hbox{\goth m}u(x)\,dx$,
\item
the \hbox{\goth t}extit{potential energy}
\,$\hbox{\goth m}athcal{P}[u](t)
=-\,\hbox{\goth h}box{\goth f}rac12{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\,
Lu(x,t)\,\hbox{\goth o}verline{u(x,t)}\,\hbox{\goth m}u(x)\,dx$,
\item
the \hbox{\goth t}extit{total energy}
\,$\hbox{\goth m}athcal{E}[u](t)
=\hbox{\goth m}athcal{K}[u](t)
+\hbox{\goth m}athcal{P}[u](t)$.
\hbox{\goth e}nd{itemize}
In this paper we prove
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{itemize}
\item
the conservation of the total energy:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}
\hbox{\goth h}box{\goth l}abel{Conservation}
\hbox{\goth t}extstyle
\hbox{\goth m}athcal{E}[u](t)=\hbox{\goth t}ext{constant},
\hbox{\goth e}nd{equation}
\item
the strict equipartition of energy,
under the assumptions that the dimension $d$ is odd
and that all the multiplicities $k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha$ are integers:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}
\hbox{\goth h}box{\goth l}abel{StrictEquipartition}
\hbox{\goth t}extstyle
\hbox{\goth m}athcal{K}[u](t)=\hbox{\goth m}athcal{P}[u](t)=\hbox{\goth h}box{\goth f}rac12\,\hbox{\goth m}athcal{E}[u]
\hbox{\goth t}ext{ \,for }|t|\hbox{\goth t}ext{ large}\,,
\hbox{\goth e}nd{equation}
\item
the asymptotic equipartition of energy, for arbitrary $d$ and $\R^+$-valued multiplicity function $k$:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}
\hbox{\goth h}box{\goth l}abel{AsymptoticEquipartition}
\hbox{\goth t}extstyle
\hbox{\goth m}athcal{K}[u](t)\hbox{\goth t}o\hbox{\goth h}box{\goth f}rac12\,\hbox{\goth m}athcal{E}[u]
\hbox{\goth t}ext{ \,and \,}
\hbox{\goth m}athcal{P}[u](t)\hbox{\goth t}o\hbox{\goth h}box{\goth f}rac12\,\hbox{\goth m}athcal{E}[u]
\hbox{\goth t}ext{ \,as \,}
|t| \hbox{\goth t}ext{ goes to } \infty.
\hbox{\goth e}nd{equation}
\hbox{\goth e}nd{itemize}
The proofs follow \cite{BOS}
and use the Fourier transform in the Dunkl-Cherednik setting,
which we will recall in next section. We mention that during the past twenty years,
several works were devoted to
Huygens' principle and equipartition of energy
for wave equations
on symmetric spaces and related settings.
See for instance \cite{BO}, \cite [ch.\,V, \S\;5]{H}, \cite{BOS},
\cite{KY}, \cite{SO}, \cite{BOP1}, \cite{BOP2}, \cite{S}.
\hbox{\goth s}ection{Generalized hypergeometric functions
and Dunkl-Cherednik transform}
\hbox{\goth h}box{\goth n}oindent
Opdam \cite{O1} introduced the following special functions,
which are deformations of exponential functions
\,$e^{\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,x\rangle}$,
\,and the associated Fourier transform.
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{theorem}
There exist a neighborhood \,$U$ of \,$0$ in $\hbox{\goth m}athfrak{a}$
and a unique holomorphic function \,$(\hbox{\goth h}box{\goth l}ambda,x)\hbox{\goth m}apsto G_\hbox{\goth h}box{\goth l}ambda(x)$
\,on \,$a_{\hbox{\goth m}athbb{C}}\!\hbox{\goth t}imes\!(\hbox{\goth m}athfrak{a}\!+\!iU)$ such that
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{cases}
\;T_\xi G_\hbox{\goth h}box{\goth l}ambda(x)=\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\xi\rangle\,G_\hbox{\goth h}box{\goth l}ambda(x)
\hbox{\goth q}uad\hbox{\goth h}box{\goth f}orall\;\xi\!\in\!\hbox{\goth m}athfrak{a}\,,\\
\;G_\hbox{\goth h}box{\goth l}ambda(0)=1\,.
\hbox{\goth e}nd{cases}
\hbox{\goth e}nd{equation*}
Moreover, the following estimate holds on
\,$a_{\hbox{\goth m}athbb{C}}\!\hbox{\goth t}imes\!\hbox{\goth m}athfrak{a}$\,:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
|G_\hbox{\goth h}box{\goth l}ambda(x)|\hbox{\goth h}box{\goth l}e|W|^{\hbox{\goth h}box{\goth f}rac12}\,e^{\|{\rm Re}\,\hbox{\goth h}box{\goth l}ambda\|\|x\|}\,.
\hbox{\goth e}nd{equation*}
\hbox{\goth e}nd{theorem}
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In this paper we prove
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{itemize}
\item
the conservation of the total energy:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}
\hbox{\goth h}box{\goth l}abel{Conservation}
\hbox{\goth t}extstyle
\hbox{\goth m}athcal{E}[u](t)=\hbox{\goth t}ext{constant},
\hbox{\goth e}nd{equation}
\item
the strict equipartition of energy,
under the assumptions that the dimension $d$ is odd
and that all the multiplicities $k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha$ are integers:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}
\hbox{\goth h}box{\goth l}abel{StrictEquipartition}
\hbox{\goth t}extstyle
\hbox{\goth m}athcal{K}[u](t)=\hbox{\goth m}athcal{P}[u](t)=\hbox{\goth h}box{\goth f}rac12\,\hbox{\goth m}athcal{E}[u]
\hbox{\goth t}ext{ \,for }|t|\hbox{\goth t}ext{ large}\,,
\hbox{\goth e}nd{equation}
\item
the asymptotic equipartition of energy, for arbitrary $d$ and $\R^+$-valued multiplicity function $k$:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}
\hbox{\goth h}box{\goth l}abel{AsymptoticEquipartition}
\hbox{\goth t}extstyle
\hbox{\goth m}athcal{K}[u](t)\hbox{\goth t}o\hbox{\goth h}box{\goth f}rac12\,\hbox{\goth m}athcal{E}[u]
\hbox{\goth t}ext{ \,and \,}
\hbox{\goth m}athcal{P}[u](t)\hbox{\goth t}o\hbox{\goth h}box{\goth f}rac12\,\hbox{\goth m}athcal{E}[u]
\hbox{\goth t}ext{ \,as \,}
|t| \hbox{\goth t}ext{ goes to } \infty.
\hbox{\goth e}nd{equation}
\hbox{\goth e}nd{itemize}
The proofs follow \cite{BOS}
and use the Fourier transform in the Dunkl-Cherednik setting,
which we will recall in next section. We mention that during the past twenty years,
several works were devoted to
Huygens' principle and equipartition of energy
for wave equations
on symmetric spaces and related settings.
See for instance \cite{BO}, \cite [ch.\,V, \S\;5]{H}, \cite{BOS},
\cite{KY}, \cite{SO}, \cite{BOP1}, \cite{BOP2}, \cite{S}.
\hbox{\goth s}ection{Generalized hypergeometric functions
and Dunkl-Cherednik transform}
\hbox{\goth h}box{\goth n}oindent
Opdam \cite{O1} introduced the following special functions,
which are deformations of exponential functions
\,$e^{\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,x\rangle}$,
\,and the associated Fourier transform.
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{theorem}
There exist a neighborhood \,$U$ of \,$0$ in $\hbox{\goth m}athfrak{a}$
and a unique holomorphic function \,$(\hbox{\goth h}box{\goth l}ambda,x)\hbox{\goth m}apsto G_\hbox{\goth h}box{\goth l}ambda(x)$
\,on \,$a_{\hbox{\goth m}athbb{C}}\!\hbox{\goth t}imes\!(\hbox{\goth m}athfrak{a}\!+\!iU)$ such that
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{cases}
\;T_\xi G_\hbox{\goth h}box{\goth l}ambda(x)=\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\xi\rangle\,G_\hbox{\goth h}box{\goth l}ambda(x)
\hbox{\goth q}uad\hbox{\goth h}box{\goth f}orall\;\xi\!\in\!\hbox{\goth m}athfrak{a}\,,\\
\;G_\hbox{\goth h}box{\goth l}ambda(0)=1\,.
\hbox{\goth e}nd{cases}
\hbox{\goth e}nd{equation*}
Moreover, the following estimate holds on
\,$a_{\hbox{\goth m}athbb{C}}\!\hbox{\goth t}imes\!\hbox{\goth m}athfrak{a}$\,:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
|G_\hbox{\goth h}box{\goth l}ambda(x)|\hbox{\goth h}box{\goth l}e|W|^{\hbox{\goth h}box{\goth f}rac12}\,e^{\|{\rm Re}\,\hbox{\goth h}box{\goth l}ambda\|\|x\|}\,.
\hbox{\goth e}nd{equation*}
\hbox{\goth e}nd{theorem}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{Definition} Let $f$ be a nice function on
$\hbox{\goth m}athfrak{a}$, say $f$ belongs to the space
$\hbox{\goth m}athcal{C}_c^\infty(\hbox{\goth m}athfrak{a}) $ of smooth functions on $\hbox{\goth m}athfrak{a}$
with compact support. Its Dunkl-Cherednik transform is defined by
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda)
=\int_{\hbox{\goth m}athfrak{a}}f(x)\,G_{-iw_0\hbox{\goth h}box{\goth l}ambda}(w_0x)\,\hbox{\goth m}u(x)\,dx.
\hbox{\goth e}nd{equation*}
Here $w_0$ denotes the longest element in the Weyl group $W.$
\hbox{\goth e}nd{Definition}
The involvement of $w_0$ in the definition of $\hbox{\goth m}athcal F$ is related to
the below skew-adjointness property of the Dunkl-Cherednik operators
with respect to the inner product
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\goth l}angle f,g\rangle=\int_{\hbox{\goth m}athfrak{a}}f(x)\,\hbox{\goth o}verline{g(x)}\,\hbox{\goth m}u(x)\,dx,
\hbox{\goth q}quad f,g\in\hbox{\goth m}athcal{C}_c^\infty(\hbox{\goth m}athfrak{a}).
\hbox{\goth e}nd{equation*}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{lemma}
The adjoint of \,$T_\xi$ is $-w_0T_{w_0\xi}w_0$\,:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\goth l}angle T_\xi f,g\rangle=\hbox{\goth h}box{\goth l}angle f,-w_0T_{w_0\xi}w_0g\rangle\,.
\hbox{\goth e}nd{equation*}
\hbox{\goth e}nd{lemma}
As an immediate consequence, we obtain:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{corollary}
For every \,$\xi,\hbox{\goth h}box{\goth l}ambda\!\in\!\hbox{\goth m}athfrak{a}$
and $f\!\in\!\hbox{\goth m}athcal{C}_c^\infty(\hbox{\goth m}athfrak{a})$,
we have
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth m}athcal{F}(T_\xi f)(\hbox{\goth h}box{\goth l}ambda)
=i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\xi\rangle\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda),
\hbox{\goth e}nd{equation*}
and therefore
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth m}athcal{F}(Lf)(\hbox{\goth h}box{\goth l}ambda)=-\|\hbox{\goth h}box{\goth l}ambda\|^{2}\,\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda)\,.
\hbox{\goth e}nd{equation*}
\hbox{\goth e}nd{corollary}
Next we will recall from \cite{O1} three main results about the Dunkl-Cherednik transform
(see also \cite{O2}). For $R>0,$ let $\hbox{\goth m}athcal{C}_R^\infty(\hbox{\goth m}athfrak{a})$ be
the space of smooth functions on $\hbox{\goth m}athfrak{a}$
\,vanishing outside the ball
\,$B_R\!=\!\{x\!\in\!\hbox{\goth m}athfrak{a}\,|\|x\|\!\hbox{\goth h}box{\goth l}e\!R\}.$
We let $\hbox{\goth m}athcal{H}_{R}(\hbox{\goth m}athfrak{a}_{\hbox{\goth m}athbb{C}})$ denote the space of holomorphic
functions $h$ on the complexification $\hbox{\goth m}athfrak{a}_{\hbox{\goth m}athbb{C}}$ of $\hbox{\goth m}athfrak{a}$
such that, for every integer $N\!>\!0$,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth t}extstyle
\hbox{\goth s}up_{\hbox{\goth h}box{\goth l}ambda\in\hbox{\goth m}athfrak{a}_{\hbox{\goth m}athbb{C}}}
(1\!+\!\|\hbox{\goth h}box{\goth l}ambda\|)^N\,
e^{-R\,\|{\rm Im}\,\hbox{\goth h}box{\goth l}ambda\|}\,|h(\hbox{\goth h}box{\goth l}ambda)|
<+\infty\,.
\hbox{\goth e}nd{equation*}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{theorem}
\hbox{\goth h}box{\goth l}abel{PaleyWiener} {\rm\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}f (Paley-Wiener)}
The transformation
$\hbox{\goth m}athcal{F}$ is an isomorphism of
$\hbox{\goth m}athcal{C}_R^\infty(\hbox{\goth m}athfrak{a})$ onto
$\hbox{\goth m}athcal{H}_{R}(\hbox{\goth m}athfrak{a}_{\hbox{\goth m}athbb{C}}),$ for every $R\!>\!0$.
\hbox{\goth e}nd{theorem}
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0.191.2
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The involvement of $w_0$ in the definition of $\hbox{\goth m}athcal F$ is related to
the below skew-adjointness property of the Dunkl-Cherednik operators
with respect to the inner product
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\goth l}angle f,g\rangle=\int_{\hbox{\goth m}athfrak{a}}f(x)\,\hbox{\goth o}verline{g(x)}\,\hbox{\goth m}u(x)\,dx,
\hbox{\goth q}quad f,g\in\hbox{\goth m}athcal{C}_c^\infty(\hbox{\goth m}athfrak{a}).
\hbox{\goth e}nd{equation*}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{lemma}
The adjoint of \,$T_\xi$ is $-w_0T_{w_0\xi}w_0$\,:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\goth l}angle T_\xi f,g\rangle=\hbox{\goth h}box{\goth l}angle f,-w_0T_{w_0\xi}w_0g\rangle\,.
\hbox{\goth e}nd{equation*}
\hbox{\goth e}nd{lemma}
As an immediate consequence, we obtain:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{corollary}
For every \,$\xi,\hbox{\goth h}box{\goth l}ambda\!\in\!\hbox{\goth m}athfrak{a}$
and $f\!\in\!\hbox{\goth m}athcal{C}_c^\infty(\hbox{\goth m}athfrak{a})$,
we have
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth m}athcal{F}(T_\xi f)(\hbox{\goth h}box{\goth l}ambda)
=i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\xi\rangle\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda),
\hbox{\goth e}nd{equation*}
and therefore
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth m}athcal{F}(Lf)(\hbox{\goth h}box{\goth l}ambda)=-\|\hbox{\goth h}box{\goth l}ambda\|^{2}\,\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda)\,.
\hbox{\goth e}nd{equation*}
\hbox{\goth e}nd{corollary}
Next we will recall from \cite{O1} three main results about the Dunkl-Cherednik transform
(see also \cite{O2}). For $R>0,$ let $\hbox{\goth m}athcal{C}_R^\infty(\hbox{\goth m}athfrak{a})$ be
the space of smooth functions on $\hbox{\goth m}athfrak{a}$
\,vanishing outside the ball
\,$B_R\!=\!\{x\!\in\!\hbox{\goth m}athfrak{a}\,|\|x\|\!\hbox{\goth h}box{\goth l}e\!R\}.$
We let $\hbox{\goth m}athcal{H}_{R}(\hbox{\goth m}athfrak{a}_{\hbox{\goth m}athbb{C}})$ denote the space of holomorphic
functions $h$ on the complexification $\hbox{\goth m}athfrak{a}_{\hbox{\goth m}athbb{C}}$ of $\hbox{\goth m}athfrak{a}$
such that, for every integer $N\!>\!0$,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth t}extstyle
\hbox{\goth s}up_{\hbox{\goth h}box{\goth l}ambda\in\hbox{\goth m}athfrak{a}_{\hbox{\goth m}athbb{C}}}
(1\!+\!\|\hbox{\goth h}box{\goth l}ambda\|)^N\,
e^{-R\,\|{\rm Im}\,\hbox{\goth h}box{\goth l}ambda\|}\,|h(\hbox{\goth h}box{\goth l}ambda)|
<+\infty\,.
\hbox{\goth e}nd{equation*}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{theorem}
\hbox{\goth h}box{\goth l}abel{PaleyWiener} {\rm\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}f (Paley-Wiener)}
The transformation
$\hbox{\goth m}athcal{F}$ is an isomorphism of
$\hbox{\goth m}athcal{C}_R^\infty(\hbox{\goth m}athfrak{a})$ onto
$\hbox{\goth m}athcal{H}_{R}(\hbox{\goth m}athfrak{a}_{\hbox{\goth m}athbb{C}}),$ for every $R\!>\!0$.
\hbox{\goth e}nd{theorem}
The Plancherel formula and the inversion formula of $\hbox{\goth m}athcal F$
involve the complex measure \,$\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda$
\,with density
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth t}extstyle
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,
=\hbox{\goth h}space{-1mm}\hbox{\goth h}box{\goth p}rod\hbox{\goth h}box{\goth l}imits_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\in\hbox{\goth m}athcal{R}_0^+}\hbox{\goth h}space{-1mm}
\hbox{\goth h}box{\goth f}rac
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,
+\,k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)
\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}\,
\hbox{\goth h}box{\goth f}rac
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(\hbox{\goth h}box{\goth f}rac{i \hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,
+\,k_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}}2+\,k_{2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(\hbox{\goth h}box{\goth f}rac{i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,
+\,k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}\,
\hbox{\goth h}box{\goth f}rac
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(-\,i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,
+\,k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(-\,i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,+\,1\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)
\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}\,
\hbox{\goth h}box{\goth f}rac
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(\hbox{\goth h}box{\goth f}rac{-i \hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,
+\,k_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}}2+\,k_{2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}+\,1\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(\hbox{\goth h}box{\goth f}rac{-i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,
+\,k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}\,,
\hbox{\goth e}nd{equation*}
where
\,$\hbox{\goth m}athcal{R}_0^+\!
=\hbox{\goth h}space{-.25mm}\{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\!\in\hbox{\goth m}athcal{R}^+
|\,\hbox{\goth h}box{\goth f}rac\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha2\!\hbox{\goth h}box{\goth n}otin\!\hbox{\goth m}athcal{R}\,\}$
\,is the set of positive indivisible roots,
\,$\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\!=\!2\|\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\|^{-2}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha$
\,the coroot corresponding to $\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha$,
and \,$k_{2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}\!=\!0$ \,if \,$2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\!\hbox{\goth h}box{\goth n}otin\!\hbox{\goth m}athcal{R}$.
Notice that $\hbox{\goth h}box{\goth n}u$ is an analytic function on $\hbox{\goth m}athfrak{a},$
with a polynomial growth
and which extends meromorphically to~$\hbox{\goth m}athfrak{a}_{\hbox{\goth m}athbb{C}}$.
It is actually a polynomial
if the multiplicity function $k$ is integer-valued
and it has poles otherwise.
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\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{theorem}
{\rm\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}f (Inversion formula)}
There is a constant \,$c_0\!>\!0$ such
that, for every $f\!\in\!\hbox{\goth m}athcal{C}_c^\infty(\hbox{\goth m}athfrak{a})$,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
f(x)\,=\,c_0\int_{\hbox{\goth m}athfrak{a}}\,
\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda)\,G_{i\hbox{\goth h}box{\goth l}ambda}(x)\,\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\,.
\hbox{\goth e}nd{equation*}
\hbox{\goth e}nd{theorem}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{theorem}\hbox{\goth h}box{\goth l}abel{Plancherel}
{\rm\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}f (Plancherel formula)}
For every
$f,g\!\in\!\hbox{\goth m}athcal{C}_c^\infty(\hbox{\goth m}athfrak{a})$,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\int_{\hbox{\goth m}athfrak{a}}\,f(x)\,\hbox{\goth o}verline{g(x)}\,\hbox{\goth m}u(x)\,dx\,
=\,c_0\int_{\hbox{\goth m}athfrak{a}}\,\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\,,
\hbox{\goth e}nd{equation*}
where \,$\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)
:=\hbox{\goth o}verline{\hbox{\goth m}athcal{F}(w_0g)(w_0\hbox{\goth h}box{\goth l}ambda)}
=\!{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
\hbox{\goth o}verline{g(x)}\,G_{i\hbox{\goth h}box{\goth l}ambda}(x)\,\hbox{\goth m}u(x)\,dx$\,.
\hbox{\goth e}nd{theorem}
\hbox{\goth s}ection{Conservation of energy}
\hbox{\goth h}box{\goth n}oindent
This section is devoted to the proof of \hbox{\goth e}qref{Conservation}.
Via the Dunkl-Cherednik transform,
the wave equation \hbox{\goth e}qref{WaveEquation} becomes
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{cases}
\,\hbox{\goth h}box{\goth p}artial_t^2\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}u(t,\hbox{\goth h}box{\goth l}ambda)
=-\|\hbox{\goth h}box{\goth l}ambda\|^2\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}u(t,\hbox{\goth h}box{\goth l}ambda),\\
\,\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}u(0,\hbox{\goth h}box{\goth l}ambda)
=\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda),
\,\hbox{\goth h}box{\goth p}artial_t\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|_{t=0}\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}u(t,\hbox{\goth h}box{\goth l}ambda)
=\hbox{\goth m}athcal{F}\!g(\hbox{\goth h}box{\goth l}ambda),
\hbox{\goth e}nd{cases}
\hbox{\goth e}nd{equation*}
and its solution satisfies
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}\hbox{\goth h}box{\goth l}abel{SolutionFourier}
\hbox{\goth t}extstyle
\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}u(t,\hbox{\goth h}box{\goth l}ambda)
=(\cos t\|\hbox{\goth h}box{\goth l}ambda\|)\,\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda)
+\hbox{\goth h}box{\goth f}rac{\hbox{\goth s}in t\|\hbox{\goth h}box{\goth l}ambda\|}{\|\hbox{\goth h}box{\goth l}ambda\|}\hbox{\goth m}athcal{F}\!g(\hbox{\goth h}box{\goth l}ambda)\,.
\hbox{\goth e}nd{equation}
By means of the Paley-Wiener Theorem 2.5, in \cite[p. 52-53]{S} the author proves
the following finite speed propagation property:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{quote}
Assume that the initial data $f$ and $g$ belong to
$\hbox{\goth m}athcal{C}_R^\infty(\hbox{\goth m}athfrak{a})$.
Then the solution $u(t,x)$ belongs to
$\hbox{\goth m}athcal{C}_{R+|t|}^{\infty}(\hbox{\goth m}athfrak{a})$
as a function of $x$.
\hbox{\goth e}nd{quote}
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0.191.4
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Let us express the potential and kinetic energies defined in the introduction
via the Dunkl-Cherednik transform. Using the Plancherel formula and Corollary 2.4,
we have
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}
\hbox{\goth h}box{\goth l}abel{PotentialFourier1}
\hbox{\goth m}athcal{P}[u](t)
=\,{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}2}{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
\|\hbox{\goth h}box{\goth l}ambda\|^2\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}u(t,\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}u(t,\hbox{\goth h}box{\goth l}ambda)\,
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\,.
\hbox{\goth e}nd{equation}
Moreover, since the Dunkl-Cherednik Laplacian is $W$-invariant, it follows that $(w_0u)(t,x)\!=\!u(t,w_0x)$ is the solution to the wave equation \hbox{\goth e}qref{WaveEquation}
with the initial data \,$w_0f$ and $w_0g$.
Thus
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}
\hbox{\goth h}box{\goth l}abel{SolutionFourierBis}
\hbox{\goth t}extstyle
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}u(t,\hbox{\goth h}box{\goth l}ambda)
=(\cos t\|\hbox{\goth h}box{\goth l}ambda\|)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)
+\hbox{\goth h}box{\goth f}rac{\hbox{\goth s}in t\|\hbox{\goth h}box{\goth l}ambda\|}{\|\hbox{\goth h}box{\goth l}ambda\|}
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)\,.
\hbox{\goth e}nd{equation}
Now, by substituting \hbox{\goth e}qref{SolutionFourier}
and \hbox{\goth e}qref{SolutionFourierBis} in \hbox{\goth e}qref{PotentialFourier1},
we get
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}\hbox{\goth h}box{\goth l}abel{PotentialFourier2}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{aligned}
\hbox{\goth m}athcal{P}[u](t)
&={\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}2}{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
\|\hbox{\goth h}box{\goth l}ambda\|^2\,(\cos t\|\hbox{\goth h}box{\goth l}ambda\|)^2\,
\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)\,
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\\
&+{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}2}{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
(\hbox{\goth s}in t\|\hbox{\goth h}box{\goth l}ambda\|)^2\,
\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\\
&+{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}4}{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
\|\hbox{\goth h}box{\goth l}ambda\|\,(\hbox{\goth s}in 2t\|\hbox{\goth h}box{\goth l}ambda\|)\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\,\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)
+\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)\,\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\,.\\
\hbox{\goth e}nd{aligned}
\hbox{\goth e}nd{equation}
Similarly to $\hbox{\goth m}athcal P[u]$, we can rewrite the kinetic energy as
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth m}athcal{K}[u](t)
=\,{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}2}{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
\hbox{\goth h}box{\goth p}artial_{t}\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}u(t,\hbox{\goth h}box{\goth l}ambda)\,
\hbox{\goth h}box{\goth p}artial_{t}\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}u(t,\hbox{\goth h}box{\goth l}ambda)\,
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\,.
\hbox{\goth e}nd{equation*}
Using the following facts
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{cases}
\;\hbox{\goth h}box{\goth p}artial_{t}\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}u(t,\hbox{\goth h}box{\goth l}ambda)
=-\,\|\hbox{\goth h}box{\goth l}ambda\|\,(\hbox{\goth s}in t\|\hbox{\goth h}box{\goth l}ambda\|)\,
\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}f(\hbox{\goth h}box{\goth l}ambda)
+(\cos t\|\hbox{\goth h}box{\goth l}ambda\|)\,
\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)\,,\\
\;\hbox{\goth h}box{\goth p}artial_{t}\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}u(t,\hbox{\goth h}box{\goth l}ambda)
=-\,\|\hbox{\goth h}box{\goth l}ambda\|\,(\hbox{\goth s}in t\|\hbox{\goth h}box{\goth l}ambda\|)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)
+(\cos t\|\hbox{\goth h}box{\goth l}ambda\|)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)\,,\\
\hbox{\goth e}nd{cases}
\hbox{\goth e}nd{equation*}
we deduce that
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}
\hbox{\goth h}box{\goth l}abel{KineticFourier1}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{aligned}
\hbox{\goth m}athcal{K}[u](t)
&={\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}2}
{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
\|\hbox{\goth h}box{\goth l}ambda\|^2\,(\hbox{\goth s}in t\|\hbox{\goth h}box{\goth l}ambda\|)^2\,
\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)\,
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\\
&+{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}2}{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
(\cos t\|\hbox{\goth h}box{\goth l}ambda\|)^2\,
\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\\
&-{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}4}{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
\|\hbox{\goth h}box{\goth l}ambda\|\,(\hbox{\goth s}in 2t\|\hbox{\goth h}box{\goth l}ambda\|)\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\,\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)
+\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)\,\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\,.\\
\hbox{\goth e}nd{aligned}
\hbox{\goth e}nd{equation}
By suming up \hbox{\goth e}qref{PotentialFourier2} and \hbox{\goth e}qref{KineticFourier1},
we obtain the conservation of the total energy\,:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth m}athcal{E}[u](t)
=\,{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}2}
{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\|\hbox{\goth h}box{\goth l}ambda\|^2\,
\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)
+\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.4mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda
=\,\hbox{\goth m}athcal{E}[u](0)\,.
\hbox{\goth e}nd{equation*}
That is $\hbox{\goth m}athcal{E}[u](t)$ is independent of $t.$
\hbox{\goth s}ection{Equipartition of energy}
\hbox{\goth h}box{\goth n}oindent
This section is devoted to the proof
of \hbox{\goth e}qref{StrictEquipartition} and \hbox{\goth e}qref{AsymptoticEquipartition}.
Using the classical trigonometric identities for double angles, we can
rewrite the identities \hbox{\goth e}qref{PotentialFourier2} and \hbox{\goth e}qref{KineticFourier1}
respectively as
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{aligned}
\hbox{\goth m}athcal{P}[u](t)
&={\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}4}
{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\|\hbox{\goth h}box{\goth l}ambda\|^2\,
\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)
+\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.4mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\\
&+{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}4}
{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
(\cos2t\|\hbox{\goth h}box{\goth l}ambda\|)\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\|\hbox{\goth h}box{\goth l}ambda\|^2
\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)
-\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.4mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\\
&+{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}4}
{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
\|\hbox{\goth h}box{\goth l}ambda\|\,(\hbox{\goth s}in 2t\|\hbox{\goth h}box{\goth l}ambda\|)\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\,\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)
+\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)\,\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\\
\hbox{\goth e}nd{aligned}
\hbox{\goth e}nd{equation*}
and
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{aligned}
\hbox{\goth m}athcal{K}[u](t)
&={\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}4}
{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\|\hbox{\goth h}box{\goth l}ambda\|^2\,
\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)
+\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.4mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\\
&-{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}4}
{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
(\cos2t\|\hbox{\goth h}box{\goth l}ambda\|)\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\|\hbox{\goth h}box{\goth l}ambda\|^2
\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)
-\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.4mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\\
&-{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}4}
{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
\|\hbox{\goth h}box{\goth l}ambda\|\,(\hbox{\goth s}in 2t\|\hbox{\goth h}box{\goth l}ambda\|)\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\,\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)
+\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)\,\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\,.\\
\hbox{\goth e}nd{aligned}
\hbox{\goth e}nd{equation*}
Hence
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0.191.5
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\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}
\hbox{\goth h}box{\goth l}abel{Difference1}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{aligned}
&\hbox{\goth m}athcal{P}[u](t)\hbox{\goth h}space{-.3mm}-\hbox{\goth h}space{-.2mm}\hbox{\goth m}athcal{K}[u](t)\,=\\
&={\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}2}
{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\,
(\cos2t\|\hbox{\goth h}box{\goth l}ambda\|)\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\|\hbox{\goth h}box{\goth l}ambda\|^2
\hbox{\goth m}athcal{F}\!f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)
-\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.4mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\\
&+\,{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}2}
{{\displaystyle}playstyle\int_{\hbox{\goth m}athfrak{a}}}\;
\|\hbox{\goth h}box{\goth l}ambda\|\,(\hbox{\goth s}in 2t\|\hbox{\goth h}box{\goth l}ambda\|)\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\,\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}f(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)
+\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}g(\hbox{\goth h}box{\goth l}ambda)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(\hbox{\goth h}box{\goth l}ambda)\,\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,d\hbox{\goth h}box{\goth l}ambda\,.\\
\hbox{\goth e}nd{aligned}
\hbox{\goth e}nd{equation}
Introducing polar coordinates in $\hbox{\goth m}athfrak{a}$,
\hbox{\goth e}qref{Difference1} becomes
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation}
\hbox{\goth h}box{\goth l}abel{Difference2}
\hbox{\goth m}athcal{P}[u](t)\hbox{\goth h}space{-.3mm}-\hbox{\goth h}space{-.2mm}\hbox{\goth m}athcal{K}[u](t)
={\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}2}
{{\displaystyle}playstyle\int_{\,0}^{+\infty}}\hbox{\goth h}space{-1mm}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\cos(2tr)\,\Phi(r)+\hbox{\goth s}in(2tr)\,r\,\Psi(r)\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,
r^{d-1}\,dr\,,
\hbox{\goth e}nd{equation}
where
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{aligned}
\Phi(r)\,&=\int_{S(\hbox{\goth m}athfrak{a})}
\hbox{\goth t}extstyle\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\,r^2\hbox{\goth m}athcal{F}\!f(r\hbox{\goth s}igma)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(r\hbox{\goth s}igma)
-\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.4mm}g(r\hbox{\goth s}igma)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.4mm}g(r\hbox{\goth s}igma)\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,
\hbox{\goth h}box{\goth n}u(r\hbox{\goth s}igma)\,d\hbox{\goth s}igma\,,\\
\Psi(r)\,&=\int_{S(\hbox{\goth m}athfrak{a})}
\hbox{\goth t}extstyle\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\hbox{\goth m}athcal{F}\!f(r\hbox{\goth s}igma)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.4mm}g(r\hbox{\goth s}igma)
+\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.4mm}g(r\hbox{\goth s}igma)\,
\widetilde{\hbox{\goth m}athcal{F}}\!f(r\hbox{\goth s}igma)\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,
\hbox{\goth h}box{\goth n}u(r\hbox{\goth s}igma)\,d\hbox{\goth s}igma\,,\\
\hbox{\goth e}nd{aligned}
\hbox{\goth e}nd{equation*}
and $d\hbox{\goth s}igma$ denotes the surface measure
on the unit sphere $S(\hbox{\goth m}athfrak{a})$ in $\hbox{\goth m}athfrak{a}$.
Let \,$\hbox{\goth h}box{\goth g}amma_0\!\in\!(0,+\infty]$ \,be the width of
the largest horizontal strip \,$|{\rm Im}\, z|\!<\!\hbox{\goth h}box{\goth g}amma_0$
\,in which \,$z\hbox{\goth m}apsto\hbox{\goth h}box{\goth n}u(z\hbox{\goth s}igma)$ \,is holomorphic
for all directions \,$\hbox{\goth s}igma\!\in\!S(\hbox{\goth m}athfrak{a})$.
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{lemma}
\hbox{\goth h}box{\goth l}abel{EstimatePhiPsi}
{\rm (i)} $\Phi(z)$ and $\Psi(z)$ extend to even holomorphic functions
in the strip \,$|{\rm Im}\, z|\!<\!\hbox{\goth h}box{\goth g}amma_0$.
{\rm (ii)} If $\hbox{\goth h}box{\goth g}amma_0\!<\!+\infty$,
the following estimate holds in every substrip \,$|{\rm Im}\, z|\!\hbox{\goth h}box{\goth l}e\!\hbox{\goth h}box{\goth g}amma$
\,with \,$\hbox{\goth h}box{\goth g}amma\!<\!\hbox{\goth h}box{\goth g}amma_0:$
\,For every $N\!>\!0$, there is a constant \,$C\!>\!0$
$($depending on $f,g\!\in\!\hbox{\goth m}athcal{C}_R^\infty(\hbox{\goth m}athfrak{a})$,
$N$ \!and $\hbox{\goth h}box{\goth g}amma$$)$ such that
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
|\Phi(z)|+|\Psi(z)|\hbox{\goth h}box{\goth l}e
C\,|z|^{|\hbox{\goth m}athcal{R}_0^+|}\,(1\!+\!|z|)^{-N}\,e^{\,2R\,|{\rm Im}\, z|}\,.
\hbox{\goth e}nd{equation*}
{\rm (iii)} If $\hbox{\goth h}box{\goth g}amma_0=+\infty$,
the previous estimate holds uniformly in $\hbox{\goth m}athbb{C}$.
\hbox{\goth e}nd{lemma}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{proof}
(i) follows from the definition of \,$\Phi$ and $\Psi$.
Let us turn to the estimates (ii) and (iii).
On one hand,
according to the Paley-Wiener Theorem
(Theorem \ref{PaleyWiener}), all transforms
\,$\hbox{\goth m}athcal{F}\!f(z\hbox{\goth s}igma)$,
$\widetilde{\hbox{\goth m}athcal{F}}\!f(z\hbox{\goth s}igma)$,
$\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.25mm}g(z\hbox{\goth s}igma)$,
$\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.25mm}g(z\hbox{\goth s}igma)$
\,are \,$\hbox{\goth t}ext{O}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(\{1\!+\!|z|\}^{-N}e^{\,R\,|{\rm Im}\, z|}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)$.
On the other hand,
let us discuss the behavior of the Plancherel measure.
Consider first the case where all multiplicities are integers.
Without loss of generality, we may assume that $k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\!\in\!\hbox{\goth m}athbb{N}^*$
and $k_{2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}\!\in\!\hbox{\goth m}athbb{N}$
\;for every indivisible root $\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha$.
Then
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{aligned}
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,
&\hbox{\goth t}extstyle
=\,{\rm const.}\,\;\hbox{\goth h}box{\goth p}rod_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\in\hbox{\goth m}athcal{R}_0^+}
\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\!
+\!i(k_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}\hbox{\goth h}space{-.75mm}+\!2k_{2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha})\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\\
&\hbox{\goth t}extstyle\hbox{\goth t}imes\;
\hbox{\goth h}box{\goth p}rod_{0<j<k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle^2\hbox{\goth h}space{-.75mm}+\!j^2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}
\;\hbox{\goth h}box{\goth p}rod_{0\hbox{\goth h}box{\goth l}e\widetilde{j}<k_{2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle^2\hbox{\goth h}space{-.75mm}
+\!(k_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}\hbox{\goth h}space{-.75mm}+\!2\widetilde{j})^2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}
\hbox{\goth e}nd{aligned}
\hbox{\goth e}nd{equation*}
is a polynomial of degree
\,$2\,|k|=2\hbox{\goth s}um_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\in\hbox{\goth m}athcal{R}^+}\!k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha$.
In general,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)={\rm const.}\,\,\hbox{\goth h}box{\goth p}i(\hbox{\goth h}box{\goth l}ambda)\,\widetilde\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,,
\hbox{\goth e}nd{equation*}
where
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where
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth t}extstyle
\hbox{\goth h}box{\goth p}i(\hbox{\goth h}box{\goth l}ambda)
=\,\hbox{\goth h}box{\goth p}rod_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\in\hbox{\goth m}athcal{R}_0^+}
\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle
\hbox{\goth e}nd{equation*}
is a homogeneous polynomial of degree $|\hbox{\goth m}athcal{R}_0^+|$
and
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth t}extstyle
\widetilde\hbox{\goth h}box{\goth n}u(\hbox{\goth h}box{\goth l}ambda)\,
=\hbox{\goth h}space{-1mm}\hbox{\goth h}box{\goth p}rod\hbox{\goth h}box{\goth l}imits_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\in\hbox{\goth m}athcal{R}_0^+}\hbox{\goth h}space{-1mm}
\hbox{\goth h}box{\goth f}rac
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,
+\,k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,+\,1\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)
\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}\,
\hbox{\goth h}box{\goth f}rac
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(\hbox{\goth h}box{\goth f}rac{i \hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,
+\,k_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}}2+\,k_{2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(\hbox{\goth h}box{\goth f}rac{i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,
+\,k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}\,
\hbox{\goth h}box{\goth f}rac
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(-\,i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,
+\,k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(-\,i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,+\,1\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)
\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}\,
\hbox{\goth h}box{\goth f}rac
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(\hbox{\goth h}box{\goth f}rac{-i \hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,
+\,k_{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}}2+\,k_{2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}+\,1\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}
{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(\hbox{\goth h}box{\goth f}rac{-i\hbox{\goth h}box{\goth l}angle\hbox{\goth h}box{\goth l}ambda,\check\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha\rangle\,
+\,k_\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}lpha}2\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)\hbox{\goth v}phantom{\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|}}
\hbox{\goth e}nd{equation*}
is an analytic function which never vanishes on $\hbox{\goth m}athfrak{a}$.
Notice that \,$z\hbox{\goth m}apsto\hbox{\goth h}box{\goth n}u(z\hbox{\goth s}igma)\hbox{\goth t}ext{ or }\widetilde\hbox{\goth h}box{\goth n}u(z\hbox{\goth s}igma)$
has poles for generic directions $\hbox{\goth s}igma\!\in\!S(\hbox{\goth m}athfrak{a})$
as soon as some multiplicities are not integers.
Using Stirling's formula
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth G}amma(\xi)\hbox{\goth s}im\hbox{\goth s}qrt{2\hbox{\goth h}box{\goth p}i}\,\xi^{\xi-\hbox{\goth h}box{\goth f}rac12}\,e^{-\xi}
\hbox{\goth q}uad\hbox{\goth t}ext{as \,}|\xi|\hbox{\goth t}o+\infty
\hbox{\goth t}ext{ \,with \,}|\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth a}rg z|\!<\!\hbox{\goth h}box{\goth p}i\!-\!\hbox{\goth v}arepsilon\,,
\hbox{\goth e}nd{equation*}
we get the following estimate for the Plancherel density, in each
strip \,$|{\rm Im}\, z|\!<\!\hbox{\goth h}box{\goth g}amma$ \,with \,$0\!<\!\hbox{\goth h}box{\goth g}amma\!<\!\hbox{\goth h}box{\goth g}amma_0$\,:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
|\hbox{\goth h}box{\goth n}u(z\hbox{\goth s}igma)|\hbox{\goth h}box{\goth l}e\,C\,|z|^{|\hbox{\goth m}athcal{R}_0^+|}\,(1\!+\!|z|)^{2|k|-|\hbox{\goth m}athcal{R}_0^+|}\,.
\hbox{\goth e}nd{equation*}
The estimates (ii) and (iii) follow easily from these considerations.
\hbox{\goth e}nd{proof}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{proposition}
\hbox{\goth h}box{\goth l}abel{DifferenceEstimate1}
Assume that the dimension $d$ is odd
and that all multiplicities are integers.
Then there exists a constant \,$C\!>\!0$
$($depending on the initial data
\hbox{\goth h}box{\goth l}inebreak
$f,g\!\in\!\hbox{\goth m}athcal{C}_R^\infty(\hbox{\goth m}athfrak{a}))$
such that, for every \,$\hbox{\goth h}box{\goth g}amma\!\hbox{\goth h}box{\goth g}e\!0$ and \,$t\!\in\!\hbox{\goth m}athbb{R}$,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
|\hbox{\goth m}athcal{P}[u](t)\hbox{\goth h}space{-.3mm}-\hbox{\goth h}space{-.2mm}\hbox{\goth m}athcal{K}[u](t)|
\hbox{\goth h}box{\goth l}e C\,e^{\,2\hbox{\goth h}box{\goth g}amma(R-|t|)}\,.
\hbox{\goth e}nd{equation*}
\hbox{\goth e}nd{proposition}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{proof}
Evenness allows us to rewrite \hbox{\goth e}qref{Difference2} as follows\,:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth m}athcal{P}[u](t)\hbox{\goth h}space{-.3mm}-\hbox{\goth h}space{-.2mm}\hbox{\goth m}athcal{K}[u](t)
={\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}4}
{{\displaystyle}playstyle\int_{-\infty}^{+\infty}}\hbox{\goth h}space{-1mm}e^{\,i2tr}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\Phi(r)\!-\!ir\Psi(r)\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,r^{d-1}\,dr\,.
\hbox{\goth e}nd{equation*}
Let us shift the contour of integration
from \,$\hbox{\goth m}athbb{R}$ \,to \,$\hbox{\goth m}athbb{R}\hbox{\goth h}space{-.75mm}\hbox{\goth h}box{\goth p}m\!i\hbox{\goth h}box{\goth g}amma$,
according to the sign of~$t$,
and estimate the resulting integral, using Lemma \ref{EstimatePhiPsi}.iii.
As a result, the difference of energy
$$\hbox{\goth m}athcal{P}[u](t)-\hbox{\goth m}athcal{K}[u](t)= \hbox{\goth h}box{\goth f}rac{c_0}4\,e^{-2\hbox{\goth h}box{\goth g}amma|t|}
{\int_{-\infty}^{+\infty}}
e^{\,i2tr}\,\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\Phi(r\!\hbox{\goth h}box{\goth p}m\!i\hbox{\goth h}box{\goth g}amma)\!
-i(r\!\hbox{\goth h}box{\goth p}m\!\hbox{\goth h}box{\goth g}amma)\,\Psi(r\!\hbox{\goth h}box{\goth p}m\!i\hbox{\goth h}box{\goth g}amma)\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,
(r\!\hbox{\goth h}box{\goth p}m\!i\hbox{\goth h}box{\goth g}amma)^{d-1} dr$$
is \,$\hbox{\goth t}ext{O}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl((1\!+\!\hbox{\goth h}box{\goth g}amma)^{-N}e^{\,2\hbox{\goth h}box{\goth g}amma(R-|t|)}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)$.
\hbox{\goth e}nd{proof}
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\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
|\hbox{\goth h}box{\goth n}u(z\hbox{\goth s}igma)|\hbox{\goth h}box{\goth l}e\,C\,|z|^{|\hbox{\goth m}athcal{R}_0^+|}\,(1\!+\!|z|)^{2|k|-|\hbox{\goth m}athcal{R}_0^+|}\,.
\hbox{\goth e}nd{equation*}
The estimates (ii) and (iii) follow easily from these considerations.
\hbox{\goth e}nd{proof}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{proposition}
\hbox{\goth h}box{\goth l}abel{DifferenceEstimate1}
Assume that the dimension $d$ is odd
and that all multiplicities are integers.
Then there exists a constant \,$C\!>\!0$
$($depending on the initial data
\hbox{\goth h}box{\goth l}inebreak
$f,g\!\in\!\hbox{\goth m}athcal{C}_R^\infty(\hbox{\goth m}athfrak{a}))$
such that, for every \,$\hbox{\goth h}box{\goth g}amma\!\hbox{\goth h}box{\goth g}e\!0$ and \,$t\!\in\!\hbox{\goth m}athbb{R}$,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
|\hbox{\goth m}athcal{P}[u](t)\hbox{\goth h}space{-.3mm}-\hbox{\goth h}space{-.2mm}\hbox{\goth m}athcal{K}[u](t)|
\hbox{\goth h}box{\goth l}e C\,e^{\,2\hbox{\goth h}box{\goth g}amma(R-|t|)}\,.
\hbox{\goth e}nd{equation*}
\hbox{\goth e}nd{proposition}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{proof}
Evenness allows us to rewrite \hbox{\goth e}qref{Difference2} as follows\,:
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth m}athcal{P}[u](t)\hbox{\goth h}space{-.3mm}-\hbox{\goth h}space{-.2mm}\hbox{\goth m}athcal{K}[u](t)
={\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{c_0}4}
{{\displaystyle}playstyle\int_{-\infty}^{+\infty}}\hbox{\goth h}space{-1mm}e^{\,i2tr}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\Phi(r)\!-\!ir\Psi(r)\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,r^{d-1}\,dr\,.
\hbox{\goth e}nd{equation*}
Let us shift the contour of integration
from \,$\hbox{\goth m}athbb{R}$ \,to \,$\hbox{\goth m}athbb{R}\hbox{\goth h}space{-.75mm}\hbox{\goth h}box{\goth p}m\!i\hbox{\goth h}box{\goth g}amma$,
according to the sign of~$t$,
and estimate the resulting integral, using Lemma \ref{EstimatePhiPsi}.iii.
As a result, the difference of energy
$$\hbox{\goth m}athcal{P}[u](t)-\hbox{\goth m}athcal{K}[u](t)= \hbox{\goth h}box{\goth f}rac{c_0}4\,e^{-2\hbox{\goth h}box{\goth g}amma|t|}
{\int_{-\infty}^{+\infty}}
e^{\,i2tr}\,\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{\Phi(r\!\hbox{\goth h}box{\goth p}m\!i\hbox{\goth h}box{\goth g}amma)\!
-i(r\!\hbox{\goth h}box{\goth p}m\!\hbox{\goth h}box{\goth g}amma)\,\Psi(r\!\hbox{\goth h}box{\goth p}m\!i\hbox{\goth h}box{\goth g}amma)\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,
(r\!\hbox{\goth h}box{\goth p}m\!i\hbox{\goth h}box{\goth g}amma)^{d-1} dr$$
is \,$\hbox{\goth t}ext{O}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl((1\!+\!\hbox{\goth h}box{\goth g}amma)^{-N}e^{\,2\hbox{\goth h}box{\goth g}amma(R-|t|)}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)$.
\hbox{\goth e}nd{proof}
As an immediate consequence of the above statement and in view of the fact that
$\hbox{\goth h}box{\goth g}amma_0=\infty$ when $k$ is integer valued, we deduce the strict equipartition of energy \hbox{\goth e}qref{StrictEquipartition}
for \,$|t|\!\hbox{\goth h}box{\goth g}e\!R,$ by letting $\hbox{\goth h}box{\goth g}amma\!\hbox{\goth t}o\!\infty.$
Henceforth, we will drop the above assumption on $k.$ By resuming the proof of Proposition \ref{DifferenceEstimate1}
and using Lemma \ref{EstimatePhiPsi}.ii
instead of Lemma \ref{EstimatePhiPsi}.iii,
we obtain the following result.
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{proposition}
\hbox{\goth h}box{\goth l}abel{EstimateDifference2}
Assume that the dimension $d$ is odd.
Then, for every \,$0\!<\!\hbox{\goth h}box{\goth g}amma\!<\!\hbox{\goth h}box{\goth g}amma_0$,
there is a constant \,$C\!>\!0$
$(\hbox{\goth t}ext{depending on the initial data
$f,g\!\in\!\hbox{\goth m}athcal{C}_R^\infty(\hbox{\goth m}athfrak{a})$})$
such that
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
|\hbox{\goth m}athcal{P}[u](t)\hbox{\goth h}space{-.3mm}-\hbox{\goth h}space{-.2mm}\hbox{\goth m}athcal{K}[u](t)|
\hbox{\goth h}box{\goth l}e C\,e^{-2\hbox{\goth h}box{\goth g}amma|t|}
\hbox{\goth q}uad\hbox{\goth h}box{\goth f}orall\;t\!\in\!\hbox{\goth m}athbb{R}\,.
\hbox{\goth e}nd{equation*}
\hbox{\goth e}nd{proposition}
As a corollary,
we obtain the asymptotic equipartition of energy
\hbox{\goth e}qref{AsymptoticEquipartition} in the odd dimensional case,
with an exponential rate of decay.
In the even dimensional case,
the expression \hbox{\goth e}qref{Difference2} cannot be handled by complex analysis
and we proceed differently.
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{proposition}
\hbox{\goth h}box{\goth l}abel{EstimateDifference2}
Assume that the dimension $d$ is even.
Then there is a constant \,$C\!>\!0$
$(\hbox{\goth t}ext{depending on the initial data
$f,g\!\in\!\hbox{\goth m}athcal{C}_R^\infty(\hbox{\goth m}athfrak{a})$})$
such that
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
|\hbox{\goth m}athcal{P}[u](t)\hbox{\goth h}space{-.3mm}-\hbox{\goth h}space{-.2mm}\hbox{\goth m}athcal{K}[u](t)|
\hbox{\goth h}box{\goth l}e C\,(1\!+\!|t|)^{-d-|\hbox{\goth m}athcal{R}_0^+|}
\hbox{\goth q}uad\hbox{\goth h}box{\goth f}orall\;t\!\in\!\hbox{\goth m}athbb{R}\,.
\hbox{\goth e}nd{equation*}
\hbox{\goth e}nd{proposition}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{proof}
The problem lies in the decay at infinity.
According to lemma \ref{EstimatePhiPsi},
$\Phi(r)$ and $\Psi(r)$ are divisible by $r^{D}$,
where $D\!=\!|\hbox{\goth m}athcal{R}_0^+|$.
Let us integrate \hbox{\goth e}qref{Difference2}
\,$d\!+\!D$ times by parts.
This way
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\int_{\,0}^{+\infty}\hbox{\goth h}space{-1mm}
\cos(2tr)\,r^{d-1}
\hbox{\goth u}nderbrace{\Bigl\{
\int_{S(\hbox{\goth m}athfrak{a})}
\hbox{\goth m}athcal{F}\hbox{\goth h}space{-.4mm}g(r\hbox{\goth s}igma)\,
\widetilde{\hbox{\goth m}athcal{F}}\hbox{\goth h}space{-.4mm}g(r\hbox{\goth s}igma)\,
\hbox{\goth h}box{\goth n}u(r\hbox{\goth s}igma)\,d\hbox{\goth s}igma
\Bigr\}}_{\widetilde\Phi(r)}
dr
\hbox{\goth e}nd{equation*}
becomes
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\hbox{\goth h}box{\goth p}m\;{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{1\hbox{\goth t}ext{ \,or \,}0}{(2t)^{d+D}}}\,
{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{(d\,+D)\,!}{(D+1)\,!}}\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl({\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac\hbox{\goth h}box{\goth p}artial{\hbox{\goth h}box{\goth p}artial r}}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)^{D+1}
\widetilde\Phi(r)\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}ig|_{r=0}\,
\hbox{\goth h}box{\goth p}m\int_{\,0}^{+\infty}\hbox{\goth h}space{-1mm}
{\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac{\cos(2tr)\hbox{\goth t}ext{ or }\hbox{\goth s}in(2tr)}{(2t)^{d+D}}}\,
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl({\hbox{\goth t}extstyle\hbox{\goth h}box{\goth f}rac\hbox{\goth h}box{\goth p}artial{\hbox{\goth h}box{\goth p}artial r}}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)^{d+D}
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl\{r^{d-1}\,\widetilde\Phi(r)\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr\}\,dr
\hbox{\goth e}nd{equation*}
which is \,$\hbox{\goth t}ext{O}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(|t|^{-d-D}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)$.
Similarly
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\int_{\,0}^{+\infty}\hbox{\goth h}space{-1mm}
\cos(2tr)\,r^{d+1}\,
\Bigl\{\int_{S(\hbox{\goth m}athfrak{a})}\!
\hbox{\goth m}athcal{F}\!f(r\hbox{\goth s}igma)\,\widetilde{\hbox{\goth m}athcal{F}}\!f(r\hbox{\goth s}igma)\,
\hbox{\goth h}box{\goth n}u(r\hbox{\goth s}igma)\,d\hbox{\goth s}igma\Bigr\}\,dr
=\,\hbox{\goth t}ext{O}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(|t|^{-d-D-2}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)
\hbox{\goth e}nd{equation*}
and
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{equation*}
\int_{\,0}^{+\infty}\hbox{\goth h}space{-1mm}
\cos(2tr)\,r^{d}\,\Psi(r)\,dr
=\,\hbox{\goth t}ext{O}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(|t|^{-d-D-1}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr).
\hbox{\goth e}nd{equation*}
This concludes the proof of Proposition \ref{EstimateDifference2}.
\hbox{\goth e}nd{proof}
As a corollary,
we obtain the asymptotic equipartition of energy
\hbox{\goth e}qref{AsymptoticEquipartition} in the even dimensional case,
with a polynomial rate of decay.
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{remark}
Our result may not be optimal.
In the $W$\hbox{\goth h}space{-1mm}-invariant case,
one obtains indeed the rate of decay
\,$\hbox{\goth t}ext{\rm O}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(\{1\!+\!|t|\}^{-d-2|\hbox{\goth m}athcal{R}_0^+\!|}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)$
as in \cite{BOS}.
\hbox{\goth e}nd{remark}
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As a corollary,
we obtain the asymptotic equipartition of energy
\hbox{\goth e}qref{AsymptoticEquipartition} in the even dimensional case,
with a polynomial rate of decay.
\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}egin{remark}
Our result may not be optimal.
In the $W$\hbox{\goth h}space{-1mm}-invariant case,
one obtains indeed the rate of decay
\,$\hbox{\goth t}ext{\rm O}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igl(\{1\!+\!|t|\}^{-d-2|\hbox{\goth m}athcal{R}_0^+\!|}\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}igr)$
as in \cite{BOS}.
\hbox{\goth e}nd{remark}
\references \hbox{\goth h}box{\goth n}extref{S}{Ben Said, S.}{\hbox{\goth e}m Huygens' principle for
the wave equation associated with the trigonometric Dunkl-Cherednik
operators} {Math. Research Letters {\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}f 13} (2006), no. 1, 43--58 }
\hbox{\goth h}box{\goth n}extref{SO} {Ben Said, S. and {\O}rsted, B.}{\hbox{\goth e}m The wave equation
for Dunkl operators} { Math. (N.S.) {\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}f 16} (2005), no. 3--4,
351--391 }
\hbox{\goth h}box{\goth n}extref{BO}{Branson, T. and \'Olafsson, G.}{\hbox{\goth e}m Equipartition of energy for waves in symmetric space}
{J. Funct. Anal. {\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}f 97} (1991), no. 2, 403--416}
\hbox{\goth h}box{\goth n}extref{BOP1} {Branson, T. \'Olafsson, G. Pasquale, A.}{\hbox{\goth e}m The
Paley--Wiener theorem and the local Huygens' principle for compact
symmetric spaces}{ Math. (N.S.) {\hbox{\goth h}box{\hbox{\goth h}box{\goth g}oth b}f 16} (2005), no. 3--4, 393--428
}
\hbox{\goth h}box{\goth n}extref{BOP2} {Branson, T. and \'Olafsson, G.} {\hbox{\goth e}m The
Paley-Wiener theorem for the Jacobi transform and the local Huygens'
principle for root systems with even multiplicities}{ Math. (N.S.)
16 (2005), no. 3--4, 429--442 }
\hbox{\goth h}box{\goth n}extref{BOS} {Branson, T. \'Olafsson, G.Schlichtkrull, H.} {\hbox{\goth e}m
Huygens' principle in Riemannian symmetric spaces} {Math. Ann. 301
(1995), no. 3, 445--462}
\hbox{\goth h}box{\goth n}extref{KY} {El Kamel, J. Yacoub, C.} {\hbox{\goth e}m Huygens' principle and
equipartition of energy for the modified wave equation associated to
a generalized radial Laplacian} {Ann. Math Blaise Pascal 12 (2005),
no. 1, 147--160}
\hbox{\goth h}box{\goth n}extref{H} { Helgason, S.} {\hbox{\goth e}m Geometric analysis on symmetric
spaces} {Math. Surveys Monographs 39, Amer. Math. Soc. (1994)}
\hbox{\goth h}box{\goth n}extref{O1} { Opdam, E. M. }{\hbox{\goth e}m Harmonic analysis for certain
representations of graded Hecke algebras} {Acta. Math. 175 (1995),
75--121}
\hbox{\goth h}box{\goth n}extref{O2} { Opdam, E. M. }{\hbox{\goth e}m Lecture notes on Dunkl operators
for real and complex reflection groups}{ Math. Soc. Japan Mem. 8
(2000) }
\hbox{\goth h}box{\goth l}astpage
\hbox{\goth e}nd{document}
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\begin{document}
\title{Sequence entropy tuples and mean sensitive tuples}
\author[J. Li]{Jie Li }
\address[Jie Li]{School of Mathematics and Statistics, Jiangsu Normal
University, Xuzhou, Jiangsu, 221116, P.R. China}
\email{[email protected]}
\author[C. Liu]{Chunlin Liu }
\address[Chunlin Liu]{CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, P.R. China}
\email{[email protected]}
\author[S. Tu]{Siming Tu }
\address[Siming Tu]{ School of Mathematics (Zhuhai), Sun Yat-sen University,
Zhuhai, Guangdong 519082, P.R. China}
\email{[email protected]}
\author[T. Yu]{Tao Yu }
\address[Tao Yu]{Department of Mathematics, Shantou University, Shantou 515063, P. R. China}
\email{[email protected]}
\begin{abstract}
Using the idea of local entropy theory, we characterize the sequence entropy tuple via mean forms of the sensitive tuple in both topological and measure-theoretical senses.
For the measure-theoretical sense, we show that for an ergodic measure-preserving system, the
$\mu$-sequence entropy tuple, the $\mu$-mean sensitive tuple and the
$\mu$-sensitive in the mean tuple coincide, and give an example to show that the ergodicity condition is necessary.
For the topological sense, we show that for a certain class of minimal systems, the
mean sensitive tuple is the sequence entropy tuple.
\end{abstract}
\date{\today}
\subjclass[2020]{37A35, 37B05}
\keywords{Sequence entropy tuples; mean sensitive tuples; sensitive in the mean tuples}
\maketitle
\section{Introduction}
By a {\it topological dynamical system} ({\it t.d.s.} for short) we mean a pair $(X,T)$, where $X$ is a compact metric space with a metric $d$ and $T$ is a homeomorphism from $X$ to itself.
A point $x\in X$ is called a \textit{transitive point} if ${\mathrm{Orb}(x,T)}=\{x,Tx,\ldots\}$ is dense in $X$. A t.d.s. $(X,T)$ is called \textit{minimal} if all points in $X$ are transitive points.
Denote by $\B_X$ all Borel measurable subsets of $X$. A Borel (probability) measure $\mu$ on $X$ is called $T$-\textit{invariant} if $\mu(T^{-1}A)=\mu(A)$ for any $A\in \mathcal{B}_X$.
A $T$-invariant measure $\mu$ on $X$ is called \textit{ergodic} if $B\in \mathcal{B}_X$
with $T^{-1}B=B$ implies $\mu(B)=0$ or $\mu(B)=1$. Denote by $M(X, T)$ (resp. $M^e(X, T)$) the collection of all $T$-invariant measures (resp. all ergodic measures) on $X$.
For $\mu \in M(X,T)$, the \textit{support} of $\mu$ is defined by
$\supp(\mu )=\{x\in X\colon \mu (U)>0\text{ for any neighbourhood }U\text{ of
}x\}$. Each measure $\mu\in M(X,T)$ induces a {\it measure-preserving system} ({\it m.p.s.} for short) $(X,\B_X,\mu, T)$.
It is well known that the entropy can be used to measure the local complexity of the structure of orbits in a given system. One may naturally ask how to characterize the entropy in a local way.
The related research started from the series of pioneering papers of Blanchard et al \cite{B1992, B1993, B1997, B1995}, in which the notions of entropy pairs and entropy pairs for a measure were introduced. From then on entropy pairs have been intensively studied by many researchers. Huang and Ye \cite{HY06} extended the notions from pairs to finite tuples, and showed that
if the entropy of a given system is positive, then there are entropy $n$-tuples
for any $n\in \mathbb{N}$ in both topological and measurable settings.
The sequence entropy was introduced by Ku\v shnirenko \cite{Kus} to establish the relation between spectrum theory and entropy theory. As in classical local entropy theory, the sequence entropy can also be localized. In \cite{HLSY03, HMY04} authors investigated the sequence entropy pairs, sequence entropy tuples and sequence
entropy tuples for a measure, respectively.
Using tools from combinatorics, Kerr and Li \cite{KL07, KL09} studied (sequence) entropy tuples, (sequence)
entropy tuples for a measure and IT-tuples via independence sets.
Huang and Ye \cite{HY09} showed that a system has a sequence entropy $n$-tuple
if and only if its maximal pattern entropy is no less than $\log n$ in both topological and measurable settings.
More introductions and applications of the local entropy theory can refer to a survey \cite{GY09}.
In addition to the entropy, the sensitivity is another candidate to describe the complexity of a system, which was first
used by Ruelle \cite{Ruelle1977}. In \cite{X05}, Xiong introduced a multi-variant version of the sensitivity, called the $n$-sensitivity.
\begin{comment}
According to Auslander and Yorke \cite{AY80}
a t.d.s. $(X,T)$ is called \emph{sensitive}
if there exists $\delta>0$ such that
for every opene (open and non-empty)
subset $U$, there exist $x_1,x_2\in U$ and $m\in\mathbb{N}$ with $d(T^mx_1,T^mx_2)>\delta$.
In \cite{X05}, Xiong introduced a multi-variate version of sensitivity, called $n$-sensitivity.
\end{comment}
Motivated by the local entropy theory, Ye and Zhang \cite{YZ08} introduced the notion of sensitive tuples. Particularly, they showed that a transitive t.d.s. is $n$-sensitive if and only if it has a sensitive $n$-tuple;
and a sequence entropy $n$-tuple of a minimal t.d.s. is a sensitive $n$-tuple.
For the converse, Maass and Shao \cite{MS07} showed that in a minimal t.d.s., if a sensitive $n$-tuple is a minimal point of the $n$-fold product t.d.s. then it is a sequence entropy $n$-tuple.
\begin{comment}
They introduced the notions of $n$-sensitivity for a measure $\mu$ and sensitive $n$-tuple for $\mu$ and showed that a t.d.s. with an ergodic measure $\mu$ is $n$-sensitive for $\mu$ if and only if it has a sensitive $n$-tuple for $\mu$;
and for a t.d.s. with an ergodic
measure $\mu$, sequence entropy $n$-tuple for $\mu$ is a sensitive $n$-tuple for $\mu$.
\end{comment}
Recently, Li, Tu and Ye \cite{LTY15} studied the sensitivity in the mean form.
Li, Ye and Yu \cite{LY21,LYY22} further studied the multi-version of mean sensitivity and its local representation, namely, the mean $n$-sensitivity and the mean $n$-sensitive tuple. One naturally wonders if there is still a characterization of sequence entropy tuples via mean sensitive tuples.
By the results of \cite{ FGJO, GJY21,KL07,LYY22} one can see that a sequence entropy tuple is not always a mean sensitive tuple even in a minimal t.d.s. Nonetheless, the works of \cite{DG16,Huang06,LTY15} yield that every minimal mean sensitive t.d.s. (i.e. has a mean sensitive pair by \cite{LYY22}) is not tame (i.e. exists an IT pair by \cite{KL07}).
So generally, we conjecture that for any minimal t.d.s., a mean sensitive $n$-tuple is an IT $n$-tuple and so a sequence entropy $n$-tuple by \cite[Theorem 5.9]{KL07}.
Now we can answer this question under an additional condition. Namely,\begin{thm}\rightarrowbel{thm:ms=>it}
Let $(X,T)$ be a minimal t.d.s. and $\pi: (X,T)\rightarrow (X_{eq},T_{eq})$ be the factor map
to its maximal equicontinuous factor which is almost one to one. Then for $2\le n\in\mathbb{N}$,
$$MS_n(X,T)\subset IT_n(X,T),$$
where $MS_n(X,T)$ denotes all the mean sensitive $n$-tuples and $IT_n(X,T)$ denotes all the IT $n$-tuples.
\end{thm}
In the parallel measure-theoretical setting, Huang, Lu and Ye \cite{HLY11} studied measurable sensitivity and its local representation.
The notion of $\mu$-mean sensitivity for an invariant measure $\mu$ on a t.d.s. was studied by Garc\'{\i}a-Ramos \cite{G17}.
Li \cite{L16} introduced the notion of the $\mu$-mean $n$-sensitivity,
and showed that
an ergodic m.p.s. is $\mu$-mean $n$-sensitive if and only if its maximal pattern entropy is no less than $\log n$.
The authors in \cite{LYY22} introduced the notion of the $\mu$-$n$-sensitivity in the mean, which was
\begin{comment}
if there is $\delta>0$ such that for any Borel subset $A$ of $X$ with $\mu(A)>0$ there are $m\in \mathbb{N}$ and $n$ pairwise distinct points $x_1^m,x_2^m,\dots,x_n^m\in A$ such that
$$
\frac{1}{m}\sum_{k=0}^{m-1}\min_{1\le i\neq j\le n} d(T^k x_i^m, T^k x_j^m)>\delta.
$$
By definitions $\mu$-sensitivity in the mean tuple seems weaker than $\mu$-mean sensitivity tuple, however, they are
\end{comment}
proved to be equivalent to the $\mu$-mean $n$-sensitivity in the ergodic case.
Using the idea of localization, the authors \cite{LY21} introduced the notion of the $\mu$-mean sensitive tuple and showed that
every $\mu$-entropy tuple of an ergodic m.p.s. is a $\mu$-mean sensitive tuple. A natural question is left open in \cite{LY21}:
\begin{ques}
Is there a characterization of $\mu$-sequence entropy tuples via $\mu$-mean sensitive tuples?
\end{ques}
The authors in \cite{LT20} introduced a weaker notion named the density-sensitive tuple and showed that every $\mu$-sequence entropy tuple of an ergodic m.p.s. is a $\mu$-density-sensitive tuple.
In this paper, we give a positive answer to this question. Namely,
\begin{thm}\rightarrowbel{cor:se=sm}
Let $(X,T)$ be a t.d.s., $\mu\in M^e(X,T)$ and $2\le n\in \mathbb{N}$. Then
the $\mu$-sequence entropy $n$-tuple, the $\mu$-mean sensitive $n$-tuple and the
$\mu$-$n$-sensitive in the mean tuple coincide.
\end{thm}
By the definitions, it is easy to see that a $\mu$-mean sensitive $n$-tuple must be a $\mu$-$n$-sensitive in the mean tuple. Thus, Theorem \ref{cor:se=sm} is a direct corollary of the following two theorems.
\begin{thm}\rightarrowbel{thm:sm=>se}
Let $(X,T)$ be a t.d.s., $\mu\in M(X,T)$ and $2\le n\in \mathbb{N}$. Then
each $\mu$-$n$-sensitive in the mean tuple is a $\mu$-sequence entropy $n$-tuple.
\end{thm}
\begin{thm}\rightarrowbel{thm:se=>ms}
Let $(X,T)$ be a t.d.s., $\mu\in M^e(X,T)$ and $2\le n\in \mathbb{N}$. Then
each $\mu$-sequence entropy $n$-tuple is a $\mu$-mean sensitive $n$-tuple.
\end{thm}
In fact, Theorem \ref{thm:sm=>se} shows a bit more than Theorem \ref{cor:se=sm}, as for a $T$-invariant measure $\mu$ which is not ergodic, every $\mu$-$n$-sensitive in the mean tuple is still a $\mu$-sequence entropy $n$-tuple.
However, the following result shows that ergodicity of $\mu$ in Theorem \ref{thm:se=>ms} is necessary.
\begin{thm}\rightarrowbel{thm:sm=/=se}
For every $2\le n\in \mathbb{N}$, there exist a t.d.s. $(X,T)$ and $\mu\in M(X,T)$ such that there is a $\mu$-sequence entropy $n$-tuple but it is not a $\mu$-$n$-sensitive in the mean tuple.
\end{thm}
It is fair to note that Garc{\'i}a-Ramos told us that at the same time, he with Mu{\~n}oz-L{\'o}pez had also got a completely independent proof of the equivalence of the sequence entropy pair and the mean sensitive pair in the ergodic case \cite{GM22}. Their proof relies on the deep equivalent characterization of measurable sequence entropy pairs developed by Kerr and Li \cite{KL09} using the combinatorial notion of independence. Our results provide more information in general case, and the proofs work on the classical definition of sequence entropy pairs introduced in \cite{HMY04}. It is worth noting that the proofs depend on a new interesting ergodic measure decomposition result (Lemma \ref{0726}), which was applied to prove the profound Erd\"os's conjectures in the number theory by Kra, Moreira, Richter and Robertson \cite{KMRR,KMRR1}. This decomposition may have more applications because it has the hybrid topological and Borel structures.
The outline of the paper is the following. In Sec. \ref{sec2}, we recall some basic notions that we
will use in the paper. In Sec. \ref{sec3}, we prove Theorem \ref{thm:sm=>se}. In Sec. \ref{sect:proof of thm se=>ms}, we show Theorem \ref{thm:se=>ms} and Theorem \ref{thm:sm=/=se}. In Sec. \ref{sec5}, we study the mean sensitive tuple and the sequence entropy in the topological sense and show Theorem \ref{thm:ms=>it}.
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\section{Preliminaries}\rightarrowbel{sec2}
Throughout the paper, denote by $\mathbb{N}$ and ${\mathbb{Z}}_{+}$ the
collections of natural numbers $\{1,2,\dots\}$ and non-negative integers $\{0,1,2,\dots\}$, respectively.
For $F\subset \mathbb{Z}_+$, denote by $\#\{F\}$ (or simply write $\#F$ when it is clear from the context) the cardinality of $F$. The \emph{upper density} $\overline{D}(F)$ of $F$ is defined by
$$
\overline{D}(F)=\limsup_{n\to\infty} \frac{\#\{F\cap[0,n-1]\}}{n}.
$$
Similarly, the \emph{lower density} $\underline{D}(F)$ of $F$ can be given by
$$
\underline{D}(F)=\liminf_{n\to\infty} \frac{\#\{F\cap[0,n-1]\}}{n}.
$$
If $\overline{D}(F)=\underline{D}(F)$, we say that the \textit{density} of $F$ exists and equals to the common value, which is written as $D(F)$.
Given a t.d.s. $(X,T)$ and $n\in \mathbb{N}$, denote by $X^{(n)}$ the $n$-fold product of $X$. Let $\Delta_n(X)=\{(x,x,\dots, x)\in X^{(n)}\colon x\in X\}$ be the diagonal of $ X^{(n)}$ and $\Delta_n^\prime(X)=\{(x_1,x_2,...,x_n)\in X^{(n)}: x_i=x_j \text{ for some } 1\le i\neq j\le n \}$.
If a closed subset $Y\subset X$ is $T$-invariant in the sense of $TY= Y$,
then the restriction $(Y, T|_Y)$ (or simply write $(Y,T)$ when it is clear from the context) is also a t.d.s., which is called a \textit{subsystem} of $(X,T)$.
Let. $(X,T)$ be a t.d.s., $x\in X$ and $U,V\subset X$. Denote by
$$
N(x,U)=\{n\in\mathbb{Z}_+ \colon T^n x\in U\} \ \text{ and }\ N(U,V)=\{n\in\mathbb{Z}_+: U\cap T^{-n}V\neq\emptyset\}.
$$
A t.d.s. $(X,T)$ is called \textit{transitive} if $N(U,V)\neq\emptyset$ for all non-empty open subsets $U,V$ of $X$. It is well known that the set of all transitive points in a transitive t.d.s. forms a dense $G_\delta$ subset of $X$ .
Given two t.d.s. $(X, T)$ and $(Y,S)$,
a map $\pi\colon X\to Y$ is called a \textit{factor map}
if $\pi$ is surjective and continuous such that $\pi\circ T=S\circ\pi$,
and in which case $(Y,S)$ is referred to be a \textit{factor} of $(X, T)$. Furthermore,
If $\pi$ is a homeomorphism, we say that $(X,T)$ is \textit{conjugate} to $(Y,S)$.
A t.d.s. $(X,T)$ is called \textit{equicontinuous} (resp. \textit{mean equicontinuous}) if for any $\epsilon>0$ there is $\delta>0$ such that if $x,y\in X$ with $d(x,y)<\delta$ then $\max_{k\in\mathbb{Z}_+}d(T^kx,T^ky)<\epsilon$ (resp. $\limsup_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}d(T^kx,T^ky)<\epsilon$).
Every t.d.s. $(X, T)$ is known to have a maximal equicontinuous factor (or a maximal mean equicontinuous factor \cite{LTY15}). More studies on mean equicontinuous systems can see the recent survey \cite{LYY}.
In the following of this section, we
fix a t.d.s. $(X,T)$ with a measure $\mu\in M(X,T)$. The {\it entropy of a finite measurable partition $\alpha=\left\{A_1, A_2, \ldots, A_k\right\}$ of $X$ } is defined by
$
H_\mu(\alpha)=-\sum_{i=1}^k \mu\left(A_i\right) \log \mu\left(A_i\right),
$
where $0 \log 0$ is defined to be 0. Moreover, we define the {\it sequence entropy of $T$ with respect to $\alpha$ along an increasing sequence $S=\left\{s_i\right\}_{i=1}^{\infty}$ of $\mathbb{Z}_+$ } by
$$
h_\mu^{S}(T, \alpha)=\limsup _{n\rightarrow \infty} \frac{1}{n} H_\mu\left(\bigvee_{i=1}^n T^{-s_i} \alpha\right).
$$
The {\it sequence entropy of $T$ along the sequence $S$} is
$$
h_\mu^{S}(T)=\sup _{\alpha} h_\mu^{S}(T, \alpha),
$$
where the supremum takes over all finite measurable partitions.
Correspondingly, the {\it topological sequence entropy of $T$ with respect to $S$ and a finite open cover $\mathcal{U}$ } is
$$
h^{S}(T, \mathcal{U})=\limsup _{n \rightarrow\infty} \frac{1}{n} \log N\left(\bigvee_{i=1}^n T^{-s_i} \mathcal{U}\right),
$$
where $N\left(\bigvee_{i=1}^n T^{-s_i} \mathcal{U}\right)$ is the minimum among the cardinalities of all sub-families of
$\bigvee_{i=1}^n T^{-s_i} \mathcal{U}$ covering $X$.
The {\it topological sequence entropy of $T$ with respect to $S$ } is defined by $$h^{S}(T)=\sup _{\mathcal{U}} h^{S}(T, \mathcal{U}),$$
where the supremum takes over all finite open covers.
Let $(x_i)_{i=1}^n\in X^{(n)}$. A finite cover $\mathcal{U}=\{U_1,U_2,\ldots,U_k\}$ of $X$ is said to be an {\it admissible cover} with respect to $(x_i)_{i=1}^n$ if for each $1\leq j\leq k$ there exists $1\leq i_j\leq n$ such that $x_{i_j}\notin\overline{U_j}$. Analogously, we define admissible partitions with respect to $(x_i)_{i=1}^n$.
\begin{defn}[\cite{HMY04},\cite{MS07}]An $n$-tuple $(x_i)_{i=1}^n\in X^{(n)}\setminus \Delta_n(X)$, $n\geq 2$ is called
\begin{itemize}
\item a sequence entropy $n$-tuple for $\mu$ if for any admissible finite Borel measurable partition $\alpha$ with respect to $(x_i)_{i=1}^n$, there exists a sequence $S=\{m_i\}_{i=1}^{\infty}$ of $\mathbb{Z}_+$ such that $h^{S}_{\mu}(T,\alpha)>0$. Denote by $SE_n^{\mu}(X,T)$ the set of all sequence entropy $n$-tuples for $\mu$.
\item a sequence entropy $n$-tuple if for any admissible finite open cover $\mathcal{U}$ with respect to $(x_i)_{i=1}^n$, there exists a sequence $S=\{m_i\}_{i=1}^{\infty}$ of $\mathbb{Z}_+$ such that $h^{S}(T,\mathcal{U})>0$. Denote by $SE_n(X,T)$ the set of all sequence entropy $n$-tuples.
\end{itemize}
\end{defn}
We say that $f\in L^2(X,\B_X,\mu)$ is {\it almost periodic} if $\{f\circ T^n : n\in \mathbb{Z}_+\}$ is precompact
in $L^2(X,\B_X,\mu)$. The set of all
almost periodic functions is denoted by $H_c$,
and there exists a $T$-invariant $\sigma$-algebra $\mathcal{K}_\mu \subset \B_X$
such that $H_c= L^2(X,\mathcal{K}_\mu,\mu)$, $\mathcal{K}_\mu$ is called the Kronecker algebra of $(X, \B_X,\mu, T )$. The product $\sigma$-algebra of $X^{(n)}$ is denoted by $\mathcal{B}_X^{(n)}$. Define the measure $\rightarrowmbda_n(\mu)$ on $\mathcal{B}_X^{(n)}$ by letting $$\rightarrowmbda_n(\mu)(\prod_{i=1}^nA_i)=\int_{X}\prod_{i=1}^n\mathbb{E}(1_{A_i}|\mathcal{K}_\mu)d\mu.$$
Note that $SE_n^{\mu}(X,T)=\supp(\rightarrowmbda_n(\mu))\setminus \Delta_n(X)$ \cite[Theorem 3.4]{HMY04}.
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\section{Proof of Theorem \ref{thm:sm=>se}}\rightarrowbel{sec3}
\begin{defn}[\cite{LY21}]\rightarrowbel{defn:mu mean n-sensitive tuple}
For $2\le n\in \mathbb{N}$ and a t.d.s. $(X,T)$ with $\mu\in M(X,T)$, we say that the $n$-tuple $(x_1,x_2,\dotsc,x_n)\in X^{(n)}\setminus \Delta_n(X)$ is
\begin{enumerate}
\item a \textit{$\mu$-mean $n$-sensitive tuple} if for any open neighbourhoods $U_i$ of $x_i$ with $i=1,2,\dotsc,n$, there is $\delta> 0$ such that for any $A\in \B_X$ with $\mu(A)>0$ there are $y_1,y_2,\dotsc,y_n\in A$ and a subset $F$ of $\mathbb{Z}_+$ with $\overline{D}(F)>\delta$ such that $T^k y_i \in U_i$ for all $i=1,2,\dots,n$ and $k\in F$.
\item a \textit{$\mu$-$n$-sensitive in the mean tuple} if for any $\tau>0$, there is $\delta=\delta(\tau)> 0$ such that for any $A\in\B_X$ with $\mu(A)>0$ there is $m\in \mathbb{N}$ and $y_1^m,y_2^m,\dotsc,y_n^m\in A$ such that
$$
\frac{\#\{0\le k\le m-1: T^ky_i^m\in B(x_i,\tau), i=1,2,\ldots,n\}}{m}>\delta.
$$
\end{enumerate}
\end{defn}
We denote the set of all $\mu$-mean $n$-sensitive tuples (resp. $\mu$-$n$-sensitive in the mean tuples) by $MS_n^\mu(X,T)$ (resp. $SM_n^\mu(X,T)$). We call an $n$-tuple $(x_1,x_2,\dotsc,x_n)\in X^{(n)}$ \textit{essential} if $x_i\neq x_j$ for each $1\le i<j\le n$, and at this time we write the collection of all essential $n$-tuples in $MS_n^\mu(X,T)$ (resp. $SM_n^\mu(X,T)$) as $MS_n^{\mu,e}(X,T)$ (resp. $SM_n^{\mu,e}(X,T)$).
\begin{comment}
\begin{defn}[\cite{LYY22}]\rightarrowbel{defn:mu-n-sensitive in the mean}
For $2\le n\in \mathbb{N}$ and a t.d.s. $(X,T)$ with $\mu\in M(X,T)$, we say that $(X,T)$ is \textit{$\mu$-$n$-sensitive in the mean} if there is $\delta>0$ such that for any Borel subset $A$ of $X$ with $\mu(A)>0$ there are $m\in \mathbb{N}$ and $n$ pairwise distinct points $x_1^m,x_2^m,\dots,x_n^m\in A$ such that
$$
\frac{1}{m}\sum_{k=0}^{m-1}\min_{1\le i\neq j\le n} d(T^k x_i^m, T^k x_j^m)>\delta.
$$
\end{defn}
\end{comment}
\begin{proof}[Proof of Theorem \ref{thm:sm=>se}]
It suffices to prove $SM_n^{\mu,e}(X,T)\subset SE_n^{\mu,e}(X,T)$.
Let $(x_1,\ldots,x_n)\in SM_n^{\mu,e}(X,T)$. Take $\alpha=\{A_1,\ldots,A_l\}$ as an admissible partition of $(x_1,\ldots,x_n)$. Then for each $1\le k\le l$, there is $i_k\in \{1,\ldots,n\}$ such that $x_{i_k}\notin \overline{A_k}$. Put $E_i=\{1\le k\le l: x_i\not\in \overline{A_k}\}$ for $1\le i\le n$. Obviously, $\cup_{i=1}^n E_i=\{1,\ldots,l\}$. Set
$$B_1=\cup_{k\in E_1}A_k, B_2=\cup_{k\in E_2\setminus E_1}A_k, \ldots, B_n=\cup_{k\in E_n\setminus(\cup_{j=1}^{n-1}E_j)}A_k.
$$
Then $\beta=\{B_1,\ldots,B_n\}$ is also an admissible partition of $(x_1,\ldots,x_n)$ such that $x_i\notin \overline{B_i}$ for all $1\le i\le n$. Without loss of generality, we assume $B_i\neq \emptyset$ for $1\le i\le n$.
It suffices to show that there exists a sequence $S=\{m_i\}_{i=1}^{\infty}$ of $\mathbb{Z}_+$ such that $h^{S}_{\mu}(T,\beta)>0,$ as $\alpha\succ\beta$.
Let $$h^*_\mu(T,\beta)=\sup \{h^{S}_{\mu}(T,\beta): S \ \text{is a sequence of } \mathbb{Z}_+ \}.$$
By \cite[Lemma 2.2 and Theorem 2.3]{HMY04}, we have $h^*_\mu(T,\beta)=H_\mu(\beta|\mathcal{K}_\mu)$,
where $\mathcal{K}_\mu$ is the Kronecker algebra of $(X,\B_X,\mu,T)$.
So it suffices to show $\beta\nsubseteq \mathcal{K}_\mu$.
We prove it by contradiction. Now we assume that $\beta\subseteq \mathcal{K}_\mu$. Then for each $i=1,\ldots,n$, $1_{B_i}$ is an almost periodic function. By \cite[Theorems 4.7 and 5.2]{Y19}, $1_{B_i}$ is a $\mu$-equicontinuous in the mean function. That is, for each $1\le i\le n$ and any $\tau>0$, there is a compact $K\subset X$ with $\mu(K)>1-\tau$ such that for any $\epsilon'>0$, there is $\delta'>0$ such that for all $m\in\mathbb{N}$, whenever $x,y\in K$ with $d(x,y)<\delta'$,
\begin{equation}\rightarrowbel{3}
\frac{1}{m}\sum_{t=0}^{m-1}|1_{B_i}(T^tx)- 1_{B_i}(T^ty)|<\epsilon'.
\end{equation}
On the other hand, take $\epsilon>0$ such that $B_\epsilon(x_i)\cap B_i=\emptyset$ for $i=1,\ldots,n$. Since $(x_1,\ldots,x_n)\in SM_n^{\mu,e}(X,T)$, there is $\delta:=\delta(\epsilon)>0$ such that for any $A\in \B_X$ with $\mu(A)>0$ there are $m\in\mathbb{N}$ and $y_1^m,\ldots,y_n^m\in A$ such that if we denote $C_m=\{0\le t\le m-1:T^ty_i^m\in B_\epsilon(x_i)\text{ for all }i=1,2,\ldots,n\}$ then $\#C_m \ge m\delta$. Since $ B_\epsilon(x_1)\cap B_1=\emptyset$, then $ B_\epsilon(x_1)\subset \cup_{i=2}^nB_i$. This implies that there is $i_0\in \{2,\ldots,n\}$ such that
$$
\# \{t\in C_m: T^ty_1^m\in B_{i_0} \}\ge \frac{\#C_m}{n-1}.
$$
For any $t\in C_m$, we have $T^ty_{i_0}^m\in B_\epsilon(x_{i_0})$, and then $T^ty_{i_0}^m\notin B_{i_0}$, as $B_\epsilon(x_{i_0})\cap B_{i_0}=\emptyset$. This implies that
\begin{equation}\rightarrowbel{e1}
\frac{1}{m}\sum_{t=0}^{m-1}|1_{B_{i_0}}(T^ty_1^m)-1_{B_{i_0}}(T^ty_{i_0}^m)|\ge\frac{\#C_m}{m(n-1)}\ge \frac{\delta}{n-1}.
\end{equation}
Choose a measurable subset $A\subset K$ such that $\mu(A)>0$ and $\diam(A)=\sup\{d(x,y):x,y\in A\}<\delta'$, and $\epsilon'=\frac{\delta}{2(n-1)}$. Then by \eqref{3}, for any $m\in\mathbb{N}$ and $x,y\in A$,
$$
\frac{1}{m}\sum_{t=0}^{m-1}|1_{B_{i_0}}(T^tx)- 1_{B_{i_0}}(T^ty)|<\frac{\delta}{2(n-1)},
$$
a contradiction with \eqref{e1}.
Thus, $SM_n^{\mu,e}(X,T)\subset SE_n^{\mu,e}(X,T)$.
\end{proof}
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\section{Proof of Theorem \ref{thm:se=>ms}}\rightarrowbel{sect:proof of thm se=>ms}
In Section 4.1, we first reduce Theorem \ref{thm:se=>ms} to just prove that it is true for the ergodic m.p.s.
with a continuous factor map to its Kronecker factor, and then we finish the proof of Theorem \ref{thm:se=>ms} under this assumption. In Section 4.2, we show the condition that $\mu$ is ergodic is necessary.
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\subsection{Ergodic case}
Throughout this section, we will use the following two types of factor maps between two m.p.s. $(X, \B_X,\mu, T)$ and $(Z, \B_Z,\nu, S)$.
\begin{enumerate}
\item \emph{Measurable factor maps:} a measurable map $\pi: X \rightarrow Z$ such that $\mu\circ\pi^{-1}=\nu$ and $\pi \circ T=S \circ \pi$ for $\mu$-a.e;
\item \emph{Continuous factor maps:} a topological factor map $\pi: X \rightarrow Z$ such that $\mu\circ\pi^{-1}=\nu$.
\end{enumerate}
If a continuous factor map $\pi$ such that $\pi^{-1}(\B_Z)=\mathcal{K}_\mu$,
$\pi$ is called a continuous factor map to its Kronecker factor.
The following result is a weaker version in \cite[Proposition 3.20]{KMRR}.
\begin{lem}\rightarrowbel{lem3}
Let $(X, \mathcal{B}_X,\mu, T)$ be an ergodic m.p.s. Then there exists an ergodic m.p.s. $(\tilde{X},\tilde{B}, \tilde{\mu}, \tilde{T})$ and a continuous factor map $\tilde{\pi}: \tilde{X} \rightarrow X$ such that $(\tilde{X},\tilde{B}, \tilde{\mu}, \tilde{T})$ has a continuous factor map to its Kronecker factor.
\end{lem}
The following result shows that we only need to prove $SE_n^{\mu}(X,T)\subset MS_n^{\mu}(X,T)$ for all ergodic m.p.s. with a continuous factor map to its Kronecker factor.
\begin{lem}\rightarrowbel{lem5}
If $SE_n^{\tilde{\mu}}(\tilde{X},\tilde T)\subset MS_n^{\tilde{\mu}}(\tilde{X},\tilde T)$
for all ergodic m.p.s. $(\tilde{X},\tilde{B}, \tilde{\mu}, \tilde{T})$ with a continuous factor map to its Kronecker factor, then $SE_n^{\mu}(X,T)\subset MS_n^{\mu}(X,T)$ for all ergodic m.p.s. $(X, \mathcal{B}_X,\mu, T)$.
\end{lem}
\begin{proof}
By Lemma \ref{lem3}, there exists an ergodic m.p.s. $(\tilde{X},\tilde{B}, \tilde{\mu}, \tilde{T})$ and a continuous factor map $\tilde{\pi}: \tilde{X} \rightarrow X$ such that $(\tilde{X},\tilde{B}, \tilde{\mu}, \tilde{T})$ has a continuous factor map to its Kronecker factor. Thus $SE_n^{\tilde\mu}(\tilde{X},\tilde T)\subset MS_n^{\tilde\mu}(\tilde{X},\tilde T)$, by the assumption.
For any $(x_1,\dotsc,x_n)\in SE_n^{\mu}(X,T)\setminus \Delta_n'(X)$, by \cite[Theorem 3.7]{HMY04} there exists an $n$-tuple $(\tilde{x_1},\dots,\tilde{x_n})\in SE_n^{\tilde\mu}(\tilde{X},\tilde T)\setminus \Delta_n'(\tilde{X})$ such that $\tilde\pi(\tilde{x_i})=x_i$. For any open neighborhood $U_1\times \dots \times U_n$ of $(x_1,\dotsc,x_n)$ with
$U_i\cap U_j=\emptyset$ for $i\neq j$, then
$\tilde\pi^{-1}(U_1)\times \dots \times \tilde\pi^{-1}(U_n)$ is an open neighborhood of $(\tilde{x_1},\dots,\tilde{x_n})$.
Since $(\tilde{x_1},\dots,\tilde{x_n})\in SE_n^{\tilde\mu}(\tilde{X},\tilde T)\setminus \Delta_n'(\tilde{X})\subset MS_n^{\tilde\mu}(\tilde{X},\tilde T)\setminus \Delta_n'(\tilde{X})$,
there exists $\delta>0$ such that for any $A\in\mathcal{B}_X$ with $\tilde{\mu}(\tilde\pi^{-1}(A))=\mu(A)>0$,
there exist $F\subset \mathbb{N}$ with $\overline{D}(F)\ge \delta$ and $\tilde{y_1},\dots,\tilde{y_n}\in \tilde\pi^{-1}(A)$ such that for any $m\in F$,
$$
(\tilde T^m\tilde{y_1},\dots,\tilde T^m\tilde{y_n})
\in \tilde\pi^{-1}(U_1)\times \dots \times \tilde\pi^{-1}(U_n)$$
and hence $(T^m\tilde\pi(\tilde{y_1}),\dots,T^m\tilde\pi(\tilde{y_n}))\in U_1\times \dots \times U_n$. Note that $\tilde\pi(\tilde{y_i})\in A$ for each $i=1,2,\ldots,n$. Thus we have $(x_1,\dotsc,x_n)\in MS_n^{\mu}(X,T)$.
\end{proof}
According to the lemma above-mentioned,
in the rest of this section, we fix an ergodic m.p.s. with a continuous factor map $\pi:(X,\mathcal{B}_X, \mu, T)\rightarrow (Z,\mathcal{B}_Z, \nu, R)$
to its Kronecker factor.
Moreover, we fix a measure disintegration $z \to \eta_{z}$ of $\mu$ over $\pi$,
i.e. $\mu = \int_Z \eta_{z} d\nu(z)$.
The following lemma plays a crucial role in our proof. In \cite[Proposition 3.11]{KMRR},
the authors proved it for $n=2$, but general cases are similar in idea. For readability, we move the complicated proof to Appendix \ref{APPENDIX}.
\begin{lem}\rightarrowbel{0726}
Let $\pi:(X,\mathcal{B}_X, \mu, T)\rightarrow (Z,\mathcal{B}_Z, \nu, R)$ be
a continuous factor map to its Kronecker factor.
Then for each $n\in\mathbb{N}$, there exists a continuous map $\textbf{x}\mapsto \rightarrowmbda_{\textbf{x}}^n$ from $X^{(n)}$ to $M(X^{(n)})$ such that
the map $\textbf{x} \mapsto \rightarrowmbda_{\textbf{x}}^n$ is an ergodic decomposition of $\mu^{(n)}$, where $\mu^{(n)}$ is the n-fold product of $\mu$ and $$
\rightarrowmbda^n_\textbf{x} = \int_Z \eta_{z + \pi(x_1)} \times\dots\times \eta_{z+\pi(x_n)} d\nu(z), \text{ for }\textbf{x}=(x_1,x_2,\ldots,x_n).$$
\end{lem}
The following two lemmas can be viewed as generalizations of Lemma 3.3 and Theorem 3.4 in \cite{HMY04}, respectively.
\begin{lem}\rightarrowbel{lem1}
Let $\pi:(X,\mathcal{B}_X, \mu, T)\rightarrow (Z,\mathcal{B}_Z, \nu, R)$
be a continuous factor map to its Kronecker factor. Assume that $\mathcal{U}=\{U_1, U_2, \dots, U_n\}$ is a measurable cover of $X$. Then for any measurable partition $\alpha$ finer than $\mathcal{U}$ as a cover, there exists an increasing sequence $S\subset\mathbb{Z}_+$ with $h_{\mu}^{S}(T,\alpha)>0$ if and only if $\rightarrowmbda_\textbf{x}^n (U_1^c\times\dots\times U_n^c)>0$ for all $\textbf{x}=(x_1,\dotsc, x_n)\in X^{(n)}$.
\end{lem}
\begin{proof}
$(\mathbb{R}ightarrow)$
By the contrary, we may assume that $\rightarrowmbda_\textbf{x}^n(U_1^c\times\dots\times U_n^c)=0$ for some $\textbf{x}=(x_1,\dotsc, x_n)\in X^{(n)}$.
Let $C_i=\{z\in Z: \eta_{z+\pi(x_i)}(U_i^c)>0\}$ for $i=1,\dotsc,n$. Then $$\mu(U_i^c\setminus \pi^{-1}(C_i))=\int_{Z}\eta_{z+\pi(x_i)}(U_i^c\cap \pi^{-1}(C_i^c))d \nu(z)=0.$$ Put $D_i=\pi^{-1}(C_i)\cup (U_i^c\setminus \pi^{-1}(C_i))$. Then $D_i\in \pi^{-1}(\mathcal{B}_Z)= \mathcal{K}_\mu$ and $D_i^c\subset U_i$, where $\mathcal{K}_\mu$ is the Kronecker factor of $X$.
For any $\textbf{s}=(s(1),\dotsc,s(n))\in \{0,1\}^n$, let $D_{\textbf{s}}=\cap_{i=1}^nD_i\left(s(i)\right)$, where $D_i(0)=D_i$ and $D_i(1)=D_i^c$. Set $E_1=\left(\cap_{i=1}^nD_i\right)\cap U_1 $ and $E_j=\left(\cap_{i=1}^nD_i\right)\cap( U_j\setminus \bigcup_{i=1}^{j-1}U_i)$ for $j=2,\dotsc,n$.
Consider the measurable partition
$$\alpha=\left\{D_\textbf{s}:\textbf{s}\in\{0,1\}^n\setminus\{(0,\dotsc,0)\}\right\}\cup\{E_1, \dotsc, E_n\}.$$
For any $\textbf{s}\in \{0,1\}^n\setminus\{(0,\dotsc,0)\}$, we have $s(i)=1$ for some $i=1,\dotsc,n$, then $D_\textbf{s}\subset D_i^c\subset U_i$. It is straightforward that for all $1\leq j\leq n$, $E_j\subset U_j$. Thus $\alpha$ is finer than $\mathcal{U}$ and by hypothesis there exists an increasing sequence $S$ of $\mathbb{Z}_+$ with $h_{\mu}^{S}(T,\alpha)>0$.
On the other hand, since $\rightarrowmbda_{\textbf{x}}^n(U_1^c\times\dots\times U_n^c)=0$, we deduce $\nu\left(\cap_{i=1}^nC_i\right)=0$ and hence $\mu\left(\cap_{i=1}^nD_i\right)=0$. Thus we have $E_1,\dotsc, E_n\in \mathcal{K}_\mu$. It is also clear that $D_\textbf{s}\in \mathcal{K}_\mu$ for all $\textbf{s}\in\{0,1\}^n\setminus\{(0,\dotsc,0)\}$, as $D_1,\dotsc,D_n\in \mathcal{K}_\mu.$ Therefore each element of $\alpha$ is $\mathcal{K}_\mu$-measurable, by \cite[Lemma 2.2]{HMY04}, $$h^{S}_{\mu}(T,\alpha)\leq H_{\mu}(\alpha|\mathcal{K}_\mu)=0,$$ a contradiction.
$(\Leftarrow)$Assume $\rightarrowmbda_\textbf{x}^n(U_1^c\times\dots\times U_n^c)>0$ for any $\textbf{x}\in X^{(n)}$. In particular, we take $\textbf{x}=(x,\ldots,x)$ such that $\pi(x)$ is the identity element of group $Z$. Without loss of generality, we may assume that any finite measurable partition $\alpha$ which is finer than $\mathcal{U}$ as a cover is of the type $\alpha=\left\{A_1, A_2, \ldots, A_n\right\}$ with $A_i \subset U_i$, for $1 \leqslant i \leqslant n$. Let $\alpha$ be one of such partitions. We observe that
\begin{equation*}
\begin{split}
\int_Z \eta_{z}({A_1^c}) \dots\eta_{z}(A_n^c) d\nu(z) \ge \int_Z \eta_{z }({U_1^c}) \dots\eta_{z}(U_n^c) d\nu(z)=\rightarrowmbda_\textbf{x}^n(U_1^c\times\dots\times U_n^c)>0.
\end{split}
\end{equation*}
Therefore, $A_j \notin \mathcal{K}_\mu$ for some $1 \leqslant j \leqslant n$. It follows from \cite[Theorem 2.3]{HMY04} that there exists a sequence $S \subset \mathbb{Z}_+$ such that $h_\mu^{S}(T, \alpha)=H_\mu\left(\alpha \mid \mathcal{K}_\mu\right)>0$. This finishes the proof.
\end{proof}
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\begin{lem}\rightarrowbel{lem2}
For any $\textbf{x}=(x_1,\dotsc,x_n)\in X^{(n)}$,
\[SE_{n}^\mu(X,T)= \operatorname{supp}\rightarrowmbda_{\textbf{x}}^n\setminus \Delta_n(X).\]
\end{lem}
\begin{proof}
On the one hand, let $\textbf{y}=(y_1,\dotsc,y_n)\in SE_n^{\mu}(X,T)$. We show that $\textbf{y}\in\operatorname{supp}\rightarrowmbda_{\textbf{x}}^n\setminus \Delta_n(X)$. It suffices to prove that for any measurable neighborhood $U_1\times \dots \times U_n$ of $\textbf{y}$, $$\rightarrowmbda_{\textbf{x}}^n\left(U_1\times U_2\times \dots \times U_n\right)> 0.$$
Without loss of generality, we assume that $U_i\cap U_j=\emptyset$ if $y_i\not= y_j$.
Then $\mathcal{U}=\{U_1^c, U_2^c, \dots, U_n^c\}$ is a finite cover of $X$.
It is clear that any finite measurable partition $\alpha$ finer than $\mathcal{U}$ as a cover is an admissible partition with respect to $\textbf{y}$. Therefore, there exists an increasing sequence $S\subset\mathbb{Z}_+$ with $h_{\mu}^{S}(T,\alpha)>0$.
By Lemma \ref{lem1}, we obtain that $$\rightarrowmbda_\textbf{x}^n\left(U_1\times U_2\times \dots \times U_n\right)> 0,$$ which implies that $\textbf{y}\in \operatorname{supp}\rightarrowmbda_{\textbf{x}}^n$. Since $\textbf{y}\notin \Delta_n(X)$, $\textbf{y}\in \operatorname{supp}\rightarrowmbda_{\textbf{x}}^n\setminus \Delta_n(X)$.
On the other hand, let $\textbf{y}=(y_1,\ldots,y_n) \in \operatorname{supp}\rightarrowmbda_\textbf{x}^n\setminus \Delta_n(X)$. We show that for any admissible partition $\alpha=\left\{A_1, A_2, \ldots, A_k\right\}$ with respect to $\textbf{y}$ there exists an increasing sequence $S \subset \mathbb{Z}_+$ such that $h_\mu^{S}(T, \alpha)>0$.
Since $\alpha$ is an admissible partition with respect to $\textbf{y}$ there exist closed neighborhoods $U_i$ of $y_i, 1 \leqslant i \leqslant n$, such that for each $j \in\{1,2, \ldots, k\}$ we find $i_j \in\{1,2, \ldots, n\}$ with $A_j \subset U_{i_j}^c$. That is, $\alpha$ is finer than $\mathcal{U}=\left\{U_1^c, U_2^c, \ldots, U_n^c\right\}$ as a cover. Since $$\rightarrowmbda_\textbf{x}^n\left(U_1\times U_2\times \dots \times U_n\right)>0,$$ by Lemma \ref{lem1}, there exists an increasing sequence $S \subset \mathbb{Z}_+$ such that $h_\mu^{S}(T, \alpha)>0$.
\end{proof}
Now we are ready to give the proof of Theorem \ref{thm:se=>ms}.
\begin{proof}[Proof of Theorem \ref{thm:se=>ms}]
We only need to prove that $SE_n^{\mu,e}(X,T)\subset MS_n^{\mu,e}(X,T)$.
We let $\pi:(X,\mathcal{B}_X, \mu, T)\rightarrow (Z,\mathcal{B}_Z, \nu, R)$
be a continuous factor map to its Kronecker factor.
For any $\textbf{y}=(y_1,\ldots,y_n)\in SE_n^{\mu,e}(X,T)$, let $U_1\times U_2\times \dots \times U_n$ be an open neighborhood of $\textbf{y}$ such that $U_i\cap U_j=\emptyset$ for $1\le i\not=j \le n$.
By Lemma \ref{lem2}, one has $\rightarrowmbda_\textbf{x}^n\left(U_1\times U_2\times \dots \times U_n\right)> 0$ for any $\textbf{x}=(x_1,\dotsc,x_n)\in X^{(n)}$.
Since the map $\textbf{x} \mapsto \rightarrowmbda_\textbf{x}^n$ is continuous, $X$ is compact and $U_1, U_2, \dotsc, U_n$ are open sets, it follows that there exists $\delta>0$ such that for any $\textbf{x}\in X^{(n)}$, $\rightarrowmbda_\textbf{x}^n\left(U_1\times U_2\times \dots \times U_n\right)\ge \delta$. As the map $\textbf{x} \mapsto \rightarrowmbda_\textbf{x}^n$ is an ergodic decomposition of $\mu^{(n)}$, there exists $B\subset X^{(n)}$ with $\mu^{(n)}(B)=1$ such that $\rightarrowmbda_\textbf{x}^n$ is ergodic on $X^{(n)}$ for any $\textbf{x}\in B$.
For any $A\in\mathcal{B}_X$ with $\mu (A)>0$, there exists a subset $C$ of $X^{(n)}$ with $\mu^{(n)}(C)>0$ such that for any $\textbf{x}\in C$,
\[\rightarrowmbda_\textbf{x}^n(A^n)>0.\]
Take $\textbf{x}\in B\cap C$, by the Birkhoff pointwise ergodic theorem, for $\rightarrowmbda_\textbf{x}^n$-a.e. $(x_1',\dotsc,x_n')\in X^{(n)}$
\[\lim_{N\to \infty}\frac{1}{N}\sum_{m=0}^{N-1}1_{U_1\times U_2\times\dots\times U_n}(T^mx_1',\dotsc,T^mx_n')=\rightarrowmbda_\textbf{x}^n\left(U_1\times U_2\times\dots\times U_n\right)\ge \delta.\]
Since $\rightarrowmbda_\textbf{x}^n\left(A^n\right)>0$, there exists $(x_1'',\dotsc,x_n'')\in A^n$ such that
\begin{equation*}
\begin{split}
&\lim_{N\to \infty}\frac{1}{N}\#\{m\in[0,N-1]:(T^mx_1'',\dotsc,T^mx_n'')\in U_1\times U_2\times\dots\times U_n\}\\
&=\lim_{N\to \infty}\frac{1}{N}\sum_{m=0}^{N-1}1_{U_1\times U_2\times\dots\times U_n}(T^mx_1'',\dotsc,T^mx_n'')\\
&=\rightarrowmbda_\textbf{x}^n\left(U_1\times U_2\times\dots\times U_n\right)\ge \delta.
\end{split}
\end{equation*}
Let $F=\{m\in\mathbb{Z}_+:(T^mx_1'',\dotsc,T^mx_n'')\in U_1\times U_2\times\dots\times U_n\}$. Then $D(F)\ge \delta$ and hence $\textbf{y}\in MS_n^{\mu,e}(X,T).$ This finishes the proof.
\end{proof}
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0.192.6
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\subsection{Non-ergodic case}
\begin{lem}\rightarrowbel{lem4}
Let $(X,T)$ be a t.d.s. For any $\mu\in M(X,T)$ with the form $\mu=\sum_{i=1}^{m}\rightarrowmbda_i\nu_i$, where $\nu_i\in M^e(X,T)$, $\sum_{i=1}^m\rightarrowmbda_i=1$ and $\rightarrowmbda_i>0$, one has
\begin{equation}\rightarrowbel{1}
\bigcup_{i=1}^mSE_n^{\nu_i}(X,T)\subset SE_n^{\mu}(X,T)
\end{equation}
and
\begin{equation}\rightarrowbel{2}
\bigcap_{i=1}^mSM_n^{\nu_i}(X,T)= SM_n^{\mu}(X,T).
\end{equation}
\end{lem}
\begin{proof}
We first prove that \eqref{1}. For any $\textbf{x}=(x_1,\dotsc,x_n)\in\bigcup_{i=1}^mSE_n^{\nu_i}(X,T)$, there exists $i\in\{1,2,\ldots,m\}$ such that $\textbf{x}\in SE_n^{\nu_i}(X,T)$ and then for any admissible partition $\alpha$ with respect to $\textbf{x}$, there exists $S=\{s_j\}_{j=1}^\infty$ such that
$h_{\nu_i}^S(T,\alpha)>0.$ By the definition of the sequence entropy
\[h_{\mu}^S(T,\alpha)=\limsup_{N\to \infty}\sum_{i=1}^m\rightarrowmbda_i\frac{1}{N}H_{\nu_i}(\bigvee_{j=0}^{N-1}T^{-s_j}\alpha)\ge \rightarrowmbda_ih_{\nu_i}^S(T,\alpha)>0.\]
So $\textbf{x}\in SE_n^{\mu}(X,T)$, which finishes the proof of \eqref{1}.
Next, we show \eqref{2}. For this, we only need to note that for any $A\in\mathcal{B}_X$, $\mu(A)>0$ if and only if $\nu_j(A)>0$ for some $j\in\{1,2,\ldots m\}.$
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:sm=/=se}]
We first claim that there is a t.d.s. $(X,T)$ with $\mu_1,\mu_2\in M^e(X,T)$ such that $SE_n^{\mu_1}(X,T)\neq SE_n^{\mu_2}(X,T)$. For example, we recall that the full shift on two symbols with the measure defined by the probability vector $(1/2,1/2)$. It has completely positive entropy and the measure has the full support. Thus every non-diagonal $n$-tuple is a sequence entropy $n$-tuple for this measure.
In particular, we consider such two full shifts $(X_1,T_1,\mu_1)=\left(\{0,1\}^{\mathbb{Z}},\sigma_1,\mu_1\right)$ and $(X_2,T_2,\mu_2)=\left(\{2,3\}^{\mathbb{Z}},\sigma_2,\mu_2\right)$
and define a new system $(X,T)$ as $X=X_1\bigsqcup X_2$, $T|_{X_i}=T_i, i=1,2$.
Then $\mu_1,\mu_2\in M^e(X,T)$ and $SE_n^{\mu_1}(X,T)=X_1^{(n)}\setminus\Delta_n(X_1)\neq X_2^{(n)}\setminus\Delta_n(X_2)=SE_n^{\mu_2}(X,T).$
Let $\mu=\frac{1}{2}\mu_1+\frac{1}{2}\mu_2\in M(X,T)$. By Lemma \ref{lem4}, if $SE_n^\mu(X,T)=SM_n^\mu(X,T)$ then we have
\[\cup_{i=1}^2SE_n^{\mu_i}(X,T)\subset SE_n^{\mu}(X,T)=SM_n^\mu(X,T)=\cap_{i=1}^2SM_n^{\mu_i}(X,T).\]
However, applying Theorem \ref{cor:se=sm} to each $\mu_i\in M^e(X,T)$, one has
\[SE_n^{\mu_i}(X,T)=SM_n^{\mu_i}(X,T), \text{ for }i=1,2.\]
So $SE_n^{\mu_1}(X,T)= SE_n^{\mu_2}(X,T)$, a contradiction with our assumption.
\end{proof}
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\section{topological sequence entropy and mean sensitive tuples}\rightarrowbel{sec5}
This section is devoted to providing some partial evidences for the conjecture that in a minimal system every mean sensitive tuple is a topological sequence entropy tuple.
It is known that the topological sequence entropy tuple has lift property \cite{MS07}. We can show that under the minimality condition, the mean sensitive tuple also has lift property. Let us begin with some notions.
For $2\le n\in\mathbb{N}$, we say that $(x_1,x_2,\dotsc,x_n)\in X^{(n)}\setminus \Delta_n(X)$ (resp. $(x_1,x_2,\dotsc,x_n)\in X^{(n)}\setminus \Delta'_n(X)$) is a \textit{mean $n$-sensitive tuple} (resp. an \textit{essential mean $n$-sensitive tuple}) if
for any $\tau>0$,
there is $\delta=\delta(\tau)> 0$ such that for
any nonempty open set $U\subset X$
there exist $y_1,y_2,\dotsc,y_n\in U$ such that if we denote
$F=\{k\in\mathbb{Z}_+\colon T^ky_i\in B(x_i,\tau),i=1,2,\ldots,n\}$ then $\overline{D}(F)>\delta$. Denote the set of all mean $n$-sensitive tuples (resp. essential mean $n$-sensitive tuples) by $MS_n(X,T)$ (resp. $MS^e_n(X,T)$).
\begin{thm}\rightarrowbel{lem:MSn-factor}
Let $\pi: (X,T)\rightarrow (Y,S)$ be a factor map between two t.d.s. Then
\begin{enumerate}
\item $\pi^{(n)} ( MS_n(X,T))\subset MS_n(Y,S)\cup \Delta_n(Y)$ for every $n\geq 2$;
\item $\pi^{(n)}\left(MS_n(X, T) \cup \Delta_n(X)\right)= MS_n(Y,S)\cup \Delta_n(Y)$ for every $n\geq 2$, provided that $(X,T)$ is minimal.
\end{enumerate}
\end{thm}
\begin{proof}
(1) is easy to be proved by the definition. We only prove (2).
Supposing that $(y_1,y_2,\cdots,y_n)\in MS_n(Y,S)$, we will show that there exists $(z_1,z_2,\cdots,z_n)\in MS_n(X,T)$ such that $\pi(z_i)=y_i$.
Fix $x\in X$ and let $U_m=B(x,\frac{1}{m})$. Since $(X,T)$ is minimal, $\operatorname{int}(\pi(U_m))\not= \emptyset$, where $\operatorname{int}(\pi(U_m))$ is the interior of $\pi(U_m)$.
Since $(y_1,y_2,\cdots,y_n)\in MS_n(Y,S)$, there exists $\delta>0$ and $y_m^1, \cdots, y_m^n\in \operatorname{int}(\pi(U_m))$ such that
$$\overline{D}(\{k\in \mathbb{Z}_+: S^ky_m^i \in \overline{B(y_i, 1)} \text{ for }i=1,\ldots,n\})\ge \delta.$$
Then there exist $x_m^1, \cdots, x_m^n\in U_m$ with $\pi(x_m^i)=y_m^i$ such that for any $m\in \mathbb{N}$,
$$\overline{D}(\{k\in \mathbb{Z}_+: T^kx_m^i \in \pi^{-1}(\overline{B(y_i, 1)})\text{ for }i=1,\ldots,n\})\ge \delta.$$
Put
$$
A=\prod_{i=1}^n \pi^{-1}(\overline{B(y_i, 1)}),
$$
and it is clear that $A$ is a compact subset of $X^{(n)}$.
We can cover $A$ with finite nonempty open sets of diameter less than $1$,
i.e., $A \subset \cup_{i=1}^{N_1}A_1^i$ and $\diam(A_1^i)<1$.
Then for each $m\in \mathbb{N}$ there is $1\leq N_1^m\leq N_1$ such that
$$\overline{D}(\{k\in \mathbb{Z}_+: (T^kx_m^1,\ldots, T^kx_m^n)\in \overline{A_1^{N_1^m}}\cap A \})\ge \delta/N_1.$$
Without loss of generality, we assume $N_1^m=1$ for all $m\in \mathbb{N}$. Namely,
$$
\overline{D}(\{k\in \mathbb{Z}_+: (T^kx_m^1,\ldots, T^kx_m^n)\in \overline{A_1^{1}}\cap A\}) \ge \delta/N_1 \text{ for all }m\in\mathbb{N}.
$$
Repeating above procedure, for $l\ge 1$ we can cover $\overline{A_l^{1}}\cap A$ with finite nonempty open sets of diameter less than $\frac{1}{l+1}$,
i.e., $\overline{A_l^{1}}\cap A \subset \cup_{i=1}^{N_{l+1}}A_{l+1}^i$ and $\diam(A_{l+1}^i)<\frac{1}{l+1}$.
Then for each $m\in \mathbb{N}$ there is $1\leq N_{l+1}^m\leq N_{l+1}$ such that
$$
\overline{D}(\{k\in \mathbb{Z}_+: (T^kx_m^1,\ldots, T^kx_m^n)\in \overline{A_{l+1}^{N_{l+1}^m} }\cap A \}) \ge \frac{\delta}{N_1N_2\cdots N_{l+1}}.
$$
Without loss of generality we assume $N_{l+1}^m=1$ for all $m\in \mathbb{N}$. Namely,
$$
\overline{D}(\{k\in \mathbb{Z}_+: (T^kx_m^1,\ldots, T^kx_m^n)\in \overline{A_{l+1}^{1}}\cap A \}) \ge\frac{\delta}{N_1N_2\cdots N_{l+1}} \text{ for all }m\in\mathbb{N}.
$$
It is clear that there is a unique point $(z_1^1,\ldots,z_n^1)\in \bigcap_{l=1}^{\infty} \overline{A_l^{1}}\cap A $.
We claim that $(z_1^1,\ldots,z_n^1)\in MS_n(X, T)$. In fact,
for any $\tau>0$, there is $l\in \mathbb{N}$ such that $\overline{A_{l}^{1}}\cap A \subset V_{1}\times\cdots \times V_{n}$, where $V_i=B(z_i^1,\tau)$ for $i=1,\ldots,n$.
By the construction, for any $m\in\mathbb{N}$ there are $x_m^1,\ldots, x_m^n\in U_m$ such that
$$
\overline{D}(\{k\in \mathbb{Z}_+: (T^kx_m^1,\ldots, T^kx_m^n)\in \overline{A_{l}^{1}}\cap A \}) \ge\frac{\delta}{N_1N_2\cdots N_{l}}
$$
and so
$$
\overline{D}(\{k\in \mathbb{Z}_+: (T^kx_m^1,\ldots, T^kx_m^n)\in V_{1}\times\cdots \times V_{n} \}) \ge \frac{\delta}{N_1N_2\cdots N_{l}}.
$$
for all $m\in \mathbb{N}$.
For any nonempty open set $U\subset X$, since $x$ is a transitive point, there is $s\in \mathbb{Z}$ such that $T^sx\in U$. We can choose $m\in \mathbb{Z}$ such that
$T^sU_{m}\subset U$. This implies that $T^sx_{m}^1,\ldots, T^sx_{m}^n\in U$ and
$$
\overline{D}(\{k\in \mathbb{Z}_+: (T^k(T^sx_{m}^1),\ldots, T^k(T^sx_{m}^n))\in V_{1}\times\cdots \times V_{n}\} ) \ge \frac{\delta}{N_1N_2\cdots N_{l}}.
$$
So we have $(z_1^1,\ldots,z_n^1)\in MS_n(X, T)$.
Similarly, for each $p\in\mathbb{N}$, there exists $(z_1^p,\ldots,z_n^p)\in MS_n(X, T)\cap \prod_{i=1}^n \pi^{-1}(\overline{B(y_i, \frac{1}{p})})$.
Set $z_i^p\rightarrow z_i$ as $p\rightarrow \infty$. Then $(z_1,\ldots,z_n)\in MS_n(X, T)\cup \Delta_n(X)$ and $\pi(z_i)=y_i$.
\end{proof}
Denote by $\mathcal{A}(MS_2(X, T))$ the smallest closed $T\times T$-invariant equivalence relation containing $MS_2(X, T)$.
\begin{cor}\rightarrowbel{cor:max-me-factor}
Let $(X,T)$ be a minimal t.d.s. Then $X/\mathcal{A}(MS_2(X, T))$ is the maximal mean equicontinuous factor of $(X,T)$.
\end{cor}
\begin{proof}
Let $Y=X/\mathcal{A}(MS_2(X, T))$ and $\pi:(X,T)\to (Y,S)$ be the corresponding factor map. We show that $(Y,S)$ is mean equicontinuous. Assume that $(Y,S)$ is not mean equicontinuous, by \cite[Corollary 5.5]{LTY15} $(Y,S)$ is mean sensitive.
Then by \cite[Theorem 4.4]{LYY22}, $MS_2(Y,S)\not=\emptyset$.
By Theorem \ref{lem:MSn-factor}, there exists $(x_1,x_2)\in MS_2(X, T)$ such that $(\pi(x_1),\pi(x_2))\in MS_2(Y,S)$.
Then $(x_1,x_2)\not \in R_\pi:=\{(x,x')\in X\times X:\pi(x)=\pi(x')\}$, a contradiction with $R_\pi=\mathcal{A}(MS_2(X, T))$.
Let $(Z,W)$ be a mean equicontinuous t.d.s. and $\theta: (X,T)\to (Z,W)$ be a factor map. Since $(X,T)$ is minimal, so is $(Z,W)$. Then by \cite[Corollary 5.5]{LTY15} and \cite[Theorem 4.4]{LYY22}, $MS_2(Z,W)=\emptyset$. By Theorem \ref{lem:MSn-factor} $MS_2(X,T)\subset R_\theta$, where $R_\theta$ is the corresponding equivalence relation with respect to $\theta$. This implies that $(Z,W)$ is a factor of $(Y,S)$ and so $(Y,S)$ is the maximal mean equicontinuous factor of $(X,T)$.
\end{proof}
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Denote by $\mathcal{A}(MS_2(X, T))$ the smallest closed $T\times T$-invariant equivalence relation containing $MS_2(X, T)$.
\begin{cor}\rightarrowbel{cor:max-me-factor}
Let $(X,T)$ be a minimal t.d.s. Then $X/\mathcal{A}(MS_2(X, T))$ is the maximal mean equicontinuous factor of $(X,T)$.
\end{cor}
\begin{proof}
Let $Y=X/\mathcal{A}(MS_2(X, T))$ and $\pi:(X,T)\to (Y,S)$ be the corresponding factor map. We show that $(Y,S)$ is mean equicontinuous. Assume that $(Y,S)$ is not mean equicontinuous, by \cite[Corollary 5.5]{LTY15} $(Y,S)$ is mean sensitive.
Then by \cite[Theorem 4.4]{LYY22}, $MS_2(Y,S)\not=\emptyset$.
By Theorem \ref{lem:MSn-factor}, there exists $(x_1,x_2)\in MS_2(X, T)$ such that $(\pi(x_1),\pi(x_2))\in MS_2(Y,S)$.
Then $(x_1,x_2)\not \in R_\pi:=\{(x,x')\in X\times X:\pi(x)=\pi(x')\}$, a contradiction with $R_\pi=\mathcal{A}(MS_2(X, T))$.
Let $(Z,W)$ be a mean equicontinuous t.d.s. and $\theta: (X,T)\to (Z,W)$ be a factor map. Since $(X,T)$ is minimal, so is $(Z,W)$. Then by \cite[Corollary 5.5]{LTY15} and \cite[Theorem 4.4]{LYY22}, $MS_2(Z,W)=\emptyset$. By Theorem \ref{lem:MSn-factor} $MS_2(X,T)\subset R_\theta$, where $R_\theta$ is the corresponding equivalence relation with respect to $\theta$. This implies that $(Z,W)$ is a factor of $(Y,S)$ and so $(Y,S)$ is the maximal mean equicontinuous factor of $(X,T)$.
\end{proof}
In the following we show Theorem \ref{thm:ms=>it}.
Let us begin with some preparations.
\begin{defn}[\cite{KL07}]Let $(X,T)$ be a t.d.s.
\begin{itemize}
\item For a tuple $(A_1,A_2,\ldots, A_n)$ of subsets of $X$,
we say that a set $J\subseteq \mathbb{Z}_+$ is an {\em independence set} for $A$ if for
every nonempty finite subset $I\subseteq J$ and function
$\sigma: I\rightarrow \{1,2,\ldots, n\}$ we have $\bigcap_{k\in I} T^{-k} A_{\sigma(k)}\neq \emptyset.$
\item For $n\ge2$, we call a tuple $\textbf{x}=(x_1,\ldots,x_n)\in X^{(n)}$ an {\em IT-tuple}
if for any product neighbourhood $U_1\times U_2\times \ldots \times U_n$ of $\textbf{x}$ in $X^{(n)}$
the tuple $(U_1,U_2,\ldots, U_n)$ has an infinite independence set. We denote the set of IT-tuples of length $n$ by ${\rm IT}_n (X, T)$.
\item For $n\ge2$, we call an IT-tuple $\textbf{x}=(x_1,\ldots,x_n)\in X^{(n)}$ an essential {\em IT-tuple}
if $x_i\neq x_j$ for any $i\neq j$. We denote the set of all essential IT-tuples of length $n$ by ${\rm IT}^e_n (X, T)$.
\end{itemize}
\end{defn}
\begin{prop}\cite[Proposition 3.2]{HLSY}\rightarrowbel{independent sets}
Let $X$ be a compact metric topological group
with the left Haar measure $\mu$, and let $n\in \mathbb{N}$ with $n\ge 2$. Suppose that
$V_{1},\ldots,V_{n}\subset X$ are compact subsets satisfying that
\begin{enumerate}
\item[(i)] $\overline{\operatorname{int} V_i}=V_i$ for $i=1,2,\cdots,n$,
\item[(ii)] $\operatorname{int}(V_{i})\cap \operatorname{int}(V_{j})=\emptyset$ for all $1\le i\neq j\le n$,
\item[(iii)] $\mu(\bigcap_{1\leq i\leq n}V_{i})>0$.
\end{enumerate}
Further, assume that $T: X\rightarrow X$ is a minimal rotation and $\mathcal{G}\subset X$
is a residual set. Then there exists an infinite set $I\subset \mathbb{Z}_+$
such that for all $a\in\{1,2,\ldots,n\}^{I}$ there exists $x \in\mathcal{G}$
with the property that
\begin{equation}\rightarrowbel{eq: in the int}
x\in \bigcap_{k\in I} T^{-k} {\rm int}(V_{a(k)}),\quad {\rm i.e.}\
T^kx\in \operatorname{int}(V_{a(k)}) \ \text{ for any }k\in I.
\end{equation}
\end{prop}
A subset $Z\subset X$ is called {\it proper} if $Z$ is a compact subset with $\overline{\operatorname{int}(Z)} = Z$. The following lemma can help us to complete the proof of Theorem \ref{thm:ms=>it}.
\begin{lem}\rightarrowbel{lem:proper one to one}
Let $(X,T)$ and $(Y,S)$ be two t.d.s., and $\pi:(X,T)\to (Y,S)$ be a factor map. Suppose that $(X,T)$ is minimal. Then the image of proper subsets of $X$ under $\pi$ are proper subset of $Y$.
\end{lem}
\begin{proof}
Given a proper subset $Z$ of $X$, we will show $\pi(Z)$ is also proper. It is clear that $\pi(Z)$ is compact, as $\pi$ is continuous. Now we prove $\overline{\operatorname{int}(\pi(Z))} = \pi(Z)$.
It follows from the closeness of $\pi(Z)$ that $\overline{\operatorname{int}(\pi(Z))} \subset \pi(Z)$. On the other hand, for any $y\in \pi(Z), $ take $x\in \pi^{-1}(y)\cap Z$. Since $\pi^{-1}(y)\cap Z=\pi^{-1}(y)\cap\overline{\operatorname{int}(Z)}$, there exists a sequence $\{x_n\}_{n\in\mathbb{N}}$ such that $x_n\in \operatorname{int}(Z)$ and $\lim_{n\to \infty}x_n=x$. Let $\{r_n\}_{n\in\mathbb{N}}$ be a sequence of $\mathbb{R}$ satisfying $$\lim_{n\to\infty}r_n=0\text{ and }B(x_n,r_n)\subset \operatorname{int}(Z).$$ By the minimality of $(X,T)$, we have $\pi$ is semi-open, and hence $\operatorname{int}(\pi(B(x_n,r_n)))\neq \emptyset$. Thus, there exists $x_n'\in B(x_n,r_n)$ such that $\pi(x_n')\in \operatorname{int}(\pi(B(x_n,r_n)))\subset\operatorname{int}(\pi(Z))$. Since $x_n'\in B(x_n,r_n)$ and $\lim_{n\to \infty}x_n=x$, one has $\lim_{n\to \infty}x_n'=x$, and hence $\lim_{n\to \infty}\pi(x_n')=\pi(x)=y.$ This implies that $y\in \overline{\operatorname{int}(\pi(Z))}$, which finishes the proof.
\end{proof}
Inspired by \cite[Proposition 3.7]{HLSY}, we can give the proof of Theorem \ref{thm:ms=>it}.
\begin{proof} [Proof of Theorem \ref{thm:ms=>it}]
It suffices to prove $MS^e_n(X,T)\subset IT_n^e(X,T)$.
Given $\textbf{x}=(x_1,\ldots,x_n)\in MS^e_n(X,T)$, we will show that $\textbf{x}\in IT^e_n(X,T).$
Since the minimal t.d.s. $(X_{eq},T_{eq})$ is the maximal equicontinuous factor of $(X,T)$, then $X_{eq}$ can be viewed as a compact metric group with a $T_{eq}$-invariant metric $d_{eq}$. Let $\mu$ be the left Haar probability measure of $X_{eq}$, which is also
the unique $T_{eq}$-invariant probability measure of $(X_{eq},T_{eq})$. Let
$$X_1=\{x\in X: \#\{\pi^{-1}(\pi(x))\}=1\}, \quad Y_1=\pi(X_1).$$
Then $Y_1$ is a dense $G_\delta$-set as $\pi$ is almost one to one.
Without loss of generality, assume that $\epsilon=\frac 14 \min_{1\le i\neq j\le n}d(x_i,x_j)$. Let $U_i=\overline{B_\epsilon(x_i)}$
for $1\le i\le n$. Then $U_i$ is proper for each $1\le i\le n$.
We will show that $U_1,U_2,\ldots,U_n$ is an infinite independent tuple of
$(X,T)$, i.e. there is some infinite set $I\subseteq \mathbb{Z}_+$ such that
$$\bigcap_{k\in I}T^{-k}U_{a(k)}\neq \emptyset, \ \text{for all} \ a\in \{1,2,\ldots,n\}^I.$$
Let $V_i=\pi(U_i)$ for $1\le i\le n$. By Lemma \ref{lem:proper one to one}, $V_i$ is proper
for each $i\in \{1,2,\ldots,n\}$.
We claim that ${\rm int }(V_i)\cap {\rm int}(V_j)=\emptyset$ for all $1\le i\neq j\le n$.
In fact, if there is some $1\le i\neq j\le n$ such that
${\rm int }(V_i)\cap {\rm int}(V_j)\not=\emptyset$, then
$${\rm int }(V_i)\cap {\rm int}(V_j)\cap Y_1\not=\emptyset,$$
as $Y_1$ is a dense $G_\d$-set. Let $y\in {\rm int }(V_i)\cap {\rm int}(V_j)\cap Y_1$.
Then there are $x_i\in U_i$ and $x_j\in U_j$ such that $y=\pi(x_i)=\pi(x_j)$, which contradicts with $y\in Y_1$.
Choose a nonempty open set $W_m\subset X$ with $\operatorname{diam}(\pi(W_m))<\frac{1}{m}$ for each $m\in \mathbb{N}$. Since $\textbf{x}\in MS^e_n(X,T)$,
there exist $\delta>0$ and $\textbf{x}^m=(x_1^m, x_2^m,\cdots, x_n^m)\in W_m\times \dots \times W_m$ such that
$\overline{D}(N(\textbf{x}^m, U_1\times U_2\times \cdots \times U_n))\ge \delta.$
Let $\textbf{y}^m=(y_1^m,y_2^m,\cdots,y_n^m)=\pi^{(n)} (\textbf{x}^m)$. Then
$$\overline{D}(N(\textbf{y}^m, V_1\times V_2\times \cdots \times V_n))\ge \delta.$$
For $p\in \overline{D}(N(\textbf{y}^m, V_1\times V_2\times \cdots \times V_n))$,
$T_{eq}^py_i^m\in V_i$.
As $\operatorname{diam}(\pi(W_m))<\frac{1}{m}$, $d_{eq}(y_1^m,y_i^m)<\frac{1}{m}$ for $1\le i\le n$.
Note that $$d_{eq}(T_{eq}^py_1^m,T_{eq}^py_i^m)=d_{eq}(y_1^m,y_i^m)<\frac{1}{m}\quad\text{ for }1\le i\le n.$$
Let $V_i^m=B_{\frac{1}{m}}(V_i)=\{y\in X_{eq}:d_{eq}(y,V_i)<\frac{1}{m}\}$.
Then $T_{eq}^py_1^m\in \cap_{i=1}^n V_i^m$
and $$\overline{D}(N(y_1^m, \cap_{i=1}^n V_i^m))\ge \delta.$$
Since $(X_{eq},T_{eq})$ is uniquely ergodic with respect to the measure $\mu$, $\mu(\cap_{i=1}^n V_i^m)\ge \delta$.
Letting $m\to \infty$, one has $\mu(\cap_{i=1}^n V_i)\ge \delta>0.$
By Proposition \ref{independent sets},
there is an infinite $I\subseteq \mathbb{Z}_+$ such that for all $a\in\{1,2,\ldots,n\}^{I}$ there exists $y_0\in Y_1$
with the property that
\begin{equation*}
y_0\in \bigcap_{k\in I} T_{eq}^{-k} {\rm int}(V_{a(k)}).
\end{equation*}
Set $\pi^{-1}(y_0)=\{x_0\}$. Then
\begin{equation*}
x_0\in \bigcap_{k\in I} T^{-k} U_{a(k)},
\end{equation*}
which implies that $(x_1,x_2,\cdots,x_n)\in IT_n(X,T)$.
\end{proof}
\section*{Acknowledgments}
We thank the referee for a very careful reading and
many useful comments, which helped us to improve the paper.
Research of Jie Li is supported by NNSF of China (Grant No. 12031019); Chunlin Liu is partially supported by NNSF of China (Grant No. 12090012); Siming Tu is supported by NNSF of China (Grant No. 11801584 and No. 12171175); and Tao Yu is supported by NNSF of China (Grant No. 12001354) and STU Scientific Research Foundation for Talents (Grant No. NTF19047).
\begin{appendix}
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\section{Proof of Lemma \ref{0726}}\rightarrowbel{APPENDIX}
In this section, we give the proof of Lemma \ref{0726}.
\begin{lem}\rightarrowbel{0724}
For a m.p.s. $(X,\mathcal{B}_X,\mu,T)$ with $\mathcal{K}_\mu$ its Kronecker factor, $n\in\mathbb{N}$
and $f_i\in L^\infty(X,\mu)$, $i=1,\dotsc,n$, we have
\[
\lim_{M \to \infty} \dfrac{1}{M} \sum_{m=1}^M \prod_{i=1}^{n} f_i( T^m x_i)
=
\lim_{M \to \infty} \dfrac{1}{M} \sum_{m=1}^M \prod_{i=1}^{n}\mathbb{E}(f_i | \mathcal{K}_\mu)(T^m x_i).
\]
\end{lem}
\begin{proof}
On the one hand, by the Birkhoff ergodic theorem, for $\textbf{x}=(x_1,\dotsc,x_n)\in X^{(n)}$,
let $F(\textbf{x})=F(x_1,\dots,x_n)=\prod_{i=1}^{n} f_i(x_i)$,
\[\lim_{M \to \infty} \dfrac{1}{M} \sum_{m=1}^M \prod_{i=1}^{n} f_i( T^m x_i)
=\lim_{M\to\infty}\dfrac{1}{M} \sum_{m=1}^{M} F\left(\left(T^{(n)}\right)^m\textbf{x}\right)=
\mathbb{E}_{\mu^{(n)}}(\prod_{i=1}^{n}f_i|I_{\mu^{(n)}})(\textbf{x}),\]
where $I_{\mu^{(n)}}=\{A\in \mathcal{B}^{(n)}_X: T^{(n)}A=A\}$.
On the other hand, following \cite[Lemma 4.4]{HMY04}, we have $(\mathcal{K}_\mu)^{\bigotimes n}=\mathcal{K}_{\mu^{(n)}}$. Then for $\textbf{x}=(x_1,\dotsc,x_n)\in X^{(n)}$,
\[\prod_{i=1}^{n}\mathbb{E}_{\mu}(f_i|\mathcal{K}_\mu)(x_i)
=\mathbb{E}_{\mu^{(n)}}(\prod_{i=1}^{n}f_i|(\mathcal{K}_\mu)^{\bigotimes n})(\textbf{x})
=\mathbb{E}_{\mu^{(n)}}(\prod_{i=1}^{n}f_i|\mathcal{K}_{\mu^{(n)}})(\textbf{x}).\]
This implies that
\begin{align*}
\lim_{M\to\infty}\dfrac{1}{M} \sum_{m=1}^{M} \prod_{i=1}^{n} \mathbb{E}_{\mu}(f_i|\mathcal{K}_\mu)(T^mx_i)
= & \mathbb{E}_{\mu^{(n)}}(\prod_{i=1}^{n}\mathbb{E}_{\mu}(f_i|\mathcal{K}_\mu)|I_{\mu^{(n)}})(\textbf{x})\\
= &\mathbb{E}_{\mu^{(n)}}(\mathbb{E}_{\mu^{(n)}}(\prod_{i=1}^{n}f_i|\mathcal{K}_{\mu^{(n)}})|I_{\mu^{(n)}})(\textbf{x})\\
= &\mathbb{E}_{\mu^{(n)}}(\prod_{i=1}^{n}f_i|I_{\mu^{(n)}})(\textbf{x}),
\end{align*}
where the last equality follows from the fact that $I_{\mu^{(n)}}\subset\mathcal{K}_{\mu^{(n)}}.$
\end{proof}
\begin{lem}\rightarrowbel{0725}
Let $(Z,\B_Z,\nu,R)$ be a minimal rotation on a compact abelian group.
Then for any $n\in\mathbb{N}$ and $\phi_i\in L^\infty(Z,\nu)$,
$i=1,\dotsc,n$,,
\[\lim_{M\to\infty}\dfrac{1}{M} \sum_{m=1}^{M} \prod_{i=1}^{n} \phi_i (R^mz_i)
= \int_Z \prod_{i=1}^{n} \phi_i (z_i+z)d \nu(z) \quad\text{ for }\nu^{(n)}\text{-a.e. }(z_1,\ldots, z_n).
\]
\end{lem}
\begin{proof}
Since $(Z,\B_Z,\nu,R)$ be a minimal rotation on a compact abelian group,
there exists $a\in Z$ such that $R^mz=z+ma$ for any $z\in Z$.
Let $F(z)=\prod_{i=1}^{n} \phi_i (z_i+z)$.
Then $F(R^me_Z)=F(ma)$ where $e_Z$ is identity element of $Z$.
Since $(Z,R)$ is minimal equicontinuous,
$(Z,\B_Z,\nu,R)$ is uniquely ergodic.
By an approximation argument, we have, for $\nu^{(n)}$-a.e. $(z_1,\ldots, z_n)$,
\begin{align*}
\lim_{M\to\infty}\dfrac{1}{M} \sum_{m=1}^{M}\prod_{i=1}^{n} \phi_i(R^mz_i)
=&\lim_{M\to\infty}\dfrac{1}{M} \sum_{m=1}^{M}\prod_{i=1}^{n} \phi_i (z_i+ma)\\
=&\lim_{M\to\infty}\dfrac{1}{M} \sum_{m=1}^{M} F(ma)
=\lim_{M\to\infty}\dfrac{1}{M} \sum_{m=1}^{M} F(R^me_Z)\\
=&\int_Z F(z) d \nu(z)
= \int_Z \prod_{i=1}^{n} \phi_i (z_i+z) d \nu(z).
\end{align*}
The proof is completed.
\end{proof}
\begin{proof}[Proof of Lemma \ref{0726}.]
Let $z \mapsto \eta_z$ be the disintegration of $\mu$ over the continuous factor map $\pi$ from $(X,\B_X,\mu,T)$ to its Kronecker factor $(Z,\B_Z,\nu,R)$. For $n\in\mathbb{N}$,
define
\begin{equation*}
\rightarrowbel{eqn:lambda_2_dim_definition_for_section_2_is_this_unqiue_yet}
\rightarrowmbda^n_{\textbf{x}} = \int_Z \eta_{z + \pi(x_1)} \times\dots\times \eta_{z+\pi(x_n)} d\nu(z)
\end{equation*}
for every $\textbf{x}=(x_1,\dotsc,x_n) \in X^{(n)}$.
We first note that for each $\textbf{x} \in X^{(n)}$ the measures $\eta_{z + \pi(x_i)}$ are defined for $\nu$-a.e. $z \in Z$ and therefore is well-defined.
To prove that $\textbf{x} \mapsto \rightarrowmbda^n_\textbf{x}$ is continuous first note that uniform continuity implies
\[
(u_1,\dotsc,u_n) \mapsto \int_Z \prod_{i=1}^{n}f_i(z + u_i) d\nu(z)
\]
from $Z^{(n)}$ to $\mathbb{C}$ is continuous whenever $f_i \colon Z \to \mathbb{C}$ are continuous.
An approximation argument then gives continuity for every $f_i \in L^\infty(Z,\nu)$.
In particular,
\[
\textbf{x} \mapsto \int_Z \prod_{i=1}^{n}\mathbb{E}(f_i \mid \B_Z)(z + \pi(x_i)) d\nu(z)
\]
from $X^{(n)}$ to $\mathbb{C}$ is continuous whenever $f_i \in L^\infty(X,\mu)$, which in turn implies continuity of $\textbf{x} \mapsto \rightarrowmbda_{\textbf{x}}^n$.
To prove that $\textbf{x}\mapsto \rightarrowmbda_{\textbf{x}}^n$ is an ergodic decomposition we first calculate
\begin{equation*}
\int_{X^{(n)}} \int_Z \prod_{i=1}^{n}\eta_{z + \pi(x_i)}d \nu(z) d \mu^{(n)}(\textbf{x})
=\int_Z \prod_{i=1}^{n}\int_X \eta_{z + \pi(x_i)} d \mu(x_i) d \nu(z),
\end{equation*}
which is equal to $\mu^{(n)}$ because all inner integrals are equal to $\mu$.
We conclude that
\begin{equation*}
\rightarrowbel{eq_continuousergodicdecompositionofmu1}
\mu^{(n)} = \int_{X^{(n)}}\rightarrowmbda^n_\textbf{x} d \mu^{(n)}(\textbf{x}),
\end{equation*}
which shows $\textbf{x} \mapsto \rightarrowmbda^n_\textbf{x}$ is a disintegration of $\mu^{(n)}$.
We are left with verifying that
\[
\int_{X^{(n)}} F d \rightarrowmbda^n_\textbf{x} = \mathbb{E}_{\mu^{(n)}}(F \mid I_{\mu^{(n)}})(\textbf{x})
\]
for $\mu^{(n)}$-a.e. $\textbf{x}\in X^{(n)}$ whenever $F \colon X^{(n)} \to \mathbb{C}$ is measurable and bounded.
Recall that $I_{\mu^{(n)}}$ denotes the $\sigma$-algebra of $T^{(n)}$-invariant sets.
Fix such an $F$.
It follows from the pointwise ergodic theorem that
\[
\lim_{M \to \infty} \dfrac{1}{M} \sum_{m=1}^M F( T^m x_1,\dotsc, T^m x_n)
=
\mathbb{E}_{\mu^{(n)}}(F \mid I_{\mu^{(n)}})(\textbf{x})
\]
for $\mu^{(n)}$-a.e. $\textbf{x}\in X^{(n)}$.
We therefore wish to prove that
\[
\int_{X^{(n)}} F d \rightarrowmbda^n_\textbf{x} = \lim_{M \to \infty} \dfrac{1}{M} \sum_{m=1}^M F( T^m x_1,\dotsc, T^m x_n)
\]
holds for $\mu^{(n)}$-a.e. $\textbf{x} \in X^{(n)}$.
By an approximation argument it suffices to verify that
\begin{equation*}
\rightarrowbel{eqn:proving_ergodic_kk}
\int_{X^{(n)}} f_1 \otimes\dots\otimes f_n d \rightarrowmbda^n_\textbf{x}
= \lim_{M \to \infty} \dfrac{1}{M} \sum_{m=1}^M \prod_{i=1}^{n} f_i( T^m x_i)
\end{equation*}
holds for $\mu^{(n)}$-a.e. $\textbf{x} \in X^{(n)}$ whenever $f_i$ belongs to $L^\infty(X,\mu)$ for $i=1,...,n$.
By Lemma \ref{0724},
\[
\lim_{M \to \infty} \dfrac{1}{M} \sum_{m=1}^M \prod_{i=1}^{n} f_i( T^m x_i)
=
\lim_{M \to \infty} \dfrac{1}{M} \sum_{m=1}^M \prod_{i=1}^{n}\mathbb{E}(f_i \mid \B_Z)(T^m x_i)
\]
for $\mu^{(n)}$-a.e. $\textbf{x}\in X^{(n)}$.
By Lemma \ref{0725}, for every $\phi_i$ in $L^\infty(Z,\nu)$,
\[\lim_{M \to \infty}\dfrac{1}{M} \sum_{m=1}^{M} \prod_{i=1}^{n} \phi_i (R^mz_i)
= \int_Z \prod_{i=1}^{n} \phi_i (z_i+z)d \nu(z)\]
for $\nu^{(n)}$-a.e. $\textbf{z}\in Z^{(n)}$.
Taking $\phi_i = \mathbb{E}(f_i \mid \B_Z)$ gives
\[
\lim_{M \to \infty} \dfrac{1}{M} \sum_{m=1}^M \prod_{i=1}^{n}\mathbb{E}(f_i \mid \B_Z)(T^m x_i)
=
\int_{X^{(n)}} f_1 \otimes \dots \otimes f_n d \rightarrowmbda^n_{\textbf{x}}
\]
for $\mu^{(n)}$-a.e. $\textbf{x}\in X^{(n)}$.
\end{proof}
\end{appendix}
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\betaegin{document}
\betaegin{frontmatter}
\title{Hyperbolicity in the corona and join of graphs}
\alphauthor[a]{Walter Carballosa\corref{x}}
\alphaddress[a]{Consejo Nacional de Ciencia y Tecnolog\'ia (CONACYT) $\&$ Universidad Aut\'onoma de Zacatecas,
Paseo la Bufa, int. Calzada Solidaridad, 98060 Zacatecas, ZAC, M\'exico}
\varepsilonad{[email protected]} \cortext[x]{Corresponding author.}
\alphauthor[b]{Jos\'e M. Rodr{\'\i}guez}
\alphaddress[b]{Department of Mathematics, Universidad Carlos III de Madrid, Av. de la Universidad 30, 28911 Legan\'es, Madrid, Spain.}
\varepsilonad{[email protected]}
\alphauthor[d]{Jos\'e M. Sigarreta}
\alphaddress[d]{Faculdad de Matem\'aticas, Universidad Aut\'onoma de Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, Mexico}
\varepsilonad{[email protected]}
\betaegin{abstract}
If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it
geodesic triangle} $T=\{x_1,x_2,x_3\}$ is the union of the three
geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space
$X$ is $\deltaelta$-\varepsilonmph{hyperbolic} $($in the Gromov sense$)$ if any side
of $T$ is contained in a $\deltaelta$-neighborhood of the union of the two
other sides, for every geodesic triangle $T$ in $X$.
If $X$ is hyperbolic, we denote by
$\deltaelta(X)$ the sharp hyperbolicity constant of $X$, i.e.
$\deltaelta(X)=\inf\{\deltaelta\gammae 0: \, X \, \text{ is $\deltaelta$-hyperbolic}\,\}\,.$
Some previous works characterize the hyperbolic product graphs (for the Cartesian product, strong product and lexicographic product) in terms of properties of the factor graphs.
In this paper we characterize the hyperbolic product graphs for graph join $G_1\uplus G_2$ and the corona $G_1\deltaiamond G_2$:
$G_1\uplus G_2$ is always hyperbolic, and $G_1\deltaiamond G_2$ is hyperbolic if and only if $G_1$ is hyperbolic.
Furthermore, we obtain simple formulae for the hyperbolicity constant of the graph join $G_1\uplus G_2$ and the corona $G_1\deltaiamond G_2$.
\varepsilonnd{abstract}
\betaegin{keyword}
Graph join \sigmaep Corona graph \sigmaep Gromov hyperbolicity \sigmaep Infinite graph
\MSC[2010] 05C69 \sigmaep 05A20 \sigmaep 05C50.
\varepsilonnd{keyword}
\varepsilonnd{frontmatter}
\sigmaection{Introduction}
Hyperbolic spaces play an important role in geometric
group theory and in the geometry of negatively curved
spaces (see \cite{ABCD,GH,G1}).
The concept of Gromov hyperbolicity grasps the essence of negatively curved
spaces like the classical hyperbolic space, Riemannian manifolds of
negative sectional curvature bounded away from $0$, and of discrete spaces like trees
and the Cayley graphs of many finitely generated groups. It is remarkable
that a simple concept leads to such a rich
general theory (see \cite{ABCD,GH,G1}).
The first works on Gromov hyperbolic spaces deal with
finitely generated groups (see \cite{G1}).
Initially, Gromov spaces were applied to the study of automatic groups in the science of computation
(see, \varepsilonmph{e.g.}, \cite{O}); indeed, hyperbolic groups are strongly geodesically automatic, \varepsilonmph{i.e.}, there is an automatic structure on the group \cite{Cha}.
The concept of hyperbolicity appears also in discrete mathematics, algorithms
and networking. For example, it has been shown empirically
in \cite{ShTa} that the internet topology embeds with better accuracy
into a hyperbolic space than into an Euclidean space
of comparable dimension; the same holds for many complex networks, see \cite{KPKVB}.
A few algorithmic problems in
hyperbolic spaces and hyperbolic graphs have been considered
in recent papers (see \cite{ChEs,Epp,GaLy,Kra}).
Another important
application of these spaces is the study of the spread of viruses through on the
internet (see \cite{K21,K22}).
Furthermore, hyperbolic spaces are useful in secure transmission of information on the
network (see \cite{K27,K21,K22,NS}).
The study of Gromov hyperbolic graphs is a subject of increasing interest; see, \varepsilonmph{e.g.}, \cite{BRS,BRSV2,BRST,BPK,BHB1,CDR,CPRS,CRS,CRSV,CDEHV,K50,K27,K21,K22,K23,K24,K56,KPKVB,MRSV,MRSV2,NS,PeRSV,PRST,PRSV,PT,R,RSVV,S,S2,T,WZ} and the references therein.
We say that the curve $\gamma$ in a metric space $X$ is a
\varepsilonmph{geodesic} if we have $L(\gamma|_{[t,s]})=d(\gamma(t),\gamma(s))=|t-s|$ for every $s,t\in [a,b]$
(then $\gammaamma$ is equipped with an arc-length parametrization).
The metric space $X$ is said \varepsilonmph{geodesic} if for every couple of points in
$X$ there exists a geodesic joining them; we denote by $[xy]$
any geodesic joining $x$ and $y$; this notation is ambiguous, since in general we do not have uniqueness of
geodesics, but it is very convenient.
Consequently, any geodesic metric space is connected.
If the metric space $X$ is
a graph, then the edge joining the vertices $u$ and $v$ will be denoted by $[u,v]$.
Along the paper we just consider graphs with every edge of length $1$.
In order to consider a graph $G$ as a geodesic metric space, identify (by an isometry)
any edge $[u,v]\in E(G)$ with the interval $[0,1]$ in the real line;
then the edge $[u,v]$ (considered as a graph with just one edge)
is isometric to the interval $[0,1]$.
Thus, the points in $G$ are the vertices and, also, the points in the interior
of any edge of $G$.
In this way, any connected graph $G$ has a natural distance
defined on its points, induced by taking shortest paths in $G$,
and we can see $G$ as a metric graph.
If $x,y$ are in different connected components of $G$, we define $d_G(x,y)=\infty$.
Throughout this paper, $G=(V,E)$ denotes a simple graph (not necessarily connected) such that every edge has length $1$
and $V\neq \varepsilonmptyset$.
These properties guarantee that any connected graph is a geodesic metric space.
Note that to exclude multiple
edges and loops is not an important loss of generality, since
\cite[Theorems 8 and 10]{BRSV2} reduce the problem of compute
the hyperbolicity constant of graphs with multiple edges and/or
loops to the study of simple graphs.
For a nonempty set $X\sigmaubseteq V$, and a vertex $v\in V$, $N_X(v)$ denotes the set of neighbors $v$ has in $X$:
$N_X(v):=\{u\in X: [u,v]\in E\},$ and the degree of $v$ in $X$ will be
denoted by $\deltaeg_{X}(v)=|N_{X}(v)|$. We denote the degree of a vertex $v\in V$ in $G$ by $\deltaeg(v)\lambdae\infty$, and
the maximum degree of $G$ by $\Delta_{G}:=\sigmaup_{v\in V}\deltaeg(v)$.
Consider a polygon $J=\{J_1,J_2,\deltaots,J_n\}$
with sides $J_j\sigmaubseteq X$ in a geodesic metric space $X$.
We say that $J$ is $\delta$-{\it thin} if for
every $x\in J_i$ we have that $d(x,\cup_{j\neq i}J_{j})\lambdae \delta$.
Let us
denote by $\delta(J)$ the sharp thin constant of $J$, \varepsilonmph{i.e.},
$\delta(J):=\inf\{\delta\gammae 0: \, J \, \text{ is $\delta$-thin}\,\}\,. $
If $x_1,x_2,x_3$ are three points in $X$, a {\it geodesic triangle} $T=\{x_1,x_2,x_3\}$ is
the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and
$[x_3x_1]$ in $X$.
We say that $X$ is $\delta$-\varepsilonmph{hyperbolic} if every geodesic
triangle in $X$ is $\delta$-thin, and we denote by $\delta(X)$ the sharp
hyperbolicity constant of $X$, \varepsilonmph{i.e.}, $\delta(X):=\sigmaup\{\delta(T): \, T \,
\text{ is a geodesic triangle in }\,X\,\}.$ We say that $X$ is
\varepsilonmph{hyperbolic} if $X$ is $\delta$-hyperbolic for some $\delta \gammae 0$; then $X$ is hyperbolic if and only if
$ \delta(X)<\infty.$
If $X$ has connected components $\{X_i\}_{i\in I}$, then we define $\delta(X):=\sigmaup_{i\in I} \delta(X_i)$, and we say that $X$ is hyperbolic if $\delta(X)<\infty$.
In the classical references on this subject (see, \varepsilonmph{e.g.}, \cite{BHB,GH})
appear several different definitions of Gromov hyperbolicity, which are equivalent in the sense
that if $X$ is $\delta$-hyperbolic with respect to one definition,
then it is $\delta'$-hyperbolic with respect to another definition (for some $\delta'$ related to $\delta$).
The definition that we have chosen has a deep geometric meaning (see, \varepsilonmph{e.g.}, \cite{GH}).
Trivially, any bounded
metric space $X$ is $((\deltaiam X)/2)$-hyperbolic.
A normed linear space is hyperbolic if and only if it has dimension one.
A geodesic space is $0$-hyperbolic if and only if it is a metric tree.
If a complete Riemannian manifold is simply connected and its sectional curvatures satisfy
$K\lambdaeq c$ for some negative constant $c$, then it is hyperbolic.
See the classical references \cite{ABCD,GH} in order to find further results.
We want to remark that the main examples of hyperbolic graphs are the trees.
In fact, the hyperbolicity constant of a geodesic metric space can be viewed as a measure of
how ``tree-like'' the space is, since those spaces $X$ with $\deltaelta(X) = 0$ are precisely the metric trees.
This is an interesting subject since, in
many applications, one finds that the borderline between tractable and intractable
cases may be the tree-like degree of the structure to be dealt with
(see, \varepsilonmph{e.g.}, \cite{CYY}).
Given a Cayley graph (of a presentation with solvable word problem)
there is an algorithm which allows to decide if it is hyperbolic.
However, for a general graph or a general geodesic metric space
deciding whether or not a space is
hyperbolic is usually very difficult.
Therefore, it is interesting to study the hyperbolicity of particular classes of graphs.
The papers \cite{BRST,BHB1,CCCR,CDR,CRSV,MRSV2,PeRSV,PRSV,R,Si} study the hyperbolicity of, respectively, complement of graphs, chordal
graphs, strong product graphs, lexicographic product graphs,
line graphs, Cartesian product graphs, cubic graphs, tessellation graphs, short graphs and median graphs.
In \cite{CCCR,CDR,MRSV2} the authors characterize the hyperbolic product graphs (for strong product, lexicographic product and Cartesian product) in terms of properties of the factor graphs.
In this paper we characterize the hyperbolic product graphs for graph join $G_1\uplus G_2$ and the corona $G_1\deltaiamond G_2$:
$G_1\uplus G_2$ is always hyperbolic, and $G_1\deltaiamond G_2$ is hyperbolic if and only if $G_1$ is hyperbolic
(see Corollaries \ref{cor:SP} and \ref{cor:sup}).
Furthermore, we obtain simple formulae for the hyperbolicity constant of the graph join $G_1\uplus G_2$ and the corona $G_1\deltaiamond G_2$ (see Theorems \ref{th:hypJoin} and \ref{th:corona}).
In particular, Theorem \ref{th:corona} states that
$\delta(G_1\deltaiamond G_2)=\max\{\delta(G_1),\delta(G_2\uplus E_1)\}$, where $E_1$ is a graph with just one vertex.
We want to remark that it is not usual at all to obtain explicit formulae for the hyperbolicity constant of large classes of graphs.
\sigmaection{Distance in graph join}
In order to estimate the hyperbolicity constant of the graph join $G_1\uplus G_2$ of $G_1$ and $G_2$, we will need an explicit formula for the distance between two arbitrary points.
We will use the definition given by Harary in \cite{H}.
\betaegin{definition}\lambdaabel{def:join}
Let $G_1=(V(G_1),E(G_1))$ and $G_2=(V(G_2),E(G_2))$ two graphs with $V(G_1)\cap V(G_2)=\varnothing$. The \varepsilonmph{graph join} $G_1\uplus G_2$ of $G_1$ and $G_2$ has $V(G_1\uplus G_2)=V(G_1) \cup V(G_2)$ and two different vertices $u$ and $v$ of $G_1\uplus G_2$ are adjacent if $u\in V(G_1)$ and $v\in V(G_2)$, or $[u,v]\in E(G_1)$ or $[u,v]\in E(G_2)$.
\varepsilonnd{definition}
From the definition, it follows that the graph join of two graphs is commutative.
Figure \ref{fig:join} shows the graph join of two graphs.
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In the classical references on this subject (see, \varepsilonmph{e.g.}, \cite{BHB,GH})
appear several different definitions of Gromov hyperbolicity, which are equivalent in the sense
that if $X$ is $\delta$-hyperbolic with respect to one definition,
then it is $\delta'$-hyperbolic with respect to another definition (for some $\delta'$ related to $\delta$).
The definition that we have chosen has a deep geometric meaning (see, \varepsilonmph{e.g.}, \cite{GH}).
Trivially, any bounded
metric space $X$ is $((\deltaiam X)/2)$-hyperbolic.
A normed linear space is hyperbolic if and only if it has dimension one.
A geodesic space is $0$-hyperbolic if and only if it is a metric tree.
If a complete Riemannian manifold is simply connected and its sectional curvatures satisfy
$K\lambdaeq c$ for some negative constant $c$, then it is hyperbolic.
See the classical references \cite{ABCD,GH} in order to find further results.
We want to remark that the main examples of hyperbolic graphs are the trees.
In fact, the hyperbolicity constant of a geodesic metric space can be viewed as a measure of
how ``tree-like'' the space is, since those spaces $X$ with $\deltaelta(X) = 0$ are precisely the metric trees.
This is an interesting subject since, in
many applications, one finds that the borderline between tractable and intractable
cases may be the tree-like degree of the structure to be dealt with
(see, \varepsilonmph{e.g.}, \cite{CYY}).
Given a Cayley graph (of a presentation with solvable word problem)
there is an algorithm which allows to decide if it is hyperbolic.
However, for a general graph or a general geodesic metric space
deciding whether or not a space is
hyperbolic is usually very difficult.
Therefore, it is interesting to study the hyperbolicity of particular classes of graphs.
The papers \cite{BRST,BHB1,CCCR,CDR,CRSV,MRSV2,PeRSV,PRSV,R,Si} study the hyperbolicity of, respectively, complement of graphs, chordal
graphs, strong product graphs, lexicographic product graphs,
line graphs, Cartesian product graphs, cubic graphs, tessellation graphs, short graphs and median graphs.
In \cite{CCCR,CDR,MRSV2} the authors characterize the hyperbolic product graphs (for strong product, lexicographic product and Cartesian product) in terms of properties of the factor graphs.
In this paper we characterize the hyperbolic product graphs for graph join $G_1\uplus G_2$ and the corona $G_1\deltaiamond G_2$:
$G_1\uplus G_2$ is always hyperbolic, and $G_1\deltaiamond G_2$ is hyperbolic if and only if $G_1$ is hyperbolic
(see Corollaries \ref{cor:SP} and \ref{cor:sup}).
Furthermore, we obtain simple formulae for the hyperbolicity constant of the graph join $G_1\uplus G_2$ and the corona $G_1\deltaiamond G_2$ (see Theorems \ref{th:hypJoin} and \ref{th:corona}).
In particular, Theorem \ref{th:corona} states that
$\delta(G_1\deltaiamond G_2)=\max\{\delta(G_1),\delta(G_2\uplus E_1)\}$, where $E_1$ is a graph with just one vertex.
We want to remark that it is not usual at all to obtain explicit formulae for the hyperbolicity constant of large classes of graphs.
\sigmaection{Distance in graph join}
In order to estimate the hyperbolicity constant of the graph join $G_1\uplus G_2$ of $G_1$ and $G_2$, we will need an explicit formula for the distance between two arbitrary points.
We will use the definition given by Harary in \cite{H}.
\betaegin{definition}\lambdaabel{def:join}
Let $G_1=(V(G_1),E(G_1))$ and $G_2=(V(G_2),E(G_2))$ two graphs with $V(G_1)\cap V(G_2)=\varnothing$. The \varepsilonmph{graph join} $G_1\uplus G_2$ of $G_1$ and $G_2$ has $V(G_1\uplus G_2)=V(G_1) \cup V(G_2)$ and two different vertices $u$ and $v$ of $G_1\uplus G_2$ are adjacent if $u\in V(G_1)$ and $v\in V(G_2)$, or $[u,v]\in E(G_1)$ or $[u,v]\in E(G_2)$.
\varepsilonnd{definition}
From the definition, it follows that the graph join of two graphs is commutative.
Figure \ref{fig:join} shows the graph join of two graphs.
\betaegin{figure}[h]
\centering
\sigmacalebox{.9}
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\uput[0](1.4,0){$\betaiguplus$}
\uput[0](3,0){\lambdaarge{=}}
\varepsilonnd{pspicture}}
\caption{Graph join of two graphs $C_3 \uplus P_3$.} \lambdaabel{fig:join}
\varepsilonnd{figure}
\betaegin{remark}\lambdaabel{r:K_nm}
For every graphs $G_1,G_2$ we have that $G_1\uplus G_2$ is a connected graph with a subgraph isomorphic to a complete bipartite graph with $V(G_1)$ and $V(G_2)$ as its parts.
\varepsilonnd{remark}
Note that, from a geometric viewpoint, the graph join $G_1\uplus G_2$ is obtained as an union of the graphs $G_1$, $G_2$ and the complete bipartite graph $K(G_1,G_2)$ linking the vertices of $V(G_1)$ and $V(G_2)$.
The following result allows to compute the distance between any two points in $G_1\uplus G_2$. Furthermore, this result provides information about the geodesics in the graph join.
\betaegin{proposition}\lambdaabel{prop:JoinDist}
For every graphs $G_1, G_2$ we have:
\betaegin{itemize}
\item[(a)] If $x,y \in G_i$ ($i\in\{1,2\}$), then
\[d_{G_1\uplus G_2}(x,y) = \min\lambdaeft\{ d_{G_i}(x,y) , d_{G_i}\betaig(x,V(G_i)\betaig)+2+d_{G_i}\betaig(V(G_i),y\betaig)\right\}.\]
\item[(b)] If $x \in G_i$ and $y \in G_j$ with $i\neq j$, then
\[d_{G_1\uplus G_2}(x,y) = d_{G_i}\betaig(x,V(G_i)\betaig)+1+d_{G_j}\betaig(V(G_j),y\betaig).\]
\item[(c)] If $x \in G_i$ and $y \in K(G_1,G_2)$, then
\[d_{G_1\uplus G_2}(x,y) = \min\lambdaeft\{ d_{G_i}(x,Y_i)+d_{G_1\uplus G_2}(Y_i,y) , d_{G_i}\betaig(x,V(G_i)\betaig)+1+d_{G_1\uplus G_2}(Y_j,y)\right\},\]
where $y\in [Y_1,Y_2]$ with $Y_i\in V(G_i)$ and $Y_j\in V(G_j)$.
\item[(d)] If $x,y \in K(G_1,G_2)$, then
\[d_{G_1\uplus G_2}(x,y) = \min\{ d_{K(G_1,G_2)}(x,y), M\},\]
where $x\in [X_1,X_2]$, $y\in [Y_1,Y_2]$ with $X_1,Y_1\in V(G_1)$ and $X_2,Y_2\in V(G_2)$, and $M=\min_{i\in\{1,2\}}\{d_{G_1\uplus G_2}(x,X_i)+d_{G_i}(X_i,Y_i)+d_{G_1\uplus G_2}(Y_i,y)\}$
\varepsilonnd{itemize}
\varepsilonnd{proposition}
\betaegin{proof}
We will prove each item separately.
In item (a), if $i\neq j$, we consider the two shortest possible paths from $x$ to $y$ such that they either is contained in $G_i$ or intersects $G_j$ (and then it intersects $G_j$ just in a single vertex).
In item (b), since any path in $G_1\uplus G_2$ joining $x$ and $y$ contains at less one edge in $K(G_1,G_2)$, we have a geodesic when the path contains an edge joining a closest vertex to $x$ in $V(G_i)$ and a closest vertex to $y$ in $V(G_j)$.
In item (c) we consider the two shortest possible paths from $x$ to $y$ containing either $Y_1$ or $Y_2$.
Finally, in item (d) we may consider the three shortest possible paths from $x$ to $y$ such that they either is contained in $K(G_1,G_2)$ or contains at lest an edge in $E(G_1)$ or contains at lest an edge in $E(G_2)$.
\varepsilonnd{proof}
We say that a subgraph $\Gamma$ of $G$ is \varepsilonmph{isometric} if $d_{\Gamma}(x,y)=d_{G}(x,y)$ for every $x,y\in \Gamma$. Proposition \ref{prop:JoinDist} gives the following result.
\betaegin{proposition}\lambdaabel{prop:IsomJoin}
Let $G_1,G_2$ be two graph and let $\Gamma_1,\Gamma_2$ be isometric subgraphs to $G_1$ and $G_2$, respectively. Then, $\Gamma_1\uplus\Gamma_2$ is an isometric subgraph to $G_1\uplus G_2$.
\varepsilonnd{proposition}
The following result allows to compute the diameter of the set of vertices in a graph join.
\betaegin{proposition}\lambdaabel{prop:vert}
For every graphs $G_1,G_2$ we have $1\lambdae\deltaiam V(G_1\uplus G_2)\lambdae 2$. Furthermore, $\deltaiam V(G_1\uplus G_2)=1$ if and only if $G_1$ and $G_2$ are complete graphs.
\varepsilonnd{proposition}
\betaegin{proof}
Since $V(G_1),V(G_2)\neq\varepsilonmptyset$, $\deltaiam V(G_1\uplus G_2)\gammae 1$. Besides, if $u,v\in V(G_1\uplus G_2)$, we have $d_{G_1\uplus G_2}(u,v)\lambdae d_{K(G_1,G_2)}(u,v)\lambdae 2$.
In order to finish the proof note that on the one hand, if $G_1$ and $G_2$ are complete graphs, then $G_1\uplus G_2$ is a complete graph with at least $2$ vertices and $\deltaiam V(G_1\uplus G_2)=1$. On the other hand, if $\deltaiam V(G_1\uplus G_2)=1$, then for every two vertices $u,v \in V(G_1)$ we have $[u,v]\in E(G_1)$; by symmetry, we have the same result for every $u,v \in V(G_2)$.
\varepsilonnd{proof}
Since $\deltaiam V(G) \lambdae \deltaiam G \lambdae \deltaiam V(G) + 1$ for every graph $G$, the previous proposition has the following consequence.
\betaegin{corollary}\lambdaabel{c:diam}
For every graphs $G_1,G_2$ we have $1\lambdae\deltaiam G_1\uplus G_2\lambdae 3$.
\varepsilonnd{corollary}
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\betaegin{proof}
We will prove each item separately.
In item (a), if $i\neq j$, we consider the two shortest possible paths from $x$ to $y$ such that they either is contained in $G_i$ or intersects $G_j$ (and then it intersects $G_j$ just in a single vertex).
In item (b), since any path in $G_1\uplus G_2$ joining $x$ and $y$ contains at less one edge in $K(G_1,G_2)$, we have a geodesic when the path contains an edge joining a closest vertex to $x$ in $V(G_i)$ and a closest vertex to $y$ in $V(G_j)$.
In item (c) we consider the two shortest possible paths from $x$ to $y$ containing either $Y_1$ or $Y_2$.
Finally, in item (d) we may consider the three shortest possible paths from $x$ to $y$ such that they either is contained in $K(G_1,G_2)$ or contains at lest an edge in $E(G_1)$ or contains at lest an edge in $E(G_2)$.
\varepsilonnd{proof}
We say that a subgraph $\Gamma$ of $G$ is \varepsilonmph{isometric} if $d_{\Gamma}(x,y)=d_{G}(x,y)$ for every $x,y\in \Gamma$. Proposition \ref{prop:JoinDist} gives the following result.
\betaegin{proposition}\lambdaabel{prop:IsomJoin}
Let $G_1,G_2$ be two graph and let $\Gamma_1,\Gamma_2$ be isometric subgraphs to $G_1$ and $G_2$, respectively. Then, $\Gamma_1\uplus\Gamma_2$ is an isometric subgraph to $G_1\uplus G_2$.
\varepsilonnd{proposition}
The following result allows to compute the diameter of the set of vertices in a graph join.
\betaegin{proposition}\lambdaabel{prop:vert}
For every graphs $G_1,G_2$ we have $1\lambdae\deltaiam V(G_1\uplus G_2)\lambdae 2$. Furthermore, $\deltaiam V(G_1\uplus G_2)=1$ if and only if $G_1$ and $G_2$ are complete graphs.
\varepsilonnd{proposition}
\betaegin{proof}
Since $V(G_1),V(G_2)\neq\varepsilonmptyset$, $\deltaiam V(G_1\uplus G_2)\gammae 1$. Besides, if $u,v\in V(G_1\uplus G_2)$, we have $d_{G_1\uplus G_2}(u,v)\lambdae d_{K(G_1,G_2)}(u,v)\lambdae 2$.
In order to finish the proof note that on the one hand, if $G_1$ and $G_2$ are complete graphs, then $G_1\uplus G_2$ is a complete graph with at least $2$ vertices and $\deltaiam V(G_1\uplus G_2)=1$. On the other hand, if $\deltaiam V(G_1\uplus G_2)=1$, then for every two vertices $u,v \in V(G_1)$ we have $[u,v]\in E(G_1)$; by symmetry, we have the same result for every $u,v \in V(G_2)$.
\varepsilonnd{proof}
Since $\deltaiam V(G) \lambdae \deltaiam G \lambdae \deltaiam V(G) + 1$ for every graph $G$, the previous proposition has the following consequence.
\betaegin{corollary}\lambdaabel{c:diam}
For every graphs $G_1,G_2$ we have $1\lambdae\deltaiam G_1\uplus G_2\lambdae 3$.
\varepsilonnd{corollary}
Proposition \ref{prop:JoinDist} and Corollary \ref{c:diam} give the following results.
Given a graph $G$, we say that $x\in G$ is a midpoint (of an edge) if $d_{G}(x,V(G))=1/2$.
\betaegin{corollary}\lambdaabel{cor:midpoint}
Let $G_1,G_2$ be two graphs. If $d_{G_1\uplus G_2}(x,y) = 3$, then $x,y$ are two midpoints in $G_i$ with $d_{G_i}(x,y)\gammae3$ for some $i\in \{1,2\}$.
\varepsilonnd{corollary}
\betaegin{corollary}\lambdaabel{r:diam3}
Let $G_1,G_2$ be two graphs. Then, $\deltaiam G_1\uplus G_2 = 3$ if and only if there are two midpoints $x,y$ in $G_i$ with $d_{G_i}(x,y)\gammae3$ for some $i\in \{1,2\}$.
\varepsilonnd{corollary}
\sigmaection{Hyperbolicity constant of the graph join of two graphs}
In this section we obtain some bounds for the hyperbolicity constant of the graph join of two graphs.
These bounds allow to prove that the joins of graphs are always hyperbolic with a small hyperbolicity constant.
The next well-known result will be useful.
\betaegin{theorem}\cite[Theorem 8]{RSVV}\lambdaabel{t:diameter1}
In any graph $G$ the inequality $\delta(G)\lambdae \deltaiam G / 2$ holds and it is sharp.
\varepsilonnd{theorem}
We have the following consequence of Corollary \ref{c:diam} and Theorem \ref{t:diameter1}.
\betaegin{corollary}\lambdaabel{cor:SP}
For every graphs $G_1,G_2$, the graph join $G_1\uplus G_2$ is hyperbolic with $\delta(G_1\uplus G_2)\lambdaeq 3/2$, and the inequality is sharp.
\varepsilonnd{corollary}
Theorem \ref{th:hyp3/2} characterizes the graph join of two graphs for which the equality in the previous corollary is attained.
The following result in \cite[Lemma 5]{RSVV} will be useful.
\betaegin{lemma}\lambdaabel{l:subgraph}
If $\Gamma$ is an isometric subgraph of $G$, then $\delta(\Gamma) \lambdae \delta(G)$.
\varepsilonnd{lemma}
\betaegin{theorem}\lambdaabel{th:HypIsomJoin}
For every graphs $G_1,G_2$, we have $$\delta(G_1\uplus G_2)=\max\{ \delta(\Gamma_1\uplus \Gamma_2) : \Gamma_i \text{ is isometric to } G_i \text{ for } i=1,2 \}.$$
\varepsilonnd{theorem}
\betaegin{proof}
By Proposition \ref{prop:IsomJoin} and Lemma \ref{l:subgraph} we have $\delta(G_1\uplus G_2)\gammae \delta(\Gamma_1\uplus \Gamma_2)$ for any isometric subgraph $\Gamma_i$ of $G_i$ for $i=1,2$. Besides, since any graph is an isometric subgraph of itself we obtain the equality by taking $\Gamma_1=G_1$ and $\Gamma_2=G_2$.
\varepsilonnd{proof}
Denote by $J(G)$ the set of vertices and midpoints of edges in $G$.
As usual, by \varepsilonmph{cycle} we mean a simple closed curve, i.e., a path with different vertices,
unless the last one, which is equal to the first vertex.
First, we collect some previous results of \cite{BRS} which will be useful.
\betaegin{theorem}\cite[Theorem 2.6]{BRS}
\lambdaabel{t:multk/4}
For every hyperbolic graph $G$, $\delta(G)$ is a multiple of $1/4$.
\varepsilonnd{theorem}
\betaegin{theorem}\cite[Theorem 2.7]{BRS}
\lambdaabel{t:TrianVMp}
For any hyperbolic graph $G$, there exists
a geodesic triangle $T = \{x, y, z\}$ that is a cycle with $x, y, z \in J(G)$ and $\delta(T) = \delta(G)$.
\varepsilonnd{theorem}
The following result characterizes the hyperbolic graphs with a small hyperbolicity constant, see \cite[Theorem 11]{MRSV}.
Let us define the \varepsilonmph{circumference} $c(G)$ of a graph $G$ which is not a tree as the supremum of the lengths of its cycles; if $G$ is a tree we define $c(G)=0$.
\betaegin{theorem}\lambdaabel{th:delt<1}
Let $G$ be any graph.
\betaegin{itemize}
\item[(a)] {$\delta(G) = 0$ if and only if $G$ is a tree.}
\item[(b)] {$\delta(G) = 1/4, 1/2$ is not satisfied for any graph $G$.}
\item[(c)] {$\delta(G) = 3/4$ if and only if $\ c(G)=3$.}
\varepsilonnd{itemize}
\varepsilonnd{theorem}
We have the following consequence for the hyperbolicity constant of the joins of graphs.
\betaegin{proposition}\lambdaabel{r:discretJoin}
For every graphs $G_1,G_2$ the graph join $G_1\uplus G_2$ is hyperbolic with hyperbolicity constant $\delta(G_1\uplus G_2)$ in $\{0, 3/4, 1, 5/4, 3/2\}$.
\varepsilonnd{proposition}
If $G_1$ and $G_2$ are \varepsilonmph{isomorphic}, then we write $G_1 \sigmaimeq G_2$. It is clear that if $G_1\sigmaimeq G_2$, then $\delta(G_1)=\delta(G_2)$.
The $n$-vertex edgeless graph ($n\gammae1$) or \varepsilonmph{empty graph} is a graph without edges and with $n$ vertices, and it is commonly denoted as $E_n$.
The following result allows to characterize the joins of graphs with hyperbolicity constant less than one in terms of its factor graphs.
Recall that $\Delta_G$ denotes the maximum degree of the vertices in $G$.
\betaegin{theorem}\lambdaabel{th:deltJoin<1}
Let $G_1,G_2$ be two graphs.
\betaegin{itemize}
\item[(1)] {$\delta(G_1\uplus G_2)=0$ if and only if $G_1$ and $G_2$ are empty graphs and one of them is isomorphic to $E_1$.}
\item[(2)] {$\delta(G_1\uplus G_2)=3/4$ if and only if $G_1\sigmaimeq E_1$ and $\Delta_{G_2}=1$, or $G_2\sigmaimeq E_1$ and $\Delta_{G_1}=1$.}
\varepsilonnd{itemize}
\varepsilonnd{theorem}
\betaegin{proof}$ $
\betaegin{itemize}
\item[(1)] {By Theorem \ref{th:delt<1} it suffices to characterize the joins of graphs which are trees. If $G_1$ and $G_2$ are empty graphs and one of them is isomorphic to $E_1$, then it is clear that $G_1\uplus G_2$ is a tree. Assume now that $G_1\uplus G_2$ is a tree. If $G_1$ and $G_2$ have at least two vertices then $G_1\uplus G_2$ has a cycle with length four. Thus, $G_1$ or $G_2$ is isomorphic to $E_1$. Without loss of generality we can assume that $G_1\sigmaimeq E_1$. Note that if $G_2$ has at least one edge then $G_1\uplus G_2$ has a cycle with length three. Then, $G_2\sigmaimeq E_n$ for some $n\in \mathbb{N}$.}
\item[(2)] {By Theorem \ref{th:delt<1} it suffices to characterize the joins of graphs with circumference three. If $G_1\sigmaimeq E_1$ and $\Delta_{G_2}=1$, or $G_2\sigmaimeq E_1$ and $\Delta_{G_1}=1$, then it is clear that $c(G_1\uplus G_2)=3$. Assume now that $c(G_1\uplus G_2)=3$. If $G_1,G_2$ both have at least two vertices then $G_1\uplus G_2$ contains a cycle with length four and so $c(G_1\uplus G_2)\gammae4$. Therefore, $G_1$ or $G_2$ is isomorphic to $E_1$. Without loss of generality we can assume that $G_1\sigmaimeq E_1$. Note that if $\Delta_{G_2}\gammae2$ then there is an isomorphic subgraph to $E_1\uplus P_3$ in $G_1\uplus G_2$; thus, $G_1\uplus G_2$ contains a cycle with length four. So, we have $\Delta_{G_2}\lambdae1$. Besides, since $G_2$ is a non-empty graph by (1), we have $\Delta_{G_2}\gammae1$.}
\varepsilonnd{itemize}
\varepsilonnd{proof}
The following result will be useful, see \cite[Theorem 11]{RSVV}.
The graph join of a cycle $C_{n-1}$ and a single vertex $E_1$ is referred to as a \varepsilonmph{wheel} with $n$ vertices and denoted by $W_n$. Notice that the complete bipartite graph $K_{n,m}$ is isomorphic to the graph join of two empty graphs $E_n,E_m$, i.e., $K_{n,m}\sigmaimeq E_n\uplus E_m$.
\betaegin{example}\lambdaabel{examples}
The following graphs have these hyperbolicity constants:
\betaegin{itemize}
\item The wheel graph with $n$ vertices $W_n$ verifies $\delta(W_4)=\delta(W_5)=1$, $\delta(W_n)=3/2$ for every $7\lambdae n\lambdae 10$, and $\delta(W_n)=5/4$ for $n=6$ and for every $n\gammae 11$.
\item The complete bipartite graphs verify $\delta(K_{1,n}) = 0$ for every $n\gammae1$, $\delta(K_{m,n}) = 1$ for every $m,n \gammae2$.
\varepsilonnd{itemize}
\varepsilonnd{example}
Theorem \ref{th:deltJoin<1} and Example \ref{examples} show that the family of graphs $E_1\uplus G$ when $G$ belongs to the set of graphs is a representative collection of joins of graphs since their hyperbolicity constants take all possible values.
The following results characterize the graphs with hyperbolicity constant one and greater than one, respectively. If $G_0$ is a subgraph of $G$ and $w\in V(G_0)$, we denote by $\deltaeg_{G_0}(w)$ the degree of $w$ in the induced subgraph by $V(G_0)$.
\betaegin{theorem}\cite[Theorem 3.10]{BRS2}\lambdaabel{th:delt=1}
Let $G$ be any graph. Then $\delta(G) = 1$ if and only if the following conditions hold:
\betaegin{itemize}
\item[(1)] {There exists a cycle isomorphic to $C_4$.}
\item[(2)] {For every cycle $\sigma$ with $L(\sigma) \gammae 5$ and for every vertex $w \in \sigma$, we have $\deltaeg_\sigma(w) \gammae3$.}
\varepsilonnd{itemize}
\varepsilonnd{theorem}
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The $n$-vertex edgeless graph ($n\gammae1$) or \varepsilonmph{empty graph} is a graph without edges and with $n$ vertices, and it is commonly denoted as $E_n$.
The following result allows to characterize the joins of graphs with hyperbolicity constant less than one in terms of its factor graphs.
Recall that $\Delta_G$ denotes the maximum degree of the vertices in $G$.
\betaegin{theorem}\lambdaabel{th:deltJoin<1}
Let $G_1,G_2$ be two graphs.
\betaegin{itemize}
\item[(1)] {$\delta(G_1\uplus G_2)=0$ if and only if $G_1$ and $G_2$ are empty graphs and one of them is isomorphic to $E_1$.}
\item[(2)] {$\delta(G_1\uplus G_2)=3/4$ if and only if $G_1\sigmaimeq E_1$ and $\Delta_{G_2}=1$, or $G_2\sigmaimeq E_1$ and $\Delta_{G_1}=1$.}
\varepsilonnd{itemize}
\varepsilonnd{theorem}
\betaegin{proof}$ $
\betaegin{itemize}
\item[(1)] {By Theorem \ref{th:delt<1} it suffices to characterize the joins of graphs which are trees. If $G_1$ and $G_2$ are empty graphs and one of them is isomorphic to $E_1$, then it is clear that $G_1\uplus G_2$ is a tree. Assume now that $G_1\uplus G_2$ is a tree. If $G_1$ and $G_2$ have at least two vertices then $G_1\uplus G_2$ has a cycle with length four. Thus, $G_1$ or $G_2$ is isomorphic to $E_1$. Without loss of generality we can assume that $G_1\sigmaimeq E_1$. Note that if $G_2$ has at least one edge then $G_1\uplus G_2$ has a cycle with length three. Then, $G_2\sigmaimeq E_n$ for some $n\in \mathbb{N}$.}
\item[(2)] {By Theorem \ref{th:delt<1} it suffices to characterize the joins of graphs with circumference three. If $G_1\sigmaimeq E_1$ and $\Delta_{G_2}=1$, or $G_2\sigmaimeq E_1$ and $\Delta_{G_1}=1$, then it is clear that $c(G_1\uplus G_2)=3$. Assume now that $c(G_1\uplus G_2)=3$. If $G_1,G_2$ both have at least two vertices then $G_1\uplus G_2$ contains a cycle with length four and so $c(G_1\uplus G_2)\gammae4$. Therefore, $G_1$ or $G_2$ is isomorphic to $E_1$. Without loss of generality we can assume that $G_1\sigmaimeq E_1$. Note that if $\Delta_{G_2}\gammae2$ then there is an isomorphic subgraph to $E_1\uplus P_3$ in $G_1\uplus G_2$; thus, $G_1\uplus G_2$ contains a cycle with length four. So, we have $\Delta_{G_2}\lambdae1$. Besides, since $G_2$ is a non-empty graph by (1), we have $\Delta_{G_2}\gammae1$.}
\varepsilonnd{itemize}
\varepsilonnd{proof}
The following result will be useful, see \cite[Theorem 11]{RSVV}.
The graph join of a cycle $C_{n-1}$ and a single vertex $E_1$ is referred to as a \varepsilonmph{wheel} with $n$ vertices and denoted by $W_n$. Notice that the complete bipartite graph $K_{n,m}$ is isomorphic to the graph join of two empty graphs $E_n,E_m$, i.e., $K_{n,m}\sigmaimeq E_n\uplus E_m$.
\betaegin{example}\lambdaabel{examples}
The following graphs have these hyperbolicity constants:
\betaegin{itemize}
\item The wheel graph with $n$ vertices $W_n$ verifies $\delta(W_4)=\delta(W_5)=1$, $\delta(W_n)=3/2$ for every $7\lambdae n\lambdae 10$, and $\delta(W_n)=5/4$ for $n=6$ and for every $n\gammae 11$.
\item The complete bipartite graphs verify $\delta(K_{1,n}) = 0$ for every $n\gammae1$, $\delta(K_{m,n}) = 1$ for every $m,n \gammae2$.
\varepsilonnd{itemize}
\varepsilonnd{example}
Theorem \ref{th:deltJoin<1} and Example \ref{examples} show that the family of graphs $E_1\uplus G$ when $G$ belongs to the set of graphs is a representative collection of joins of graphs since their hyperbolicity constants take all possible values.
The following results characterize the graphs with hyperbolicity constant one and greater than one, respectively. If $G_0$ is a subgraph of $G$ and $w\in V(G_0)$, we denote by $\deltaeg_{G_0}(w)$ the degree of $w$ in the induced subgraph by $V(G_0)$.
\betaegin{theorem}\cite[Theorem 3.10]{BRS2}\lambdaabel{th:delt=1}
Let $G$ be any graph. Then $\delta(G) = 1$ if and only if the following conditions hold:
\betaegin{itemize}
\item[(1)] {There exists a cycle isomorphic to $C_4$.}
\item[(2)] {For every cycle $\sigma$ with $L(\sigma) \gammae 5$ and for every vertex $w \in \sigma$, we have $\deltaeg_\sigma(w) \gammae3$.}
\varepsilonnd{itemize}
\varepsilonnd{theorem}
\betaegin{theorem}\cite[Theorem 3.2]{BRS2}\lambdaabel{th:delt>=5/4}
Let $G$ be any graph. Then $\delta(G) \gammae 5/4$ if and only if there exist a cycle $\sigma$ in $G$ with length $L(\sigma) \gammae 5$ and a vertex $w \in V(\sigma)$ such that $\deltaeg_\sigma(w) = 2$.
\varepsilonnd{theorem}
Theorem \ref{th:delt>=5/4} has the following consequence for joins of graphs.
\betaegin{lemma}\lambdaabel{l:Fact_Delt>1}
Let $G_1,G_2$ be two graphs. If $\delta(G_1)>1$, then $\delta(G_1\uplus G_2)>1$.
\varepsilonnd{lemma}
\betaegin{proof}
By Theorem \ref{th:delt>=5/4}, there exist a cycle $\sigma$ in $G_1\uplus G_2$ (contained in $G_1$) with length $L(\sigma) \gammae 5$ and a vertex $w \in\sigma$ such that $\deltaeg_\sigma(w) = 2$. Thus, Theorem \ref{th:delt>=5/4} gives $\delta(G_1\uplus G_2)>1$.
\varepsilonnd{proof}
Note that the converse of Lemma \ref{l:Fact_Delt>1} does not hold, since $\delta(E_1)=\delta(P_4)=0$ and we can check that $\delta(E_1\uplus P_4)=5/4$.
\betaegin{corollary}\lambdaabel{c:FactDelt>1}
Let $G_1,G_2$ be two graphs. Then $$\delta(G_1\uplus G_2)\gammae \min\betaig\{5/4,\max\{\delta(G_1),\delta(G_2)\}\betaig\}.$$
\varepsilonnd{corollary}
\betaegin{proof}
By symmetry, it suffices to show $\delta(G_1\uplus G_2)\gammae \min\{5/4,\delta(G_1)\}$.
If $\delta(G_1)>1$, then the inequality holds by Lemma \ref{l:Fact_Delt>1}.
If $\delta(G_1)=1$, then there exists a cycle isomorphic to $C_4$ in $G_1\sigmaubset G_1\uplus G_2$; hence, $\delta(G_1\uplus G_2)\gammae1$.
If $\delta(G_1)=3/4$, then there exists a cycle isomorphic to $C_3$ in $G_1\sigmaubset G_1\uplus G_2$; hence, $\delta(G_1\uplus G_2)\gammae3/4$.
The inequality is direct if $\delta(G_1)=0$.
\varepsilonnd{proof}
The following results allow to characterize the joins of graphs with hyperbolicity constant one in terms of $G_1$ and $G_2$.
\betaegin{lemma}\lambdaabel{l:EmptyJoin}
Let $G$ be any graph. Then, $\delta(E_1\uplus G)\lambdae1$ if and only if every path $\varepsilonta$ joining two vertices of $G$ with $L(\varepsilonta) = 3$ satisfies $\deltaeg_\varepsilonta(w)\gammae2$ for every vertex $w \in V(\varepsilonta)$.
\varepsilonnd{lemma}
Note that if every path $\varepsilonta$ joining two vertices of $G$ with $L(\varepsilonta) = 3$ satisfies $\deltaeg_\varepsilonta(w)$ $\gammae2$ for every vertex $w \in V(\varepsilonta)$, then the same result holds for $L(\varepsilonta)\gammae3$ instead of $L(\varepsilonta)=3$.
\betaegin{proof}
Let $v$ be the vertex in $E_1$.
Assume first that $\delta(E_1\uplus G)\lambdae1$. Seeking for a contradiction, assume that there is a path $\varepsilonta$ joining two vertices of $G$ with $L(\varepsilonta) = 3$ and one vertex $w' \in V(\varepsilonta)$ with $\deltaeg_\varepsilonta(w')=1$. Consider now the cycle $\sigma$ obtained by joining the endpoints of $\varepsilonta$ with $v$. Note that $w'\in \sigma$ and $\deltaeg_\sigma(w')=2$; therefore, Theorem \ref{th:delt>=5/4} gives $\delta(E_1\uplus G)>1$, which is a contradiction.
Assume now that every path $\varepsilonta$ joining two vertices of $G$ with $L(\varepsilonta) = 3$ satisfies $\deltaeg_\varepsilonta(w)\gammae2$ for every vertex $w \in V(\varepsilonta)$.
Note that if $G$ does not have paths isomorphic to $P_4$ then there is no cycle in $E_1\uplus G$ with length greater than $4$ and so, $\delta(E_1\uplus G)\lambdae1$.
We are going to prove now that for every cycle $\sigma$ in $G$ with $L(\sigma) \gammae 5$ we have $\deltaeg_{\sigma}(w)\gammae3$ for every vertex $w \in V(\sigma)$.
Let $\sigma$ be any cycle in $E_1\uplus G$ with $L(\sigma) \gammae 5$.
If $v\in \sigma$, then $\sigma\cap G$ is a subgraph of $G$ isomorphic to $P_{n}$ for $n=L(\sigma)-1$, and $\deltaeg_\sigma(v)=n\gammae4$.
Since $L(\sigma\cup G)\gammae3$, $\deltaeg_{\sigma\cap G}(w)\gammae2$ for every $w\in V(\sigma\cap G)$ by hypothesis, and we conclude $\deltaeg_\sigma(w)\gammae3$ for every $w\in V(\sigma)\sigmaetminus\{v\}$.
If $v\notin \sigma$, let $w$ be any vertex in $\sigma$ and let $P(w)$ be a path with length $3$ contained in $\sigma$ and such that $w$ is an endpoint of $P(w)$. By hypothesis $\deltaeg_{P(w)}(w)\gammae2$; since $w$ has a neighbor $w'\in V(\sigma\sigmaetminus P(w))$, $\deltaeg_{\sigma}(w)\gammae3$ for any $w\in V(\sigma)$.
Then, Theorem \ref{th:delt>=5/4} gives the result.
\varepsilonnd{proof}
Note that if a graph $G$ verifies $\deltaiam G\lambdae2$ then every path $\varepsilonta$ joining two vertices of $G$ with $L(\varepsilonta) = 3$ satisfies $\deltaeg_\varepsilonta(w)\gammae2$ for every vertex $w \in V(\varepsilonta)$. The converse does not hold, since in the disjoint union $C_3\cup C_3$ of two cycles $C_3$ any path with length $3$ is a cycle and $\deltaiam C_3\cup C_3 = \infty$. However, these two conditions are equivalent if $G$ is connected.
If $G$ is a graph with connected components $\{G_j\}$, we define
$$
\deltaiam^* G :=\sigmaup_{j} \, \deltaiam G_j.
$$
Note that $\deltaiam^* G = \deltaiam G$ if $G$ is connected; otherwise, $\deltaiam G=\infty$.
Also, $\deltaiam^* G$ $>1$ is equivalent to $\Delta_{G}\gammae2$.
We also have the following result:
\betaegin{lemma}\lambdaabel{lemaX}
Let $G$ be any graph. Then $\deltaiam^* G\lambdae2$ if and only if every $\varepsilonta$ joining two vertices of $G$ with $L(\varepsilonta)=3$ satisfy $\deltaeg_\varepsilonta(w)\gammae2$ for every $w\in V(G)$.
\varepsilonnd{lemma}
\betaegin{lemma}\lambdaabel{l:Fact2Vert}
Let $G_1$ and $G_2$ be two graphs with at least two vertices. Then, $\delta(G_1\uplus G_2)=1$ if and only if $\deltaiam G_i\lambdae2$ or $G_i$ is an empty graph for $i=1,2$.
\varepsilonnd{lemma}
\betaegin{proof}
Assume that $\delta(G_1\uplus G_2)=1$. Seeking for a contradiction, assume that $\deltaiam G_1$ $\gammae5/2$ and $G_1$ is a non-empty graph or $\deltaiam G_2 \gammae 5/2$ and $G_2$ is a non-empty graph. By symmetry, without loss of generality we can assume that $\deltaiam G_1\gammae5/2$ and $G_1$ is a non-empty graph; hence, there are a vertex $v\in V(G_1)$ and a midpoint $p\in [w_1,w_2]$ with $d_{G_1}(v,p)\gammae5/2$. Consider a cycle $\sigma$ in $G_1\uplus G_2$ containing the vertex $v$, the edge $[w_1,w_2]$ and two vertices of $G_2$, with $L(\sigma)=5$. We have $\deltaeg_{\sigma}(v)=2$. Thus, Theorem \ref{th:delt>=5/4} gives $\delta(G_1\uplus G_2)>1$. This contradicts our assumption, and so, we obtain $\deltaiam G_1\lambdae2$.
Assume now that $\deltaiam G_i\lambdae2$ or $G_i$ is an empty graph for $i=1,2$.
Since $G_1$ and $G_2$ have at least two vertices, there exists a cycle isomorphic to $C_4$ in $G_1\uplus G_2$.
First of all, if $G_1$ and $G_2$ are empty graphs then Example \ref{examples} gives $\delta(G_1\uplus G_2)=1$.
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Assume now that every path $\varepsilonta$ joining two vertices of $G$ with $L(\varepsilonta) = 3$ satisfies $\deltaeg_\varepsilonta(w)\gammae2$ for every vertex $w \in V(\varepsilonta)$.
Note that if $G$ does not have paths isomorphic to $P_4$ then there is no cycle in $E_1\uplus G$ with length greater than $4$ and so, $\delta(E_1\uplus G)\lambdae1$.
We are going to prove now that for every cycle $\sigma$ in $G$ with $L(\sigma) \gammae 5$ we have $\deltaeg_{\sigma}(w)\gammae3$ for every vertex $w \in V(\sigma)$.
Let $\sigma$ be any cycle in $E_1\uplus G$ with $L(\sigma) \gammae 5$.
If $v\in \sigma$, then $\sigma\cap G$ is a subgraph of $G$ isomorphic to $P_{n}$ for $n=L(\sigma)-1$, and $\deltaeg_\sigma(v)=n\gammae4$.
Since $L(\sigma\cup G)\gammae3$, $\deltaeg_{\sigma\cap G}(w)\gammae2$ for every $w\in V(\sigma\cap G)$ by hypothesis, and we conclude $\deltaeg_\sigma(w)\gammae3$ for every $w\in V(\sigma)\sigmaetminus\{v\}$.
If $v\notin \sigma$, let $w$ be any vertex in $\sigma$ and let $P(w)$ be a path with length $3$ contained in $\sigma$ and such that $w$ is an endpoint of $P(w)$. By hypothesis $\deltaeg_{P(w)}(w)\gammae2$; since $w$ has a neighbor $w'\in V(\sigma\sigmaetminus P(w))$, $\deltaeg_{\sigma}(w)\gammae3$ for any $w\in V(\sigma)$.
Then, Theorem \ref{th:delt>=5/4} gives the result.
\varepsilonnd{proof}
Note that if a graph $G$ verifies $\deltaiam G\lambdae2$ then every path $\varepsilonta$ joining two vertices of $G$ with $L(\varepsilonta) = 3$ satisfies $\deltaeg_\varepsilonta(w)\gammae2$ for every vertex $w \in V(\varepsilonta)$. The converse does not hold, since in the disjoint union $C_3\cup C_3$ of two cycles $C_3$ any path with length $3$ is a cycle and $\deltaiam C_3\cup C_3 = \infty$. However, these two conditions are equivalent if $G$ is connected.
If $G$ is a graph with connected components $\{G_j\}$, we define
$$
\deltaiam^* G :=\sigmaup_{j} \, \deltaiam G_j.
$$
Note that $\deltaiam^* G = \deltaiam G$ if $G$ is connected; otherwise, $\deltaiam G=\infty$.
Also, $\deltaiam^* G$ $>1$ is equivalent to $\Delta_{G}\gammae2$.
We also have the following result:
\betaegin{lemma}\lambdaabel{lemaX}
Let $G$ be any graph. Then $\deltaiam^* G\lambdae2$ if and only if every $\varepsilonta$ joining two vertices of $G$ with $L(\varepsilonta)=3$ satisfy $\deltaeg_\varepsilonta(w)\gammae2$ for every $w\in V(G)$.
\varepsilonnd{lemma}
\betaegin{lemma}\lambdaabel{l:Fact2Vert}
Let $G_1$ and $G_2$ be two graphs with at least two vertices. Then, $\delta(G_1\uplus G_2)=1$ if and only if $\deltaiam G_i\lambdae2$ or $G_i$ is an empty graph for $i=1,2$.
\varepsilonnd{lemma}
\betaegin{proof}
Assume that $\delta(G_1\uplus G_2)=1$. Seeking for a contradiction, assume that $\deltaiam G_1$ $\gammae5/2$ and $G_1$ is a non-empty graph or $\deltaiam G_2 \gammae 5/2$ and $G_2$ is a non-empty graph. By symmetry, without loss of generality we can assume that $\deltaiam G_1\gammae5/2$ and $G_1$ is a non-empty graph; hence, there are a vertex $v\in V(G_1)$ and a midpoint $p\in [w_1,w_2]$ with $d_{G_1}(v,p)\gammae5/2$. Consider a cycle $\sigma$ in $G_1\uplus G_2$ containing the vertex $v$, the edge $[w_1,w_2]$ and two vertices of $G_2$, with $L(\sigma)=5$. We have $\deltaeg_{\sigma}(v)=2$. Thus, Theorem \ref{th:delt>=5/4} gives $\delta(G_1\uplus G_2)>1$. This contradicts our assumption, and so, we obtain $\deltaiam G_1\lambdae2$.
Assume now that $\deltaiam G_i\lambdae2$ or $G_i$ is an empty graph for $i=1,2$.
Since $G_1$ and $G_2$ have at least two vertices, there exists a cycle isomorphic to $C_4$ in $G_1\uplus G_2$.
First of all, if $G_1$ and $G_2$ are empty graphs then Example \ref{examples} gives $\delta(G_1\uplus G_2)=1$.
Without loss of generality we can assume that $G_1$ is a non-empty graph, then $G_1$ satisfies $\deltaiam G_1\lambdae2$.
Assume that $G_2$ is an empty graph. Let $\sigma$ be any cycle in $G_1\uplus G_2$ with $L(\sigma)\gammae5$. Since $\sigma$ contains at least three vertices in $G_1$, we have $\deltaeg_{\sigma}(v)=|V(G_1)\cap\sigma|\gammae3$ for every $v\in V(G_2)\cap\sigma$.
Besides, if $|V(G_2)\cap\sigma|\gammae3$ then $\deltaeg_{\sigma}(w)\gammae|V(G_2)\cap\sigma|\gammae3$ for every $w\in V(G_1)\cap\sigma$. If $|V(G_2)\cap\sigma|=1$, then $\varepsilonta:=\sigma\cap G_1$ is a path in $G_1$ with $L(\varepsilonta)\gammae3$, and so, $\deltaeg_\varepsilonta(w)\gammae2$ and $\deltaeg_\sigma(w)\gammae3$ for every $w \in V(\varepsilonta)$.
If $|V(G_2)\cap\sigma|=2$, then $\sigma\cap G_1$ is the union of two paths and $|V(G_1)\cap\sigma|\gammae3$; since $\deltaiam G_1\lambdae2$, we have $\deltaeg_{G_1\cap\sigma}(w)\gammae1$ for every $w\in V(G_1)\cap\sigma$ (otherwise there are a vertex $w\in V(G_1)\cap\sigma$ and a midpoint $p\in G_1\cap \sigma$ with $d_{G_1}(w,p)>2$).
Then, we have $\deltaeg_{\sigma}(v)\gammae3$ for every $v\in V(\sigma)$ and so, we obtain $\delta(G_1\uplus G_2)=1$ by Theorem \ref{th:delt=1}.
\sigmamallskip
Finally, assume that $\deltaiam G_2\lambdae2$.
By Theorem \ref{t:TrianVMp} it suffices to consider geodesic triangles $T=\{x,y,z\}$ in $G_1\uplus G_2$ that are cycles with $x,y,z\in J(G_1\uplus G_2)$. So, since $\deltaiam G_1,\deltaiam G_2\lambdae2$, Proposition \ref{prop:JoinDist} gives that $L([xy]),L([yz]),L([zx]) \lambdae 2$; thus, for every $\alpha\in[xy]$, $d_{G_1\uplus G_2}(\alpha,[yz]\cup[zx])\lambdae d_{G_1\uplus G_2}(\alpha,\{x,y\})\lambdae L([xy])/2$. Hence, $\delta(T)\lambdae \max\{L([xy]),L([yz]),L([zx])\}/2\lambdae1$ and so, $\delta(G_1\uplus G_2)\lambdae1$. Since $G_1$ and $G_2$ have at least two vertices, by Theorem \ref{th:delt<1} we have $\delta(G_1\uplus G_2)\gammae1$ and we conclude $\delta(G_1\uplus G_2)=1$.
\varepsilonnd{proof}
The following result characterizes the joins of graphs with hyperbolicity constant one.
\betaegin{theorem}\lambdaabel{th:hypJoin1}
Let $G_1,G_2$ be any two graphs. Then the following statements hold:
\betaegin{itemize}
\item {Assume that $G_1\sigmaimeq E_1$. Then $\delta(G_1\uplus G_2)=1$ if and only if $1<\deltaiam^* G_2\lambdae2$.}
\item {Assume that $G_1$ and $G_2$ have at least two vertices. Then $\delta(G_1\uplus G_2)=1$ if and only if $\deltaiam G_i\lambdae2$ or $G_i$ is an empty graph for $i=1,2$.}
\varepsilonnd{itemize}
\varepsilonnd{theorem}
\betaegin{proof}
We have the first statement by Theorem \ref{th:deltJoin<1} and Lemmas \ref{l:EmptyJoin} and \ref{lemaX}.
The second statement is just Lemma \ref{l:Fact2Vert}.
\varepsilonnd{proof}
In order to compute the hyperbolicity constant of any graph join we are going to characterize the joins of graphs with hyperbolicity constant $3/2$.
\betaegin{lemma}\lambdaabel{l:hyp3/2Fact}
Let $G_1,G_2$ be any two graphs. If $\delta(G_1\uplus G_2)=3/2$, then each geodesic triangle $T=\{x,y,z\}$ in $G_1\uplus G_2$ that is a cycle with $x,y,z \in J(G_1\uplus G_2)$ and $\delta(T)=3/2$ is contained in either $G_1$ or $G_2$.
\varepsilonnd{lemma}
\betaegin{proof}
Seeking for a contradiction assume that there is a geodesic triangle $T=\{x,y,z\}$ in $G_1\uplus G_2$ that is a cycle with $x,y,z \in J(G_1\uplus G_2)$ and $\delta(T)=3/2$ which contains vertices in both factors $G_1,G_2$.
Without loss of generality we can assume that there is $p\in[xy]$ with $d_{G_1\uplus G_2}(p, [yz]\cup[zx]) = 3/2$, and so, $L([xy])\gammae3$. Hence, $d_{G_1\uplus G_2}(x,y)=3$ by Corollary \ref{c:diam}, and by Corollary \ref{cor:midpoint} we have that $x,y$ are midpoints either in $G_1$ or in $G_2$, and so, $p$ is a vertex in $G_1\uplus G_2$. Without loss of generality we can assume that $x,y\in G_1$.
Let $V_x$ be the closest vertex to $x$ in $[xz]\cup[zy]$.
If $p\in V(G_2)$ then $d_{G_1\uplus G_2}(p,[yz]\cup[zx]) \lambdae d_{G_1\uplus G_2}(p,V_x) =1$. This contradicts our assumption.
If $p\in V(G_1)$ then since $T$ contains vertices in both factors, we have $d_{G_1\uplus G_2}(p,[yz]\cup[zx]) \lambdae d_{G_1\uplus G_2}\betaig((p,V(G_2)\cap\{[yz]\cup[zx]\}\betaig) =1$. This also contradicts our assumption, and so, we have the result.
\varepsilonnd{proof}
\betaegin{corollary}\lambdaabel{c:3/2}
Let $G_1,G_2$ be any two graphs. If $\delta(G_1\uplus G_2)=3/2$, then $$\max\{\delta(G_1),\delta(G_2)\}\gammae3/2.$$
\varepsilonnd{corollary}
\sigmamallskip
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\sigmamallskip
Finally, assume that $\deltaiam G_2\lambdae2$.
By Theorem \ref{t:TrianVMp} it suffices to consider geodesic triangles $T=\{x,y,z\}$ in $G_1\uplus G_2$ that are cycles with $x,y,z\in J(G_1\uplus G_2)$. So, since $\deltaiam G_1,\deltaiam G_2\lambdae2$, Proposition \ref{prop:JoinDist} gives that $L([xy]),L([yz]),L([zx]) \lambdae 2$; thus, for every $\alpha\in[xy]$, $d_{G_1\uplus G_2}(\alpha,[yz]\cup[zx])\lambdae d_{G_1\uplus G_2}(\alpha,\{x,y\})\lambdae L([xy])/2$. Hence, $\delta(T)\lambdae \max\{L([xy]),L([yz]),L([zx])\}/2\lambdae1$ and so, $\delta(G_1\uplus G_2)\lambdae1$. Since $G_1$ and $G_2$ have at least two vertices, by Theorem \ref{th:delt<1} we have $\delta(G_1\uplus G_2)\gammae1$ and we conclude $\delta(G_1\uplus G_2)=1$.
\varepsilonnd{proof}
The following result characterizes the joins of graphs with hyperbolicity constant one.
\betaegin{theorem}\lambdaabel{th:hypJoin1}
Let $G_1,G_2$ be any two graphs. Then the following statements hold:
\betaegin{itemize}
\item {Assume that $G_1\sigmaimeq E_1$. Then $\delta(G_1\uplus G_2)=1$ if and only if $1<\deltaiam^* G_2\lambdae2$.}
\item {Assume that $G_1$ and $G_2$ have at least two vertices. Then $\delta(G_1\uplus G_2)=1$ if and only if $\deltaiam G_i\lambdae2$ or $G_i$ is an empty graph for $i=1,2$.}
\varepsilonnd{itemize}
\varepsilonnd{theorem}
\betaegin{proof}
We have the first statement by Theorem \ref{th:deltJoin<1} and Lemmas \ref{l:EmptyJoin} and \ref{lemaX}.
The second statement is just Lemma \ref{l:Fact2Vert}.
\varepsilonnd{proof}
In order to compute the hyperbolicity constant of any graph join we are going to characterize the joins of graphs with hyperbolicity constant $3/2$.
\betaegin{lemma}\lambdaabel{l:hyp3/2Fact}
Let $G_1,G_2$ be any two graphs. If $\delta(G_1\uplus G_2)=3/2$, then each geodesic triangle $T=\{x,y,z\}$ in $G_1\uplus G_2$ that is a cycle with $x,y,z \in J(G_1\uplus G_2)$ and $\delta(T)=3/2$ is contained in either $G_1$ or $G_2$.
\varepsilonnd{lemma}
\betaegin{proof}
Seeking for a contradiction assume that there is a geodesic triangle $T=\{x,y,z\}$ in $G_1\uplus G_2$ that is a cycle with $x,y,z \in J(G_1\uplus G_2)$ and $\delta(T)=3/2$ which contains vertices in both factors $G_1,G_2$.
Without loss of generality we can assume that there is $p\in[xy]$ with $d_{G_1\uplus G_2}(p, [yz]\cup[zx]) = 3/2$, and so, $L([xy])\gammae3$. Hence, $d_{G_1\uplus G_2}(x,y)=3$ by Corollary \ref{c:diam}, and by Corollary \ref{cor:midpoint} we have that $x,y$ are midpoints either in $G_1$ or in $G_2$, and so, $p$ is a vertex in $G_1\uplus G_2$. Without loss of generality we can assume that $x,y\in G_1$.
Let $V_x$ be the closest vertex to $x$ in $[xz]\cup[zy]$.
If $p\in V(G_2)$ then $d_{G_1\uplus G_2}(p,[yz]\cup[zx]) \lambdae d_{G_1\uplus G_2}(p,V_x) =1$. This contradicts our assumption.
If $p\in V(G_1)$ then since $T$ contains vertices in both factors, we have $d_{G_1\uplus G_2}(p,[yz]\cup[zx]) \lambdae d_{G_1\uplus G_2}\betaig((p,V(G_2)\cap\{[yz]\cup[zx]\}\betaig) =1$. This also contradicts our assumption, and so, we have the result.
\varepsilonnd{proof}
\betaegin{corollary}\lambdaabel{c:3/2}
Let $G_1,G_2$ be any two graphs. If $\delta(G_1\uplus G_2)=3/2$, then $$\max\{\delta(G_1),\delta(G_2)\}\gammae3/2.$$
\varepsilonnd{corollary}
\sigmamallskip
The following families of graphs allow to characterize the joins of graphs with hyperbolicity constant $3/2$.
Denote by $C_n$ the cycle graph with $n\gammae3$ vertices and by $V(C_n):=\{v_1^{(n)},\lambdadots,v_n^{(n)}\}$ the set of their vertices such that $[v_n^{(n)},v_1^{(n)}]\in E(C_n)$ and $[v_i^{(n)},v_{i+1}^{(n)}]\in E(C_n)$ for $1\lambdae i\lambdae n-1$.
Let us consider $\mathcal{C}_6^{(1)}$ the set of graphs obtained from $C_6$ by addying a (proper or not) subset of the set of edges $\{[v_2^{(6)},v_6^{(6)}]$, $[v_4^{(6)},v_6^{(6)}]\}$.
Let us define the set of graphs
$$\mathcal{F}_6:=\{ G \ \text{\sigmamall containing, as induced subgraph, an isomorphic graph to some element of } \mathcal{C}_6^{(1)}\}.$$
Let us consider $\mathcal{C}_7^{(1)}$ the set of graphs obtained from $C_7$ by addying a (proper or not) subset of the set of edges $\{[v_2^{(7)},v_6^{(7)}]$, $[v_2^{(7)},v_7^{(7)}]$, $[v_4^{(7)},v_6^{(7)}]$, $[v_4^{(7)},v_7^{(7)}]\}$.
Define
$$\mathcal{F}_7:=\{ G \ \text{\sigmamall containing, as induced subgraph, an isomorphic graph to some element of } \mathcal{C}_7^{(1)}\}.$$
Let us consider $\mathcal{C}_8^{(1)}$ the set of graphs obtained from $C_8$ by addying a (proper or not) subset of the set $\{[v_2^{(8)},v_6^{(8)}]$, $[v_2^{(8)},v_8^{(8)}]$, $[v_4^{(8)},v_6^{(8)}]$, $[v_4^{(8)},v_8^{(8)}]\}$.
Also, consider $\mathcal{C}_8^{(2)}$ the set of graphs obtained from $C_8$ by addying a (proper or not) subset of $\{[v_2^{(8)},v_8^{(8)}]$, $[v_4^{(8)},v_6^{(8)}]$, $[v_4^{(8)},v_7^{(8)}]$, $[v_4^{(8)},v_8^{(8)}]\}$.
Define
$$\mathcal{F}_8:=\{ G \ \text{\sigmamall containing, as induced subgraph, an isomorphic graph to some element of } \mathcal{C}_8^{(1)}\cup \mathcal{C}_8^{(2)}\}.$$
Let us consider $\mathcal{C}_9^{(1)}$ the set of graphs obtained from $C_9$ by addying a (proper or not) subset of the set of edges $\{[v_2^{(9)},v_6^{(9)}]$, $[v_2^{(9)},v_9^{(9)}]$, $[v_4^{(9)},v_6^{(9)}]$, $[v_4^{(9)},v_9^{(9)}]\}$.
Define
$$\mathcal{F}_9:=\{ G \ \text{\sigmamall containing, as induced subgraph, an isomorphic graph to some element of } \mathcal{C}_9^{(1)}\}.$$
Finally, we define the set $\mathcal{F}$ by
$$\mathcal{F}:=\mathcal{F}_6\cup\mathcal{F}_7\cup\mathcal{F}_8\cup\mathcal{F}_9.$$
Note that $\mathcal{F}_6$, $\mathcal{F}_7$, $\mathcal{F}_8$ and $\mathcal{F}_9$ are not disjoint sets of graphs.
The following theorem characterizes the joins of graphs $G_1$ and $G_2$ with $\delta(G_1\uplus G_2)=3/2$.
For any non-empty set $S\sigmaubset V(G)$, the induced subgraph of $S$ will be denoted by $\lambdaangle S\rangle$.
\betaegin{theorem}\lambdaabel{th:hyp3/2}
Let $G_1,G_2$ be any two graphs. Then, $\delta(G_1\uplus G_2)=3/2$ if and only if $G_1\in \mathcal{F}$ or $G_2\in\mathcal{F}$.
\varepsilonnd{theorem}
\betaegin{proof}
Assume first that $\delta(G_1\uplus G_2)=3/2$.
By Theorem \ref{t:TrianVMp} there is a geodesic triangle $T=\{x,y,z\}$ in $G_1\uplus G_2$ that is a cycle with $x,y,z \in J(G_1)$ and $\delta(T)=3/2$. By Lemma \ref{l:hyp3/2Fact}, $T$ is contained either in $G_1$ or in $G_2$.
Without loss of generality we can assume that $T$ is contained in $G_1$.
Without loss of generality we can assume that there is $p\in[xy]$ with $d_{G_1\uplus G_2}(p, [yz]\cup[zx]) = 3/2$, and by Corollary \ref{c:diam}, $L([xy])=3$.
Hence, by Corollary \ref{cor:midpoint} we have that $x,y$ are midpoints in $G_1$, and so, $p\in V(G_1)$.
Since $L([yz])\lambdae3$, $L([zx])\lambdae3$ and $L([yz])+L([zx])\gammae L([xy])$, we have $6\lambdae L(T)\lambdae9$.
\sigmamallskip
Assume that $L(T)=6$.
Denote by $\{v_1,\lambdadots,v_6\}$ the vertices in $T$ such that $T=\betaigcup_{i=1}^{6}[v_i,v_{i+1}]$ with $v_7:=v_1$. Without loss of generality we can assume that $x\in[v_1,v_2]$, $y\in[v_4,v_5]$ and $p=v_3$. Since $d_{G_1\uplus G_2}(x,y)=3$, we have that $\lambdaangle\{v_1,\lambdadots,v_6\}\rangle$ contains neither $[v_1,v_4]$, $[v_1,v_5]$, $[v_2,v_4]$ nor $[v_2,v_5]$; besides, since $d_{G_1\uplus G_2}(p,[yz]\cup[zx])>1$ we have that $\lambdaangle\{v_1,\lambdadots,v_6\}\rangle$ contains neither $[v_3,v_1]$, $[v_3,v_5]$ nor $[v_3,v_6]$.
Note that $[v_2,v_6]$, $[v_4,v_6]$ may be contained in $\lambdaangle\{v_1,\lambdadots,v_6\}\rangle$. Therefore, $G_1\in \mathcal{F}_6$.
Assume that $L(T)=7$ and $G_1\notin \mathcal{F}_6$.
Denote by $\{v_1,\lambdadots,v_7\}$ the vertices in $T$ such that $T=\betaigcup_{i=1}^{7}[v_i,v_{i+1}]$ with $v_8:=v_1$. Without loss of generality we can assume that $x\in[v_1,v_2]$, $y\in[v_4,v_5]$ and $p=v_3$. Since $d_{G_1\uplus G_2}(x,y)=3$, we have that $\lambdaangle\{v_1,\lambdadots,v_7\}\rangle$ contains neither $[v_1,v_4]$, $[v_1,v_5]$, $[v_2,v_4]$ nor $[v_2,v_5]$; besides, since $d_{G_1\uplus G_2}(p,[yz]\cup[zx])>1$ we have that $\lambdaangle\{v_1,\lambdadots,v_7\}\rangle$ contains neither $[v_3,v_1]$, $[v_3,v_5]$, $[v_3,v_6]$ nor $[v_3,v_7]$.
Since $G_1\notin \mathcal{F}_6$, $[v_1,v_6]$ and $[v_5,v_7]$ are not contained in $\lambdaangle\{v_1,\lambdadots,v_7\}\rangle$.
Note that $[v_2,v_6]$, $[v_2,v_7]$, $[v_4,v_6]$, $[v_4,v_7]$ may be contained in $\lambdaangle\{v_1,\lambdadots,v_7\}\rangle$. Hence, $G_1\in \mathcal{F}_7$.
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The following theorem characterizes the joins of graphs $G_1$ and $G_2$ with $\delta(G_1\uplus G_2)=3/2$.
For any non-empty set $S\sigmaubset V(G)$, the induced subgraph of $S$ will be denoted by $\lambdaangle S\rangle$.
\betaegin{theorem}\lambdaabel{th:hyp3/2}
Let $G_1,G_2$ be any two graphs. Then, $\delta(G_1\uplus G_2)=3/2$ if and only if $G_1\in \mathcal{F}$ or $G_2\in\mathcal{F}$.
\varepsilonnd{theorem}
\betaegin{proof}
Assume first that $\delta(G_1\uplus G_2)=3/2$.
By Theorem \ref{t:TrianVMp} there is a geodesic triangle $T=\{x,y,z\}$ in $G_1\uplus G_2$ that is a cycle with $x,y,z \in J(G_1)$ and $\delta(T)=3/2$. By Lemma \ref{l:hyp3/2Fact}, $T$ is contained either in $G_1$ or in $G_2$.
Without loss of generality we can assume that $T$ is contained in $G_1$.
Without loss of generality we can assume that there is $p\in[xy]$ with $d_{G_1\uplus G_2}(p, [yz]\cup[zx]) = 3/2$, and by Corollary \ref{c:diam}, $L([xy])=3$.
Hence, by Corollary \ref{cor:midpoint} we have that $x,y$ are midpoints in $G_1$, and so, $p\in V(G_1)$.
Since $L([yz])\lambdae3$, $L([zx])\lambdae3$ and $L([yz])+L([zx])\gammae L([xy])$, we have $6\lambdae L(T)\lambdae9$.
\sigmamallskip
Assume that $L(T)=6$.
Denote by $\{v_1,\lambdadots,v_6\}$ the vertices in $T$ such that $T=\betaigcup_{i=1}^{6}[v_i,v_{i+1}]$ with $v_7:=v_1$. Without loss of generality we can assume that $x\in[v_1,v_2]$, $y\in[v_4,v_5]$ and $p=v_3$. Since $d_{G_1\uplus G_2}(x,y)=3$, we have that $\lambdaangle\{v_1,\lambdadots,v_6\}\rangle$ contains neither $[v_1,v_4]$, $[v_1,v_5]$, $[v_2,v_4]$ nor $[v_2,v_5]$; besides, since $d_{G_1\uplus G_2}(p,[yz]\cup[zx])>1$ we have that $\lambdaangle\{v_1,\lambdadots,v_6\}\rangle$ contains neither $[v_3,v_1]$, $[v_3,v_5]$ nor $[v_3,v_6]$.
Note that $[v_2,v_6]$, $[v_4,v_6]$ may be contained in $\lambdaangle\{v_1,\lambdadots,v_6\}\rangle$. Therefore, $G_1\in \mathcal{F}_6$.
Assume that $L(T)=7$ and $G_1\notin \mathcal{F}_6$.
Denote by $\{v_1,\lambdadots,v_7\}$ the vertices in $T$ such that $T=\betaigcup_{i=1}^{7}[v_i,v_{i+1}]$ with $v_8:=v_1$. Without loss of generality we can assume that $x\in[v_1,v_2]$, $y\in[v_4,v_5]$ and $p=v_3$. Since $d_{G_1\uplus G_2}(x,y)=3$, we have that $\lambdaangle\{v_1,\lambdadots,v_7\}\rangle$ contains neither $[v_1,v_4]$, $[v_1,v_5]$, $[v_2,v_4]$ nor $[v_2,v_5]$; besides, since $d_{G_1\uplus G_2}(p,[yz]\cup[zx])>1$ we have that $\lambdaangle\{v_1,\lambdadots,v_7\}\rangle$ contains neither $[v_3,v_1]$, $[v_3,v_5]$, $[v_3,v_6]$ nor $[v_3,v_7]$.
Since $G_1\notin \mathcal{F}_6$, $[v_1,v_6]$ and $[v_5,v_7]$ are not contained in $\lambdaangle\{v_1,\lambdadots,v_7\}\rangle$.
Note that $[v_2,v_6]$, $[v_2,v_7]$, $[v_4,v_6]$, $[v_4,v_7]$ may be contained in $\lambdaangle\{v_1,\lambdadots,v_7\}\rangle$. Hence, $G_1\in \mathcal{F}_7$.
Assume that $L(T)=8$ and $G_1\notin \mathcal{F}_6\cup\mathcal{F}_7$.
Denote by $\{v_1,\lambdadots,v_8\}$ the vertices in $T$ such that $T=\betaigcup_{i=1}^{8}[v_i,v_{i+1}]$ with $v_9:=v_1$. Without loss of generality we can assume that $x\in[v_1,v_2]$, $y\in[v_4,v_5]$ and $p=v_3$. Since $d_{G_1\uplus G_2}(x,y)=3$, we have that $\lambdaangle\{v_1,\lambdadots,v_8\}\rangle$ contains neither $[v_1,v_4]$, $[v_1,v_5]$, $[v_2,v_4]$ nor $[v_2,v_5]$; besides, since $d_{G_1\uplus G_2}(p,[yz]\cup[zx])>1$ we have that $\lambdaangle\{v_1,\lambdadots,v_8\}\rangle$ contains neither $[v_3,v_1]$, $[v_3,v_5]$, $[v_3,v_6]$, $[v_3,v_7]$ nor $[v_3,v_8]$.
Since $G_1\notin \mathcal{F}_6\cup\mathcal{F}_7$, $[v_1,v_6]$, $[v_1,v_7]$, $[v_5,v_7]$, $[v_5,v_8]$ and $[v_6,v_8]$ are not contained in $\lambdaangle\{v_1,\lambdadots,v_8\}\rangle$.
Since $T$ is a geodesic triangle we have that $z\in\{v_{6,7},v_7,v_{7,8}\}$ with $v_{6,7}$ and $v_{7,8}$ the midpoints of $[v_6,v_7]$ and $[v_7,v_8]$, respectively.
If $z=v_7$ then $\lambdaangle\{v_1,\lambdadots,v_8\}\rangle$ contains neither $[v_2,v_7]$ nor $[v_4,v_7]$.
Note that $[v_2,v_6]$, $[v_2,v_8]$, $[v_4,v_6]$, $[v_4,v_8]$ may be contained in $\lambdaangle\{v_1,\lambdadots,v_8\}\rangle$.
If $z=v_{6,7}$ then $\lambdaangle\{v_1,\lambdadots,v_8\}\rangle$ contains neither $[v_2,v_6]$ nor $[v_2,v_7]$.
Note that $[v_2,v_8]$, $[v_4,v_6]$, $[v_4,v_7]$, $[v_4,v_8]$ may be contained in $\lambdaangle\{v_1,\lambdadots,v_8\}\rangle$.
By symmetry, we obtain an equivalent result for $z=v_{7,8}$. Therefore, $G_1\in \mathcal{F}_8$.
Assume that $L(T)=9$ and $G_1\notin \mathcal{F}_6\cup\mathcal{F}_7\cup\mathcal{F}_8$.
Denote by $\{v_1,\lambdadots,v_9\}$ the vertices in $T$ such that $T=\betaigcup_{i=1}^{9}[v_i,v_{i+1}]$ with $v_{10}:=v_1$. Without loss of generality we can assume that $x\in[v_1,v_2]$, $y\in[v_4,v_5]$ and $p=v_3$. Since $d_{G_1\uplus G_2}(x,y)=3$, we have that $\lambdaangle\{v_1,\lambdadots,v_9\}\rangle$ contains neither $[v_1,v_4]$, $[v_1,v_5]$, $[v_2,v_4]$ nor $[v_2,v_5]$; besides, since $d_{G_1\uplus G_2}(p,[yz]\cup[zx])>1$ we have that $\lambdaangle\{v_1,\lambdadots,v_9\}\rangle$ contains neither $[v_3,v_1]$, $[v_3,v_5]$, $[v_3,v_6]$, $[v_3,v_7]$, $[v_3,v_8]$ nor $[v_3,v_9]$.
Since $T$ is a geodesic triangle we have that $z$ is the midpoint of $[v_7,v_8]$.
Since $d_{G_1\uplus G_2}(y,z)=d_{G_1\uplus G_2}(z,x)=3$, we have that $\lambdaangle\{v_1,\lambdadots,v_9\}\rangle$ contains neither $[v_1,v_7]$, $[v_1,v_8]$, $[v_2,v_7]$, $[v_2,v_8]$, $[v_4,v_7]$, $[v_4,v_8]$, $[v_5,v_7]$ nor $[v_5,v_8]$.
Since $G_1\notin \mathcal{F}_6\cup\mathcal{F}_7\cup\mathcal{F}_8$, $[v_1,v_6]$, $[v_5,v_9]$, $[v_6,v_8]$, $[v_6,v_9]$ and $[v_7,v_9]$ are not contained in $\lambdaangle\{v_1,\lambdadots,v_9\}\rangle$.
Note that $[v_2,v_6]$, $[v_2,v_9]$, $[v_4,v_6]$, $[v_4,v_9]$ may be contained in $\lambdaangle\{v_1,\lambdadots,v_9\}\rangle$.
Hence, $G_1\in \mathcal{F}_9$.
\sigmamallskip
Finally, one can check that if $G_1\in \mathcal{F}$ or $G_2\in \mathcal{F}$, then $\delta(G_1\uplus G_2)=3/2$, by following the previous arguments.
\varepsilonnd{proof}
These results allow to compute, in a simple way, the hyperbolicity constant of every graph join:
\betaegin{theorem}\lambdaabel{th:hypJoin}
Let $G_1,G_2$ be any two graphs. Then,
\[
\delta(G_1\uplus G_2)=\lambdaeft\{
\betaegin{array}{ll}
0,&\text{if } G_i \sigmaimeq E_1 \text{ and } G_j \sigmaimeq E_n \text{ for } i\neq j \text{ and } n\in \mathbb{N},\\
3/4,&\text{if } G_i\sigmaimeq E_1 \text{ and } \Delta_{G_j}=1 \text{ for } i\neq j,\\
1,&\text{if } G_i\sigmaimeq E_1 \text{ and } 1< \deltaiam^* G_j\lambdae2 \text{ for } i\neq j;\text{ or}\\
& n_i\gammae2 \text{ and } \deltaiam G_i\lambdae2 \text{ or}\\
& G_i \text{ is an empty graph for } i=1,2;\\
3/2,&\text{if } G_1\in \mathcal{F} \text{ or } G_2\in\mathcal{F},\\
5/4,&\text{otherwise}.
\varepsilonnd{array}
\right.
\]
\varepsilonnd{theorem}
\betaegin{corollary}\lambdaabel{c:emptyUplus}
Let $G$ be any graph. Then,
\[
\delta(E_1\uplus G)=\lambdaeft\{
\betaegin{array}{ll}
0,\quad &\mbox{if } \deltaiam^* G=0,\\
3/4,\quad &\mbox{if } \deltaiam^* G=1,\\
1,\quad &\mbox{if } 1< \deltaiam^* G\lambdae2,\\
5/4,\quad &\mbox{if } \deltaiam^* G>2 \text{ and } G\notin \mathcal{F},\\
3/2,\quad &\mbox{if } G\in \mathcal{F}.
\varepsilonnd{array}
\right.
\]
\varepsilonnd{corollary}
\sigmaection{Hyperbolicity of corona of two graphs}\lambdaabel{Sect4}
In this section we study the hyperbolicity of the corona of two graphs, defined by Frucht and Harary in 1970, see \cite{FH}.
\betaegin{definition}\lambdaabel{def:crown}
Let $G_1$ and $G_2$ be two graphs with $V(G_1)\cap V(G_2)=\varepsilonmptyset$. The \varepsilonmph{corona} of $G_1$ and $G_2$, denoted by $G_1\deltaiamond G_2$, is defined as the graph obtained by taking one copy of $G_1$ and a copy of $G_2$ for each vertex $v\in V(G_1)$, and then joining each
vertex $v\in V(G_1)$ to every vertex in the $v$-th copy of $G_2$.
\varepsilonnd{definition}
From the definition, it clearly follows that the corona product of two graphs is a non-commutative and non-associative operation.
Figure \ref{fig:crown} show the corona of two graphs.
\betaegin{figure}[h]
\centering
\sigmacalebox{1.2}
{\betaegin{pspicture}(-.75,-1.9)(6.5,1.9)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(-.5,-.5)(.5,-.5)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(.5,-.5)(.5,.5)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(.5,.5)(-.5,.5)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(-.5,.5)(-.5,-.5)
\uput[0](1,0){$\deltaiamond$}
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\sigmamallskip
Finally, one can check that if $G_1\in \mathcal{F}$ or $G_2\in \mathcal{F}$, then $\delta(G_1\uplus G_2)=3/2$, by following the previous arguments.
\varepsilonnd{proof}
These results allow to compute, in a simple way, the hyperbolicity constant of every graph join:
\betaegin{theorem}\lambdaabel{th:hypJoin}
Let $G_1,G_2$ be any two graphs. Then,
\[
\delta(G_1\uplus G_2)=\lambdaeft\{
\betaegin{array}{ll}
0,&\text{if } G_i \sigmaimeq E_1 \text{ and } G_j \sigmaimeq E_n \text{ for } i\neq j \text{ and } n\in \mathbb{N},\\
3/4,&\text{if } G_i\sigmaimeq E_1 \text{ and } \Delta_{G_j}=1 \text{ for } i\neq j,\\
1,&\text{if } G_i\sigmaimeq E_1 \text{ and } 1< \deltaiam^* G_j\lambdae2 \text{ for } i\neq j;\text{ or}\\
& n_i\gammae2 \text{ and } \deltaiam G_i\lambdae2 \text{ or}\\
& G_i \text{ is an empty graph for } i=1,2;\\
3/2,&\text{if } G_1\in \mathcal{F} \text{ or } G_2\in\mathcal{F},\\
5/4,&\text{otherwise}.
\varepsilonnd{array}
\right.
\]
\varepsilonnd{theorem}
\betaegin{corollary}\lambdaabel{c:emptyUplus}
Let $G$ be any graph. Then,
\[
\delta(E_1\uplus G)=\lambdaeft\{
\betaegin{array}{ll}
0,\quad &\mbox{if } \deltaiam^* G=0,\\
3/4,\quad &\mbox{if } \deltaiam^* G=1,\\
1,\quad &\mbox{if } 1< \deltaiam^* G\lambdae2,\\
5/4,\quad &\mbox{if } \deltaiam^* G>2 \text{ and } G\notin \mathcal{F},\\
3/2,\quad &\mbox{if } G\in \mathcal{F}.
\varepsilonnd{array}
\right.
\]
\varepsilonnd{corollary}
\sigmaection{Hyperbolicity of corona of two graphs}\lambdaabel{Sect4}
In this section we study the hyperbolicity of the corona of two graphs, defined by Frucht and Harary in 1970, see \cite{FH}.
\betaegin{definition}\lambdaabel{def:crown}
Let $G_1$ and $G_2$ be two graphs with $V(G_1)\cap V(G_2)=\varepsilonmptyset$. The \varepsilonmph{corona} of $G_1$ and $G_2$, denoted by $G_1\deltaiamond G_2$, is defined as the graph obtained by taking one copy of $G_1$ and a copy of $G_2$ for each vertex $v\in V(G_1)$, and then joining each
vertex $v\in V(G_1)$ to every vertex in the $v$-th copy of $G_2$.
\varepsilonnd{definition}
From the definition, it clearly follows that the corona product of two graphs is a non-commutative and non-associative operation.
Figure \ref{fig:crown} show the corona of two graphs.
\betaegin{figure}[h]
\centering
\sigmacalebox{1.2}
{\betaegin{pspicture}(-.75,-1.9)(6.5,1.9)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(-.5,-.5)(.5,-.5)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(.5,-.5)(.5,.5)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(.5,.5)(-.5,.5)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(-.5,.5)(-.5,-.5)
\uput[0](1,0){$\deltaiamond$}
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(1.5,-.5)(2.65,-.5)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(2.65,-.5)(2.075,.5)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(2.075,.5)(1.5,-.5)
\uput[0](2.8,0){\lambdaarge{=}}
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-}(3.25,-.75)(4.2,-.7)(3.75,-1.61)(3.25,-.75)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-}(5.8,-.7)(6.75,-.75)(6.25,-1.61)(5.8,-.7)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-}(5.8,.7)(6.75,.75)(6.25,1.61)(5.8,.7)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-}(3.25,.75)(4.2,.7)(3.75,1.61)(3.25,.75)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-}(4.5,-.5)(5.5,-.5)(5.5,.5)(4.5,.5)(4.5,-.5)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{*-*}(4.5,-.5)(3.25,-.75)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(4.5,-.5)(4.2,-.7)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(4.5,-.5)(3.75,-1.61)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{*-*}(5.5,-.5)(5.8,-.7)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(5.5,-.5)(6.75,-.75)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(5.5,-.5)(6.25,-1.61)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{*-*}(5.5,.5)(5.8,.7)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(5.5,.5)(6.75,.75)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(5.5,.5)(6.25,1.61)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{*-*}(4.5,.5)(3.25,.75)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(4.5,.5)(4.2,.7)
\partialsline[linewidth=0.01cm,dotsize=0.07055555cm 2.5]{-*}(4.5,.5)(3.75,1.61)
\varepsilonnd{pspicture}}
\caption{Corona of two graphs $C_4 \deltaiamond C_3$.} \lambdaabel{fig:crown}
\varepsilonnd{figure}
Many authors deal just with corona of finite graphs; however, our results hold for finite or infinite graphs.
If $G$ is a connected graph, we say that $v\in V(G)$ is a \varepsilonmph{connection vertex} if
$G \sigmaetminus \{v\}$ is not connected.
Given a connected graph $G$, a family of subgraphs $\{G_n\}_{n\in \Lambdaambda}$ of
$G$ is a \varepsilonmph{T-decomposi\-tion} of $G$ if $\cup_n G_n=G$ and $G_n\cap G_m$ is either a connection vertex or the empty set for each $n\neq m$.
We will need the following result (see \cite[Theorem 5]{BRSV2}),
which allows to obtain global information about the
hyperbolicity of a graph from local information.
\betaegin{theorem}
\lambdaabel{t:treedec}
Let $G$ be any connected graph and let $\{G_n\}_n$ be any
T-decomposition of $G$. Then $\delta(G)=\sigmaup_n \delta(G_n)$.
\varepsilonnd{theorem}
We remark that the corona $G_1\deltaiamond G_2$ of two graphs is connected if and only if $G_1$ is connected.
The following result characterizes the hyperbolicity of the corona of two graphs and provides the precise value of its hyperbolicity constant.
\betaegin{theorem}\lambdaabel{th:corona}
Let $G_1,G_2$ be any two graphs. Then $\delta(G_1\deltaiamond G_2)=\max\{\delta(G_1),\delta(E_1\uplus G_2)\}$.
\varepsilonnd{theorem}
\betaegin{proof}
Assume first that $G_1$ is connected.
The formula follows from Theorem \ref{t:treedec}, since $\{G_1\}\cup \betaig\{\{v\}\uplus G_2\betaig\}_{v\in V(G_1)}$ is a T-decomposition of $G_1\deltaiamond G_2$.
Finally, note that if $G_1$ is a non-connected graph, then we can apply the previous argument to each connected component.
\varepsilonnd{proof}
Note that Corollary \ref{c:emptyUplus} provides the precise value of $\delta(E_1\uplus G_2)$.
\betaegin{corollary}\lambdaabel{cor:sup}
Let $G_1,G_2$ be any two graphs. Then $G_1\deltaiamond G_2$ is hyperbolic if and only if $G_1$ is hyperbolic.
\varepsilonnd{corollary}
\betaegin{proof}
By Theorem \ref{th:corona} we have $\delta(G_1\deltaiamond G_2)=\max\{\delta(G_1),\delta(E_1\uplus G_2)\}$. Then, by Corollary \ref{cor:SP} we have $\delta(G_1) \lambdae \delta(G_1\deltaiamond G_2) \lambdae \max\{\delta(G_1),3/2\}$.
\varepsilonnd{proof}
\betaegin{thebibliography}{99}
\betaibitem{ABCD}
Alonso, J., Brady, T., Cooper, D., Delzant, T., Ferlini, V.,
Lustig, M., Mihalik, M., Shapiro, M. and Short, H.,
Notes on word hyperbolic groups,
in: E. Ghys, A. Haefliger, A. Verjovsky (Eds.),
Group Theory from a Geometrical Viewpoint,
World Scientific, Singapore, 1992.
\betaibitem{BRS2} Bermudo, S., Rodr\'{\i}guez, J. M. and Sigarreta, J. M.,
Small values of the hyperbolicity constant in graphs.
Submitted.
\betaibitem{BRS} Bermudo, S., Rodr\'{\i}guez, J. M. and Sigarreta, J. M.,
Computing the hyperbolicity constant,
{\it Comput. Math. Appl.} {\betaf 62} (2011), 4592-4595.
\betaibitem{BRSV2} Bermudo, S., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Vilaire, J.-M.,
Gromov hyperbolic graphs,
{\it Discr. Math.} {\betaf 313} (2013), 1575-1585.
\betaibitem{BRST} Bermudo, S., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Tour{\'\i}s, E.,
Hyperbolicity and complement of graphs,
{\it Appl. Math. Letters} {\betaf 24} (2011), 1882-1887.
\betaibitem{BPK} Boguna, M., Papadopoulos, F. and Krioukov, D., Sustaining the Internet with Hyperbolic
Mapping, {\it Nature Commun.} {\betaf 1}(62) (2010), 18 p.
\betaibitem{BHB} Bowditch, B. H., Notes on Gromov's hyperbolicity criterion for path-metric spaces. Group theory from a
geometrical viewpoint, Trieste, 1990 (ed. E. Ghys, A. Haefliger and A. Verjovsky; World Scientific, River
Edge, NJ, 1991) 64-167.
\betaibitem{BHB1} Brinkmann, G., Koolen J. and Moulton ,V., On the hyperbolicity of chordal
graphs, {\it Ann. Comb.} {\betaf 5} (2001), 61-69.
\betaibitem{CCCR} Carballosa, W., Casablanca, R. M., de la Cruz, A. and Rodríguez, J. M., Gromov hyperbolicity in strong product graphs,
{\it Electr. J. Comb.} {\betaf 20}(3) (2013), P2.
\betaibitem{CDR} Carballosa, W., de la Cruz, A. and Rodr\'{\i}guez, J. M.,
Gromov hyperbolicity in lexicographic product graphs.
Submitted.
\betaibitem{CPRS} Carballosa, W., Pestana, D., Rodr\'{\i}guez, J. M. and Sigarreta, J. M.,
Distortion of the hyperbolicity constant of a graph, {\it Electr. J. Comb.} {\betaf 19} (2012), P67.
| 3,999 | 36,288 |
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|
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|
0.193.8
|
\betaegin{theorem}
\lambdaabel{t:treedec}
Let $G$ be any connected graph and let $\{G_n\}_n$ be any
T-decomposition of $G$. Then $\delta(G)=\sigmaup_n \delta(G_n)$.
\varepsilonnd{theorem}
We remark that the corona $G_1\deltaiamond G_2$ of two graphs is connected if and only if $G_1$ is connected.
The following result characterizes the hyperbolicity of the corona of two graphs and provides the precise value of its hyperbolicity constant.
\betaegin{theorem}\lambdaabel{th:corona}
Let $G_1,G_2$ be any two graphs. Then $\delta(G_1\deltaiamond G_2)=\max\{\delta(G_1),\delta(E_1\uplus G_2)\}$.
\varepsilonnd{theorem}
\betaegin{proof}
Assume first that $G_1$ is connected.
The formula follows from Theorem \ref{t:treedec}, since $\{G_1\}\cup \betaig\{\{v\}\uplus G_2\betaig\}_{v\in V(G_1)}$ is a T-decomposition of $G_1\deltaiamond G_2$.
Finally, note that if $G_1$ is a non-connected graph, then we can apply the previous argument to each connected component.
\varepsilonnd{proof}
Note that Corollary \ref{c:emptyUplus} provides the precise value of $\delta(E_1\uplus G_2)$.
\betaegin{corollary}\lambdaabel{cor:sup}
Let $G_1,G_2$ be any two graphs. Then $G_1\deltaiamond G_2$ is hyperbolic if and only if $G_1$ is hyperbolic.
\varepsilonnd{corollary}
\betaegin{proof}
By Theorem \ref{th:corona} we have $\delta(G_1\deltaiamond G_2)=\max\{\delta(G_1),\delta(E_1\uplus G_2)\}$. Then, by Corollary \ref{cor:SP} we have $\delta(G_1) \lambdae \delta(G_1\deltaiamond G_2) \lambdae \max\{\delta(G_1),3/2\}$.
\varepsilonnd{proof}
\betaegin{thebibliography}{99}
\betaibitem{ABCD}
Alonso, J., Brady, T., Cooper, D., Delzant, T., Ferlini, V.,
Lustig, M., Mihalik, M., Shapiro, M. and Short, H.,
Notes on word hyperbolic groups,
in: E. Ghys, A. Haefliger, A. Verjovsky (Eds.),
Group Theory from a Geometrical Viewpoint,
World Scientific, Singapore, 1992.
\betaibitem{BRS2} Bermudo, S., Rodr\'{\i}guez, J. M. and Sigarreta, J. M.,
Small values of the hyperbolicity constant in graphs.
Submitted.
\betaibitem{BRS} Bermudo, S., Rodr\'{\i}guez, J. M. and Sigarreta, J. M.,
Computing the hyperbolicity constant,
{\it Comput. Math. Appl.} {\betaf 62} (2011), 4592-4595.
\betaibitem{BRSV2} Bermudo, S., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Vilaire, J.-M.,
Gromov hyperbolic graphs,
{\it Discr. Math.} {\betaf 313} (2013), 1575-1585.
\betaibitem{BRST} Bermudo, S., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Tour{\'\i}s, E.,
Hyperbolicity and complement of graphs,
{\it Appl. Math. Letters} {\betaf 24} (2011), 1882-1887.
\betaibitem{BPK} Boguna, M., Papadopoulos, F. and Krioukov, D., Sustaining the Internet with Hyperbolic
Mapping, {\it Nature Commun.} {\betaf 1}(62) (2010), 18 p.
\betaibitem{BHB} Bowditch, B. H., Notes on Gromov's hyperbolicity criterion for path-metric spaces. Group theory from a
geometrical viewpoint, Trieste, 1990 (ed. E. Ghys, A. Haefliger and A. Verjovsky; World Scientific, River
Edge, NJ, 1991) 64-167.
\betaibitem{BHB1} Brinkmann, G., Koolen J. and Moulton ,V., On the hyperbolicity of chordal
graphs, {\it Ann. Comb.} {\betaf 5} (2001), 61-69.
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{\it Electr. J. Comb.} {\betaf 20}(3) (2013), P2.
\betaibitem{CDR} Carballosa, W., de la Cruz, A. and Rodr\'{\i}guez, J. M.,
Gromov hyperbolicity in lexicographic product graphs.
Submitted.
\betaibitem{CPRS} Carballosa, W., Pestana, D., Rodr\'{\i}guez, J. M. and Sigarreta, J. M.,
Distortion of the hyperbolicity constant of a graph, {\it Electr. J. Comb.} {\betaf 19} (2012), P67.
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\betaibitem{KPKVB} Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A. and Boguñ\'a, M.,
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\betaibitem{MRSV} Michel, J., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Villeta, M.,
Hyperbolicity and parameters of graphs, {\it Ars Comb.} {\betaf 100} (2011), 43-63.
\betaibitem{MRSV2} Michel, J., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Villeta, M.,
Gromov hyperbolicity in cartesian product graphs, {\it Proc. Indian Acad. Sci. Math. Sci.} {\betaf 120} (2010), 1-17.
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Large-scale curvature of networks,
{\it Physical Review E} {\betaf 84}, 066108 (2011).
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\betaibitem{PeRSV} Pestana, D., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Villeta, M.,
Gromov hyperbolic cubic graphs, {\it Central Europ. J. Math.} {\betaf 10(3)} (2012), 1141-1151.
\betaibitem{PRST} Portilla, A., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Tour{\'\i}s, E.,
Gromov hyperbolic directed graphs, to appear in {\it Acta Math. Appl. Sinica.}
\betaibitem{PRSV} Portilla, A., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Vilaire, J.-M.,
Gromov hyperbolic tessellation graphs, to appear in {\it Utilitas Math.}
Preprint in http://gama.uc3m.es/index.php/jomaro.html
\betaibitem{PT} Portilla, A. and Tour{\'\i}s, E.,
A characterization of Gromov hyperbolicity of surfaces with variable
negative curvature, {\it Publ. Mat.} {\betaf 53} (2009), 83-110.
\betaibitem{R} Rodr\'{\i}guez, J. M.,
Characterization of Gromov hyperbolic short graphs. To appear in Acta Mathematica Sinica.
Preprint in http://gama.uc3m.es/index.php/jomaro.html
\betaibitem{RSVV} Rodr\'{\i}guez, J. M., Sigarreta, J. M., Vilaire, J.-M.
and Villeta, M., On the hyperbolicity constant in graphs, {\it Discr. Math.} {\betaf 311} (2011), 211-219.
\betaibitem{S} Shang, Y.,
Lack of Gromov-hyperbolicity in small-world networks,
{\it Cent. Eur. J. Math.} {\betaf 10} (2012), 1152-1158.
\betaibitem{S2} Shang, Y.,
Non-hyperbolicity of random graphs with given expected degrees,
{\it Stoch. Models} {\betaf 29} (2013), 451-462.
\betaibitem{ShTa} Shavitt, Y., Tankel, T., On internet embedding in hyperbolic
spaces for overlay construction and distance estimation, INFOCOM 2004.
\betaibitem{Si} Sigarreta, J. M., Hyperbolicity in median graphs,
{\it Proc. Indian Acad. Sci. Math. Sci.} {\betaf} (2013). In Press.
\betaibitem{T} Tour{\'\i}s, E., Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces,
{\it J. Math. Anal. Appl.} {\betaf 380} (2011), 865-881.
| 4,028 | 36,288 |
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\betaibitem{K22} Jonckheere, E. A. and Lohsoonthorn, P., Geometry of network security,
{\it Amer. Control Conf.} {\betaf ACC} (2004), 111-151.
\betaibitem{K23} Jonckheere, E. A., Lohsoonthorn, P. and Ariaesi, F, Upper bound on scaled
Gromov-hyperbolic delta, {\it Appl. Math. Comp.} {\betaf 192} (2007), 191-204.
\betaibitem{K24} Jonckheere, E. A., Lohsoonthorn, P. and Bonahon, F., Scaled Gromov
hyperbolic graphs, {\it J. Graph Theory} {\betaf 2} (2007), 157-180.
\betaibitem{K56} Koolen, J. H. and Moulton, V., Hyperbolic Bridged
Graphs, {\it Europ. J. Comb.} {\betaf 23} (2002), 683-699.
\betaibitem{Kra} Krauthgamer, R. and Lee, J. R., Algorithms on negatively curved spaces, FOCS 2006.
\betaibitem{KPKVB} Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A. and Boguñ\'a, M.,
Hyperbolic geometry of complex networks,
{\it Physical Review E} {\betaf 82}, 036106 (2010).
\betaibitem{MRSV} Michel, J., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Villeta, M.,
Hyperbolicity and parameters of graphs, {\it Ars Comb.} {\betaf 100} (2011), 43-63.
\betaibitem{MRSV2} Michel, J., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Villeta, M.,
Gromov hyperbolicity in cartesian product graphs, {\it Proc. Indian Acad. Sci. Math. Sci.} {\betaf 120} (2010), 1-17.
\betaibitem{NS} Narayan, O. and Saniee, I.,
Large-scale curvature of networks,
{\it Physical Review E} {\betaf 84}, 066108 (2011).
\betaibitem{O} Oshika, K., Discrete groups, AMS Bookstore, 2002.
\betaibitem{PeRSV} Pestana, D., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Villeta, M.,
Gromov hyperbolic cubic graphs, {\it Central Europ. J. Math.} {\betaf 10(3)} (2012), 1141-1151.
\betaibitem{PRST} Portilla, A., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Tour{\'\i}s, E.,
Gromov hyperbolic directed graphs, to appear in {\it Acta Math. Appl. Sinica.}
\betaibitem{PRSV} Portilla, A., Rodr\'{\i}guez, J. M., Sigarreta, J. M. and Vilaire, J.-M.,
Gromov hyperbolic tessellation graphs, to appear in {\it Utilitas Math.}
Preprint in http://gama.uc3m.es/index.php/jomaro.html
\betaibitem{PT} Portilla, A. and Tour{\'\i}s, E.,
A characterization of Gromov hyperbolicity of surfaces with variable
negative curvature, {\it Publ. Mat.} {\betaf 53} (2009), 83-110.
\betaibitem{R} Rodr\'{\i}guez, J. M.,
Characterization of Gromov hyperbolic short graphs. To appear in Acta Mathematica Sinica.
Preprint in http://gama.uc3m.es/index.php/jomaro.html
\betaibitem{RSVV} Rodr\'{\i}guez, J. M., Sigarreta, J. M., Vilaire, J.-M.
and Villeta, M., On the hyperbolicity constant in graphs, {\it Discr. Math.} {\betaf 311} (2011), 211-219.
\betaibitem{S} Shang, Y.,
Lack of Gromov-hyperbolicity in small-world networks,
{\it Cent. Eur. J. Math.} {\betaf 10} (2012), 1152-1158.
\betaibitem{S2} Shang, Y.,
Non-hyperbolicity of random graphs with given expected degrees,
{\it Stoch. Models} {\betaf 29} (2013), 451-462.
\betaibitem{ShTa} Shavitt, Y., Tankel, T., On internet embedding in hyperbolic
spaces for overlay construction and distance estimation, INFOCOM 2004.
\betaibitem{Si} Sigarreta, J. M., Hyperbolicity in median graphs,
{\it Proc. Indian Acad. Sci. Math. Sci.} {\betaf} (2013). In Press.
\betaibitem{T} Tour{\'\i}s, E., Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces,
{\it J. Math. Anal. Appl.} {\betaf 380} (2011), 865-881.
\betaibitem{WZ} Wu, Y. and Zhang, C.,
Chordality and hyperbolicity of a graph, {\it Electr. J. Comb.} {\betaf 18} (2011), P43.
\varepsilonnd{thebibliography}
\varepsilonnd{document}
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\begin{document}
\mathbf{t}itle[ Equiaffine Characterization of Lagrangian Surfaces]
{Equiaffine Characterization of Lagrangian \mathbf{n}ewline Surfaces in $\mathbb{R}^4$}
\mathbf{a}uthor[M.Craizer]{Marcos Craizer}
\mathbf{a}ddress{
Catholic University \br
Rio de Janeiro\br
Brazil}
\mathbf{e}mail{[email protected]}
\mathbf{t}hanks{The author want to thank CNPq for financial support during the preparation of this manuscript.}
\keywords{ Shape Operators, Cubic Forms, Affine Normal Plane Bundle}
\subjclass{ 53A15, 53D12}
\date{December 23, 2014}
\begin{abstract}
For non-degenerate surfaces in $\mathbb{R}^4$, a distinguished transversal bundle called affine normal plane bundle was proposed in \cite{Nomizu93}.
Lagrangian surfaces have remarkable properties with respect to this normal bundle, like for example, the normal bundle being Lagrangian.
In this paper we characterize those surfaces which are Lagrangian with respect to some parallel symplectic form in $\mathbb{R}^4$.
\mathbf{e}nd{abstract}
\maketitle
\section{Introduction}
We consider non-degenerate surfaces $M^2\subset\mathbb{R}^4$. For such surfaces, there are many possible choices of the transversal bundle,
and we consider here the {\it affine normal plane bundle} proposed in \cite{Nomizu93}. For affine mean curvature, umbilical surfaces and some other properties of this bundle we
refer to \cite{Dillen}, \cite{Magid}, \cite{Verstraelen} and \cite{Vrancken}.
In this paper, considering the affine normal plane bundle, we give an equiaffine characterization of the Lagrangian surfaces. The results can be compared with
\cite{Morvan87}, where a characterization of Lagrangian surfaces is given in terms of euclidean invariants of the surface.
Consider the affine $4$-space $\mathbb{R}^4$ with the standard connection $D$ and a parallel volume form $[\cdot,\cdot,\cdot,\cdot]$.
Let $M\subset\mathbb{R}^4$ be a surface with a non-degenerate {\it Burstin-Mayer metric} $g$ (\cite{Burstin27}).
For a definite metric $g$, we write $\mathbf{e}psilon=1$, while for an indefinite metric $g$, we write $\mathbf{e}psilon=-1$.
For a given transversal plane bundle $\sigma$ and $X,Y$ tangent vector fields, write
\begin{equation}
D_XY=\mathbf{n}abla_XY+h(x,y),
\mathbf{e}nd{equation}
where $\mathbf{n}abla_XY$ is tangent to $M$ and $h(X,Y)\in\sigma$. Then $\mathbf{n}abla$ is a torsion free affine connection and $h$ is a symmetric bilinear form.
For local vector fields $\{\mathbf{x}i_1,\mathbf{x}i_2\}$ defining a basis of $\sigma$, define the symmetric bilinear forms $h^1$ and $h^2$ by
\begin{equation}
h(X,Y)=h^1(X,Y)\mathbf{x}i_1+h^2(X,Y)\mathbf{x}i_2.
\mathbf{e}nd{equation}
Let $\{X_1,X_2\}$ be a local $g$-orthonormal tangent frame, i.e., $g(X_1,X_1)=\mathbf{e}psilon$, $g(X_1,X_2)=0$, $g(X_2,X_2)=1$. For an arbitrary transversal plane bundle $\sigma$, it is proved in \cite{Nomizu93} that there exists a unique local basis $\{\mathbf{x}i_1,\mathbf{x}i_2\}$
of $\sigma$ such that
$[X_1,X_2,\mathbf{x}i_1,\mathbf{x}i_2]=1$ and
\begin{equation}\label{eq:Normalxi}
\begin{array}{c}
h^1(X_1,X_1)=1,\\
h^1(X_1,X_2)=0,\\
h^1(X_2,X_2)=-\mathbf{e}psilon,
\mathbf{e}nd{array}
\ \ \ \
\begin{array}{c}
h^2(X_1,X_1)=0,\\
h^2(X_1,X_2)=1,\\
h^2(X_2,X_2)=0.
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
There are some transversal plane bundles $\sigma$ with distinguished properties, and we shall consider here the {\it affine normal plane bundle}
proposed in \cite{Nomizu93}.
Assuming that $M$ is Lagrangian with respect to a parallel symplectic form $\Omega$, we shall verify the following remarkable facts: (1) The affine normal plane bundle is $\Omega$-Lagrangian;
(2) $\Omega\mathbf{w}edge\Omega=c[\cdot,\cdot,\cdot,\cdot]$, for some constant c; (3) $\Omega(X_1,\mathbf{x}i_2)-\Omega(X_2,\mathbf{x}i_1)=0$ and $\Omega(X_1,\mathbf{x}i_1)+\mathbf{e}psilon\Omega(X_2,\mathbf{x}i_2)=0$. Based on these
facts, we shall describe the equiaffine conditions for a surface to be Lagrangian with respect to a parallel symplectic form.
Given a transversal bundle $\sigma$ and a local basis $\{\mathbf{x}i_1,\mathbf{x}i_2\}$, define the $1$-forms $\mathbf{t}au_i^j$, $i=1,2$, $j=1,2$, and the shape operators $S_i$ by
\begin{equation}\label{eq:shape}
D_X\mathbf{x}i_i=-S_iX+\mathbf{t}au_i^1(X)\mathbf{x}i_1+\mathbf{t}au_i^2(X)\mathbf{x}i_2,
\mathbf{e}nd{equation}
where $S_iX$ is in the tangent space. Writing
\begin{equation}\label{eq:definelambda}
S_iX_j=\lambda_{ij}^1X_1+\lambda_{ij}^2X_2,
\mathbf{e}nd{equation}
define
\begin{equation}\label{eq:defineL}
\begin{array}{c}
L_{11}=\lambda_{11}^1-\lambda_{21}^2;\ \ L_{12}=-\mathbf{e}psilon\lambda_{11}^2-\lambda_{21}^1\\
L_{21}=\lambda_{12}^1-\lambda_{22}^2;\ \ L_{22}=-\mathbf{e}psilon\lambda_{12}^2-\lambda_{22}^1,
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
and the $2\mathbf{t}imes 2$ matrix
\begin{equation*}
L=\left[
\begin{array}{cc}
L_{11} & L_{12}\\
L_{21} & L_{22}
\mathbf{e}nd{array}
\right].
\mathbf{e}nd{equation*}
We shall verify that the rank of $L$ is independent of the choice of the $g$-orthonormal local frame $\{X_1,X_2\}$.
Consider the cubic forms $C^i$, $i=1,2$ given by
\begin{equation}\label{eq:defineCubic}
C^i(X,Y,Z)=\mathbf{n}abla_Xh^{i}(Y,Z)+\mathbf{t}au_1^{i}(X)h^1(Y,Z)+\mathbf{t}au_2^{i}(X)h^2(Y,Z).
\mathbf{e}nd{equation}
and define
\begin{equation}\label{eq:defineF}
\begin{array}{c}
F_{11}=3C^1(X_1,X_1,X_2)-\mathbf{e}psilon C^1(X_2,X_2,X_2)\\
F_{12}=\mathbf{e}psilon C^1(X_1,X_1,X_1)-3C^1(X_1,X_2,X_2)\\
F_{21}=3C^2(X_1,X_1,X_2)-\mathbf{e}psilon C^2(X_2,X_2,X_2)\\
F_{22}=\mathbf{e}psilon C^2(X_1,X_1,X_1)-3C^2(X_1,X_2,X_2).
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
We shall verify that the rank of the matrix
\begin{equation*}
F=\left[
\begin{array}{cc}
F_{11} & F_{12}\\
F_{21} & F_{22}
\mathbf{e}nd{array}
\right]
\mathbf{e}nd{equation*}
is also independent of the choice of the local $g$-orthonormal tangent frame $\{X_1,X_2\}$. In fact we shall prove that
the rank of the $2\mathbf{t}imes 4$ matrix
$$
H=\left[\ L\ | \ F\ \right]
$$
is independent of the choice of the local frame.
In case $rank(H)=1$, denote by $[A,B]^t$ a column-vector in the kernel of $H$ and let $\mathbf{e}ta=\mathbf{t}an^{-1}(B/A)$, if $\mathbf{e}psilon=1$, and $\mathbf{e}ta=\mathbf{t}anh^{-1}(B/A)$, if $\mathbf{e}psilon=-1$.
Define
\begin{equation}\label{eq:defineG}
\begin{array}{c}
G_1=\Gamma_{22}^2-\mathbf{e}psilon\Gamma_{11}^2+\mathbf{t}au_1^1(X_2)-\mathbf{e}psilon\mathbf{t}au_1^2(X_1)\\
G_2=\Gamma_{11}^1-\mathbf{e}psilon\Gamma_{22}^1-\mathbf{t}au_2^2(X_1)+\mathbf{t}au_1^2(X_2),
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
where
\begin{equation}\label{eq:defineChristoffel}
\mathbf{n}abla_{X_i}X_j=\Gamma_{ij}^1X_1+\Gamma_{ij}^2X_2.
\mathbf{e}nd{equation}
We shall verify that, for the affine normal plane bundle, the conditions
\begin{equation}\label{eq:DerivEta}
\begin{array}{c}
d\mathbf{e}ta(X_1)+\mathbf{e}psilon G_1=0\\
d\mathbf{e}ta(X_2)-G_2=0,
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
are independent of the choice of the local frame.
Our main theorem is the following:
\begin{thm}\label{thm1}
Given a surface $M\subset\mathbb{R}^4$, consider a local tangent frame $\{X_1,X_2\}$ and a local basis $\{\mathbf{x}i_1,\mathbf{x}i_2\}$ of the affine normal plane bundle $\sigma$
satisfying equations \mathbf{e}qref{eq:Normalxi}.
\begin{enumerate}
\item Assume that there exists a parallel symplectic form $\Omega$ such that $L$ is $\Omega$-Lagrangian. Then the affine normal plane bundle is $\Omega$-Lagrangian,
$\Omega\mathbf{w}edge\Omega=c[\cdot,\cdot,\cdot,\cdot]$, for some constant $c$ and $[A,B]^t$ belongs to the kernel of $H$, where $A=\Omega(X_1,\mathbf{x}i_2)=\Omega(X_2,\mathbf{x}i_1)$ and
$B=\Omega(X_1,\mathbf{x}i_1)=-\mathbf{e}psilon\Omega(X_2,\mathbf{x}i_2)$. Moreover $\mathbf{e}ta$ satisfies equations \mathbf{e}qref{eq:DerivEta}.
\item If $rank(H)=1$ and $ker(H)$ satisfies equations \mathbf{e}qref{eq:DerivEta}, then there exists a parallel symplectic form $\Omega$ such that
$M$ is $\Omega$-Lagrangian.
\mathbf{e}nd{enumerate}
\mathbf{e}nd{thm}
In order to complete the picture, it remains to consider what occurs under the hypothesis $H=0$. It is proved in \cite{Nomizu93} that, under the weaker hypothesis $F=0$, $M$ must be a complex curve,
if the metric $g$ is definite, or a product of planar curves, if $g$ is indefinite. In any case, it is well-known that there are two linearly independent parallel symplectic forms under which $M$ is Lagrangian
(see \cite{Craizer14}, \cite{Martinez05}). Thus we can write the following:
\begin{corollary}
A surface $M^2\subset\mathbb{R}^4$ is Lagrangian with respect to a parallel symplectic form if and only if $rank(H)=1$ and equations \mathbf{e}qref{eq:DerivEta} hold or $rank(H)=0$.
\mathbf{e}nd{corollary}
The paper is organized as follows: In section 2 we describe the equiaffine invariants of a surface in $\mathbb{R}^4$, showing that $rank(H)$ is independent of the choice of the local frame. In section 3, we give a characterization of the affine normal bundle in terms of the cubic forms and show that equations \mathbf{e}qref{eq:DerivEta} are independent of the choice of the local frame.
In section 4 we prove the main theorem.
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\section{Shape Operators and Cubic Forms}
\subsection{The affine metric and local frames}
We begin by recalling the definition of the affine metric $g$ of a surface $M\subset\mathbb{R}^4$ (\cite{Burstin27},\cite{Nomizu93}). For a local frame $u=\{X_1,X_2\}$ of the tangent plane, let
$$
G_u(Y,Z)=\frac{1}{2}\left( [X_1,X_2,D_YX_1,D_ZX_2]+ [X_1,X_2,D_ZX_1,D_YX_2] \right).
$$
Denoting
$$
\Delta(u)=G_u(X_1,X_1)G_u(X_2,X_2)-G_u(X_1,X_2)^2,
$$
one can verify that the condition $\Delta(u)\mathbf{n}eq 0$ is independent of the choice of the basis $u$. When this condition holds, we say that the surface in {\it non-degenerate}.
Along this paper, we shall always assume that the surface $M$ is non-degenerate. For a non-degenerate surface, define
$$
g(Y,Z)=\frac{1}{\Delta(u)^{1/3}}G_u(Y,Z).
$$
Then $g$ is independent of $u$ and is called the {\it affine metric} of the surface.
Consider a $g$-orthonormal local frame $\{X_1,X_2\}$ of $M$. Any other $g$-orthonormal local frame $\{Y_1,Y_2\}$ is related to $\{X_1,X_2\}$ by
\begin{equation}\label{eq:ChangeFrame1}
\begin{array}{c}
Y_1=\cos(\mathbf{t}heta)X_1+\sin(\mathbf{t}heta)X_2\\
Y_2=-\sin(\mathbf{t}heta)X_1+\cos(\mathbf{t}heta)X_2,
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
for $\mathbf{e}psilon=1$ and
\begin{equation}\label{eq:ChangeFrame2}
\begin{array}{c}
Y_1=\cosh(\mathbf{t}heta)X_1+\sinh(\mathbf{t}heta)X_2\\
Y_2=\sinh(\mathbf{t}heta)X_1+\cosh(\mathbf{t}heta)X_2,
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
for $\mathbf{e}psilon=-1$, for some $\mathbf{t}heta$.
It is verified in \cite{Nomizu93}, lemmas 4.1 and 4.2, that the corresponding local frame $\{\overline{\mathbf{x}i}_1,\overline{\mathbf{x}i}_2\}$ for $\sigma$ satisfying \mathbf{e}qref{eq:Normalxi} is given by
\begin{equation}
\begin{array}{c}\label{eq:Changexi1}
\overline{\mathbf{x}i}_1=\cos(2\mathbf{t}heta)\mathbf{x}i_1+\sin(2\mathbf{t}heta)\mathbf{x}i_2\\
\overline{\mathbf{x}i}_2=-\sin(2\mathbf{t}heta)\mathbf{x}i_1+\cos(2\mathbf{t}heta)\mathbf{x}i_2,
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
for $\mathbf{e}psilon=1$ and
\begin{equation}\label{eq:Changexi2}
\begin{array}{c}
\overline{\mathbf{x}i}_1=\cosh(2\mathbf{t}heta)\mathbf{x}i_1+\sinh(2\mathbf{t}heta)\mathbf{x}i_2\\
\overline{\mathbf{x}i}_2=\sinh(2\mathbf{t}heta)\mathbf{x}i_1+\cosh(2\mathbf{t}heta)\mathbf{x}i_2,
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
for $\mathbf{e}psilon=-1$.
\subsection{Shape operators}
The shape operators $S_1$ and $S_2$ are defined by equation \mathbf{e}qref{eq:shape} and its components $\lambda_{ij}^k$ are defined by \mathbf{e}qref{eq:definelambda}.
In this section we show how the matrix $L$ defined by \mathbf{e}qref{eq:defineL} changes by a change of the $g$-orthonormal local frame $\{X_1,X_2\}$.
In order to have a more compact notation, consider the matrices $R_{\mathbf{e}psilon}$, $\mathbf{e}psilon=\pm 1$, given by
$$
R_{1}(\mathbf{t}heta)=
\left[
\begin{array}{cc}
\cos(\mathbf{t}heta) & \sin(\mathbf{t}heta)\\
-\sin(\mathbf{t}heta) & \cos(\mathbf{t}heta)
\mathbf{e}nd{array}
\right];\ \
R_{-1}(\mathbf{t}heta)=
\left[
\begin{array}{cc}
\cosh(\mathbf{t}heta) & \sinh(\mathbf{t}heta)\\
\sinh(\mathbf{t}heta) & \cosh(\mathbf{t}heta)
\mathbf{e}nd{array}
\right].
$$
\begin{lemma}\label{lemma:ChangeL}
Denote by $\overline{L}$ the matrix $L$ associated with the local frame $\{Y_1,Y_2\}$ defined by \mathbf{e}qref{eq:ChangeFrame1} and \mathbf{e}qref{eq:ChangeFrame2}. Then
\begin{equation}\label{eq:ChangeL}
\overline{L}=R_{\mathbf{e}psilon}(\mathbf{t}heta)LR_{\mathbf{e}psilon}(3\mathbf{e}psilon\mathbf{t}heta).
\mathbf{e}nd{equation}
\mathbf{e}nd{lemma}
\begin{proof}
The proof are long but straightforward calculations. For example, in case $\mathbf{e}psilon=-1$, we can calculate the first row of $\overline{L}$ as follows: From equation \mathbf{e}qref{eq:Changexi2} we have that
\begin{equation*}
\begin{array}{c}
\overline{S}_1(Y_1)=\cosh(\mathbf{t}heta)\cosh(2\mathbf{t}heta)S_1(X_1)+\cosh(\mathbf{t}heta)\sinh(2\mathbf{t}heta)S_2(X_1)+\\
+\sinh(\mathbf{t}heta)\cosh(2\mathbf{t}heta)S_1(X_2)+\sinh(\mathbf{t}heta)\sinh(2\mathbf{t}heta)S_2(X_2)
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
and
\begin{equation*}
\begin{array}{c}
\overline{S}_2(Y_1)=\cosh(\mathbf{t}heta)\sinh(2\mathbf{t}heta)S_1(X_1)+\cosh(\mathbf{t}heta)\cosh(2\mathbf{t}heta)S_2(X_1)+\\
+\sinh(\mathbf{t}heta)\sinh(2\mathbf{t}heta)S_1(X_2)+\sinh(\mathbf{t}heta)\cosh(2\mathbf{t}heta)S_2(X_2)
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
Now using again equations \mathbf{e}qref{eq:ChangeFrame2} and comparing the coefficients we obtain after some calculations
\begin{equation*}
\begin{array}{c}
\overline{L}_{11}=\cosh(\mathbf{t}heta)\cosh(3\mathbf{t}heta)L_{11}-\cosh(\mathbf{t}heta)\sinh(3\mathbf{t}heta)L_{12}+\\
+\sinh(\mathbf{t}heta)\cosh(3\mathbf{t}heta)L_{21}-\sinh(\mathbf{t}heta)\sinh(3\mathbf{t}heta)L_{22}
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
and
\begin{equation*}
\begin{array}{c}
\overline{L}_{12}=-\cosh(\mathbf{t}heta)\sinh(3\mathbf{t}heta)L_{11}+\cosh(\mathbf{t}heta)\cosh(3\mathbf{t}heta)L_{12}-\\
-\sinh(\mathbf{t}heta)\sinh(3\mathbf{t}heta)L_{21}+\sinh(\mathbf{t}heta)\cosh(3\mathbf{t}heta)L_{22},
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
which agree with equation \mathbf{e}qref{eq:ChangeL}.
\mathbf{e}nd{proof}
\subsection{Cubic forms}
Consider the cubic forms $C^1$ and $C^2$ defined by equation \mathbf{e}qref{eq:defineCubic} and the matrix $F$ whose entries are defined by equations \mathbf{e}qref{eq:defineF}.
\begin{lemma}\label{lemma:ChangeF}
Denote by $\overline{F}$ the matrix $F$ associated with the local frame $\{Y_1,Y_2\}$ defined by \mathbf{e}qref{eq:ChangeFrame1} and \mathbf{e}qref{eq:ChangeFrame2}. Then
\begin{equation}\label{eq:MudF}
\overline{F}=R_{\mathbf{e}psilon}(2\mathbf{e}psilon\mathbf{t}heta)FR_{\mathbf{e}psilon}(3\mathbf{e}psilon\mathbf{t}heta).
\mathbf{e}nd{equation}
\mathbf{e}nd{lemma}
\begin{proof}
We give a proof in case $\mathbf{e}psilon=1$, the case $\mathbf{e}psilon=-1$ being similar. Using complex notation, observe that
$$
C^1(X_1+iX_2, X_1+iX_2,X_1+iX_2)=F_{12}+iF_{11};
$$
$$
C^2(X_1+iX_2, X_1+iX_2,X_1+iX_2)=F_{22}+iF_{21};
$$
By lemma 6.2 of \cite{Nomizu93},
$$
e^{3i\mathbf{t}heta}\overline{C}^1(Y_1+iY_2,Y_1+iY_2,Y_1+iY_2)=
$$
$$
\cos(2\mathbf{t}heta)C^1(X_1+iX_2, X_1+iX_2,X_1+iX_2)+\sin(2\mathbf{t}heta)C^2(X_1+iX_2, X_1+iX_2,X_1+iX_2).
$$
$$
e^{3i\mathbf{t}heta}\overline{C}^2(Y_1+iY_2,Y_1+iY_2,Y_1+iY_2)=
$$
$$
-\sin(2\mathbf{t}heta)C^1(X_1+iX_2, X_1+iX_2,X_1+iX_2)+\cos(2\mathbf{t}heta)C^2(X_1+iX_2, X_1+iX_2,X_1+iX_2).
$$
Thus
$$
\begin{array}{c}
\overline{F}_{12}+i\overline{F}_{11}=e^{-3i\mathbf{t}heta} \left[ \cos(2\mathbf{t}heta)(F_{12}+iF_{11})+\sin(2\mathbf{t}heta)(F_{22}+iF_{21}) \right]\\
\overline{F}_{22}+i\overline{F}_{21}=e^{-3i\mathbf{t}heta} \left[ -\sin(2\mathbf{t}heta)(F_{12}+iF_{11})+\cos(2\mathbf{t}heta)(F_{22}+iF_{21}) \right],
\mathbf{e}nd{array}
$$
which can be written as in equation \mathbf{e}qref{eq:MudF}.
\mathbf{e}nd{proof}
Now we can prove the following lemma:
\begin{lemma}\label{lemma:rankH}
The rank of $H$ is independent of the choice of the local frame $\{X_1,X_2\}$. Moreover, if $rank(H)=1$, then $\overline{\mathbf{e}ta}=\mathbf{e}ta+3\mathbf{t}heta$.
\mathbf{e}nd{lemma}
\begin{proof}
By lemmas \ref{lemma:ChangeL} and \ref{lemma:ChangeF}, the column-vector $[\overline{A},\overline{B}]^t$ belongs to the kernel of $\overline{H}$ if and only if $[A,B]^t$ belongs to the kernel of $H$,
where $[\overline{A},\overline{B}]^t=R_{\mathbf{e}psilon}(-3\mathbf{e}psilon\mathbf{t}heta)[A,B]^t$, which implies the invariance of $rank(H)$. In case $\mathbf{e}psilon=1$, we have
that
\begin{equation*}
\mathbf{t}an(\mathbf{e}ta+3\mathbf{t}heta)=\frac{\sin(\mathbf{e}ta)\cos(3\mathbf{t}heta)+\cos(\mathbf{e}ta)\sin(3\mathbf{t}heta)}{\cos(\mathbf{e}ta)\cos(3\mathbf{t}heta)-\sin(\mathbf{e}ta)\sin(3\mathbf{t}heta)}=\frac{B\cos(3\mathbf{t}heta)+A\sin(3\mathbf{t}heta)}{A\cos(3\mathbf{t}heta)-B\sin(3\mathbf{t}heta)}=\frac{\overline{B}}{\overline{A}},
\mathbf{e}nd{equation*}
thus proving that $\overline{\mathbf{e}ta}=\mathbf{e}ta+3\mathbf{t}heta$. Similarly, in case $\mathbf{e}psilon=-1$,
\begin{equation*}
\mathbf{t}anh(\mathbf{e}ta+3\mathbf{t}heta)=\frac{B\cosh(3\mathbf{t}heta)+A\sinh(3\mathbf{t}heta)}{A\cosh(3\mathbf{t}heta)+B\sinh(3\mathbf{t}heta)}=\frac{\overline{B}}{\overline{A}},
\mathbf{e}nd{equation*}
again proving that $\overline{\mathbf{e}ta}=\mathbf{e}ta+3\mathbf{t}heta$.
\mathbf{e}nd{proof}
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\subsection{Cubic forms}
Consider the cubic forms $C^1$ and $C^2$ defined by equation \mathbf{e}qref{eq:defineCubic} and the matrix $F$ whose entries are defined by equations \mathbf{e}qref{eq:defineF}.
\begin{lemma}\label{lemma:ChangeF}
Denote by $\overline{F}$ the matrix $F$ associated with the local frame $\{Y_1,Y_2\}$ defined by \mathbf{e}qref{eq:ChangeFrame1} and \mathbf{e}qref{eq:ChangeFrame2}. Then
\begin{equation}\label{eq:MudF}
\overline{F}=R_{\mathbf{e}psilon}(2\mathbf{e}psilon\mathbf{t}heta)FR_{\mathbf{e}psilon}(3\mathbf{e}psilon\mathbf{t}heta).
\mathbf{e}nd{equation}
\mathbf{e}nd{lemma}
\begin{proof}
We give a proof in case $\mathbf{e}psilon=1$, the case $\mathbf{e}psilon=-1$ being similar. Using complex notation, observe that
$$
C^1(X_1+iX_2, X_1+iX_2,X_1+iX_2)=F_{12}+iF_{11};
$$
$$
C^2(X_1+iX_2, X_1+iX_2,X_1+iX_2)=F_{22}+iF_{21};
$$
By lemma 6.2 of \cite{Nomizu93},
$$
e^{3i\mathbf{t}heta}\overline{C}^1(Y_1+iY_2,Y_1+iY_2,Y_1+iY_2)=
$$
$$
\cos(2\mathbf{t}heta)C^1(X_1+iX_2, X_1+iX_2,X_1+iX_2)+\sin(2\mathbf{t}heta)C^2(X_1+iX_2, X_1+iX_2,X_1+iX_2).
$$
$$
e^{3i\mathbf{t}heta}\overline{C}^2(Y_1+iY_2,Y_1+iY_2,Y_1+iY_2)=
$$
$$
-\sin(2\mathbf{t}heta)C^1(X_1+iX_2, X_1+iX_2,X_1+iX_2)+\cos(2\mathbf{t}heta)C^2(X_1+iX_2, X_1+iX_2,X_1+iX_2).
$$
Thus
$$
\begin{array}{c}
\overline{F}_{12}+i\overline{F}_{11}=e^{-3i\mathbf{t}heta} \left[ \cos(2\mathbf{t}heta)(F_{12}+iF_{11})+\sin(2\mathbf{t}heta)(F_{22}+iF_{21}) \right]\\
\overline{F}_{22}+i\overline{F}_{21}=e^{-3i\mathbf{t}heta} \left[ -\sin(2\mathbf{t}heta)(F_{12}+iF_{11})+\cos(2\mathbf{t}heta)(F_{22}+iF_{21}) \right],
\mathbf{e}nd{array}
$$
which can be written as in equation \mathbf{e}qref{eq:MudF}.
\mathbf{e}nd{proof}
Now we can prove the following lemma:
\begin{lemma}\label{lemma:rankH}
The rank of $H$ is independent of the choice of the local frame $\{X_1,X_2\}$. Moreover, if $rank(H)=1$, then $\overline{\mathbf{e}ta}=\mathbf{e}ta+3\mathbf{t}heta$.
\mathbf{e}nd{lemma}
\begin{proof}
By lemmas \ref{lemma:ChangeL} and \ref{lemma:ChangeF}, the column-vector $[\overline{A},\overline{B}]^t$ belongs to the kernel of $\overline{H}$ if and only if $[A,B]^t$ belongs to the kernel of $H$,
where $[\overline{A},\overline{B}]^t=R_{\mathbf{e}psilon}(-3\mathbf{e}psilon\mathbf{t}heta)[A,B]^t$, which implies the invariance of $rank(H)$. In case $\mathbf{e}psilon=1$, we have
that
\begin{equation*}
\mathbf{t}an(\mathbf{e}ta+3\mathbf{t}heta)=\frac{\sin(\mathbf{e}ta)\cos(3\mathbf{t}heta)+\cos(\mathbf{e}ta)\sin(3\mathbf{t}heta)}{\cos(\mathbf{e}ta)\cos(3\mathbf{t}heta)-\sin(\mathbf{e}ta)\sin(3\mathbf{t}heta)}=\frac{B\cos(3\mathbf{t}heta)+A\sin(3\mathbf{t}heta)}{A\cos(3\mathbf{t}heta)-B\sin(3\mathbf{t}heta)}=\frac{\overline{B}}{\overline{A}},
\mathbf{e}nd{equation*}
thus proving that $\overline{\mathbf{e}ta}=\mathbf{e}ta+3\mathbf{t}heta$. Similarly, in case $\mathbf{e}psilon=-1$,
\begin{equation*}
\mathbf{t}anh(\mathbf{e}ta+3\mathbf{t}heta)=\frac{B\cosh(3\mathbf{t}heta)+A\sinh(3\mathbf{t}heta)}{A\cosh(3\mathbf{t}heta)+B\sinh(3\mathbf{t}heta)}=\frac{\overline{B}}{\overline{A}},
\mathbf{e}nd{equation*}
again proving that $\overline{\mathbf{e}ta}=\mathbf{e}ta+3\mathbf{t}heta$.
\mathbf{e}nd{proof}
\subsection{Some formulas}
For further references, we write some formulas that hold for any transversal bundle $\sigma$.
The symmetry conditions on the cubic forms imply that
\begin{equation}\label{eq:SymmetryC1}
\begin{array}{c}
2\Gamma_{22}^2+\mathbf{t}au_1^1(X_2)=-\Gamma_{12}^1+\mathbf{e}psilon\Gamma_{11}^2+\mathbf{t}au_2^1(X_1)\\
-2\mathbf{e}psilon\Gamma_{11}^1-\mathbf{e}psilon\mathbf{t}au_1^1(X_1)=\mathbf{e}psilon\Gamma_{21}^2-\Gamma_{22}^1+\mathbf{t}au_2^1(X_2)
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
and
\begin{equation}\label{eq:SymmetryC2}
\begin{array}{c}
-2\Gamma_{12}^1-\mathbf{e}psilon\mathbf{t}au_1^2(X_1)=\mathbf{t}au_2^2(X_2)\\
-2\Gamma_{21}^2+\mathbf{t}au_1^2(X_2)=\mathbf{t}au_2^2(X_1)
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
On the other hand, the condition $[X_1,X_2,\mathbf{x}i_1,\mathbf{x}i_2]=1$ implies that
\begin{equation}\label{eq:Det1}
\begin{array}{c}
\Gamma_{11}^1+\Gamma_{12}^2+\mathbf{t}au_1^1(X_1)+\mathbf{t}au_2^2(X_1)=0\\
\Gamma_{21}^1+\Gamma_{22}^2+\mathbf{t}au_1^1(X_2)+\mathbf{t}au_2^2(X_2)=0,
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
(see \cite{Nomizu93}).
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\section{ The affine normal plane bundle}
\subsection{Definition and some relations}
Consider a $g$-ortonormal local frame $\{X_1,X_2\}$ of the tangent bundle. We say that a transversal bundle $\sigma$ is equiaffine if
\begin{equation*}
\begin{array}{c}
\mathbf{e}psilon\mathbf{n}abla(g)(X_1,X_1,X_1)+\mathbf{n}abla(g)(X_1,X_2,X_2)=0\\
\mathbf{e}psilon\mathbf{n}abla(g)(X_2,X_1,X_1)+\mathbf{n}abla(g)(X_2,X_2,X_2)=0
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
The affine normal plane bundle is an equiaffine bundle $\sigma$ satisfying
\begin{equation*}
\begin{array}{c}
\mathbf{n}abla(g)(X_2,X_1,X_1)+\mathbf{n}abla(g)(X_1,X_2,X_1)=0\\
\mathbf{n}abla(g)(X_1,X_2,X_2)+\mathbf{n}abla(g)(X_2,X_1,X_2)=0
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
Lemma 7.3 of \cite{Nomizu93} says that the affine normal plane is characterized by the conditions
\begin{equation}\label{eq:AffineBundle1}
\Gamma_{12}^2=-\Gamma_{11}^1;\ \Gamma_{21}^1=-\Gamma_{22}^2;
\mathbf{e}nd{equation}
and
\begin{equation}\label{eq:AffineBundle2}
2\Gamma_{11}^1=\Gamma_{21}^2+\mathbf{e}psilon\Gamma_{22}^1;\ 2\Gamma_{22}^2=\Gamma_{12}^1+\mathbf{e}psilon\Gamma_{11}^2.
\mathbf{e}nd{equation}
As a consequence of equations \mathbf{e}qref{eq:Det1} and \mathbf{e}qref{eq:AffineBundle1} we obtain
\begin{equation}\label{eq:SumTau}
\mathbf{t}au_1^1+\mathbf{t}au_2^2=0.
\mathbf{e}nd{equation}
It is proved in \cite{Nomizu93} that a non-degenerate immersion admits a unique affine normal bundle.
\subsection{Characterization of the affine normal bundle in terms of the cubic forms}
Define
\begin{equation*}\label{eq:E1E2}
\begin{array}{c}
E_1=\mathbf{e}psilon C^1(X_1,X_1,X_1)+C^1(X_1,X_2,X_2)-\mathbf{e}psilon C^2(X_1,X_1,X_2)-C^2(X_2,X_2,X_2)\\
E_2=\mathbf{e}psilon C^1(X_1,X_1,X_2)+C^1(X_2,X_2,X_2)+\mathbf{e}psilon C^2(X_1,X_2,X_2)+C^2(X_1,X_1,X_1)\\
E_3=3C^1(X_1,X_1,X_1)-\mathbf{e}psilon C^1(X_1,X_2,X_2)+3C^2(X_1,X_1,X_2)-\mathbf{e}psilon C^2(X_2,X_2,X_2)\\
E_4=C^1(X_1,X_1,X_2)-3\mathbf{e}psilon C^1(X_2,X_2,X_2)+3C^2(X_1,X_2,X_2)-\mathbf{e}psilon C^2(X_1,X_1,X_1).
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
\begin{Proposition}\label{prop:NormalCubic}
$\sigma$ is the affine normal plane bundle if and only if $E_1=E_2=E_3=E_4=0$.
\mathbf{e}nd{Proposition}
\begin{proof}
For a general transversal bundle $\sigma$, the components of the cubic form are given by
\begin{equation}\label{eq:CubicFormulas}
\begin{array}{c}
C^1(X_1,X_1,X_1)=-2\Gamma_{11}^1+\mathbf{t}au_1^1(X_1),\\
C^1(X_1,X_1,X_2)=-2\Gamma_{21}^1+\mathbf{t}au_1^1(X_2),\\
C^1(X_1,X_2,X_2)=2\mathbf{e}psilon\Gamma_{12}^2-\mathbf{e}psilon\mathbf{t}au_1^1(X_1),\\
C^1(X_2,X_2,X_2)=2\mathbf{e}psilon\Gamma_{22}^2-\mathbf{e}psilon\mathbf{t}au_1^1(X_2),
\mathbf{e}nd{array}
\begin{array}{c}
C^2(X_1,X_1,X_1)=-2\Gamma_{11}^2+\mathbf{t}au_1^2(X_1),\\
C^2(X_2,X_1,X_1)=-2\Gamma_{21}^2+\mathbf{t}au_1^2(X_2),\\
C^2(X_1,X_2,X_2)=-2\Gamma_{12}^1-\mathbf{e}psilon\mathbf{t}au_1^2(X_1),\\
C^2(X_2,X_2,X_2)=-2\Gamma_{22}^1-\mathbf{e}psilon\mathbf{t}au_1^2(X_2).
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
Assuming that $\sigma$ is the affine normal plane bundle, equations \mathbf{e}qref{eq:AffineBundle1} and \mathbf{e}qref{eq:AffineBundle2} easily imply that $E_1=E_2=0$. Moreover, it is not difficult
to verify that equations \mathbf{e}qref{eq:AffineBundle1} and \mathbf{e}qref{eq:AffineBundle2} together with equations \mathbf{e}qref{eq:SymmetryC1}, \mathbf{e}qref{eq:SymmetryC2} and \mathbf{e}qref{eq:Det1} imply that
$E_3=E_4=0$.
Assume now that $E_1=E_2=E_3=E_4=0$. Then we can write
\begin{equation*}
\begin{array}{c}
-\Gamma_{11}^1+\Gamma_{12}^2+\Gamma_{21}^2+\mathbf{e}psilon\Gamma_{22}^1=0\\
-3\Gamma_{11}^1-\Gamma_{12}^2-3\Gamma_{21}^2+\mathbf{e}psilon\Gamma_{22}^1=-2(\mathbf{t}au_1^1(X_1)+\mathbf{t}au_1^2(X_2)).
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
and
\begin{equation*}
\begin{array}{c}
-\Gamma_{21}^1+\Gamma_{22}^2-\mathbf{e}psilon\Gamma_{11}^2-\Gamma_{12}^1=0\\
\mathbf{e}psilon\Gamma_{11}^2-3\Gamma_{12}^1-\Gamma_{21}^1-3\Gamma_{22}^2=-2(\mathbf{t}au_1^1(X_2)-\mathbf{e}psilon\mathbf{t}au_1^2(X_1)).
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
By using equations \mathbf{e}qref{eq:SymmetryC2} we obtain
\begin{equation*}
\begin{array}{c}
-3\Gamma_{11}^1-\Gamma_{12}^2+\Gamma_{21}^2+\mathbf{e}psilon\Gamma_{22}^1=-2(\mathbf{t}au_1^1+\mathbf{t}au_2^2)(X_1)\\
\Gamma_{11}^1+\Gamma_{12}^2=(\mathbf{t}au_1^1+\mathbf{t}au_2^2)(X_1),
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
and
\begin{equation*}
\begin{array}{c}
-3\Gamma_{22}^2-\Gamma_{21}^1+\Gamma_{12}^1+\mathbf{e}psilon\Gamma_{11}^2=-2(\mathbf{t}au_1^1+\mathbf{t}au_2^2)(X_2)\\
\Gamma_{22}^2+\Gamma_{21}^1=(\mathbf{t}au_1^1+\mathbf{t}au_2^2)(X_2).
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
Now we use equations \mathbf{e}qref{eq:Det1} to conclude that equations \mathbf{e}qref{eq:AffineBundle1} and \mathbf{e}qref{eq:AffineBundle2} hold, which proves that $\sigma$ is the affine normal plane bundle.
\mathbf{e}nd{proof}
\begin{remark}
There is another choice of the transversal bundle $\sigma$ introduced by Klingenberg (\cite{Klingenberg51}) that is characterized by four conditions
involving the cubic forms $C^1$ and $C^2$ (see lemma 6.1. of \cite{Nomizu93}). Two of these conditions are $E_1=E_2=0$.
\mathbf{e}nd{remark}
When we choose the affine normal bundle as the transversal bundle $\sigma$, the elements $F_{ij}$ of the matrix $F$ assume a remarkable simple form.
\begin{Proposition} For the affine normal plane bundle
\begin{equation}\label{eq:FNormal}
F=\left[
\begin{array}{cc}
F_{11}& F_{12}\\
F_{21}& F_{22}
\mathbf{e}nd{array}
\right]=4\left[
\begin{array}{cc}
\Gamma_{22}^2+\mathbf{t}au_1^1(X_2) & \mathbf{e}psilon\Gamma_{11}^1+\mathbf{e}psilon\mathbf{t}au_1^1(X_1) \\
\Gamma_{11}^1-\mathbf{t}au_1^1(X_1) & -\Gamma_{22}^2+\mathbf{t}au_1^1(X_2)
\mathbf{e}nd{array}
\right]
\mathbf{e}nd{equation}
\mathbf{e}nd{Proposition}
\begin{proof}
We shall check these formulas for $F_{12}$, the other cases being similar. From equations \mathbf{e}qref{eq:CubicFormulas}, we have
$$
F_{12}=\mathbf{e}psilon C^1(X_1,X_1,X_1)-3C^1(X_1,X_2,X_2)=
$$
$$
=\mathbf{e}psilon \left[-2\Gamma_{11}^1+\mathbf{t}au_1^1(X_1) \right]-3\mathbf{e}psilon\left[ 2\Gamma_{12}^2 -\mathbf{t}au_1^1(X_1) \right]=4\mathbf{e}psilon\left[ \Gamma_{11}^1+\mathbf{t}au_1^1(X_1) \right],
$$
where in last equality we have used equations \mathbf{e}qref{eq:AffineBundle1}.
\mathbf{e}nd{proof}
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\subsection{Invariance of equations \mathbf{e}qref{eq:DerivEta} under the choice of the local frame}
Consider $G_1$ and $G_2$ defined by equations \mathbf{e}qref{eq:defineG}.
\begin{lemma}
When $\sigma$ is the affine normal plane bundle we can write
\begin{equation}
\begin{array}{c}
G_1=5\Gamma_{22}^2-3\mathbf{e}psilon\Gamma_{11}^2\\
G_2=5\Gamma_{11}^1-3\mathbf{e}psilon\Gamma_{22}^1.
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
\mathbf{e}nd{lemma}
\begin{proof}
We shall prove the above formula for $G_1$, the proof for $G_2$ is similar.
We have
$$
G_1=\Gamma_{22}^2-\mathbf{e}psilon\Gamma_{11}^2+\mathbf{t}au_1^1(X_2)-\mathbf{e}psilon\mathbf{t}au_1^2(X_1)=\Gamma_{22}^2-\mathbf{e}psilon\Gamma_{11}^2+2\Gamma_{12}^1,
$$
where we have used formulas \mathbf{e}qref{eq:SymmetryC2} and \mathbf{e}qref{eq:SumTau}.
Now using equations \mathbf{e}qref{eq:AffineBundle2}, we obtain the desired formula.
\mathbf{e}nd{proof}
\begin{lemma}
We have that
\begin{equation}
\left[
\begin{array}{c}
\overline{G}_1\\
\overline{G}_2
\mathbf{e}nd{array}
\right]
=R_{\mathbf{e}psilon}(-\mathbf{e}psilon\mathbf{t}heta)
\left[
\begin{array}{c}
G_1\\
G_2
\mathbf{e}nd{array}
\right]
+3
\left[
\begin{array}{cc}
-\mathbf{e}psilon & 0 \\
0 & 1
\mathbf{e}nd{array}
\right]
R_{\mathbf{e}psilon}(\mathbf{t}heta)
\left[
\begin{array}{c}
d\mathbf{t}heta(X_1)\\
d\mathbf{t}heta(X_2)
\mathbf{e}nd{array}
\right].
\mathbf{e}nd{equation}
\mathbf{e}nd{lemma}
\begin{proof}
We consider the case $\mathbf{e}psilon=1$, the case $\mathbf{e}psilon=-1$ being similar. From equations \mathbf{e}qref{eq:ChangeFrame1} we obtain
\begin{equation*}
\begin{array}{c}
\mathbf{n}abla_{Y_1}Y_1=\cos^2(\mathbf{t}heta)\mathbf{n}abla_{X_1}X_1+\sin(\mathbf{t}heta)\cos(\mathbf{t}heta)(\mathbf{n}abla_{X_1}X_2+\mathbf{n}abla_{X_2}X_1)+\sin^2(\mathbf{t}heta)\mathbf{n}abla_{X_2}X_2\\
+\left[ d\mathbf{t}heta(X_1)\cos(\mathbf{t}heta)+d\mathbf{t}heta(X_2)\sin(\mathbf{t}heta) \right]Y_2
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
\begin{equation*}
\begin{array}{c}
\mathbf{n}abla_{Y_2}Y_2=\sin^2(\mathbf{t}heta)\mathbf{n}abla_{X_1}X_1-\sin(\mathbf{t}heta)\cos(\mathbf{t}heta)(\mathbf{n}abla_{X_1}X_2+\mathbf{n}abla_{X_2}X_1)+\cos^2(\mathbf{t}heta)\mathbf{n}abla_{X_2}X_2\\
-\left[ -d\mathbf{t}heta(X_1)\sin(\mathbf{t}heta)+d\mathbf{t}heta(X_2)\cos(\mathbf{t}heta) \right]Y_1
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
Using again equations \mathbf{e}qref{eq:ChangeFrame1} we obtain
$$
\overline{G}_1=-3\left(\cos(\mathbf{t}heta)d\mathbf{t}heta(X_1)+\sin(\mathbf{t}heta)d\mathbf{t}heta(X_2)\right)+\cos^3(\mathbf{t}heta)G_1-\sin^3(\mathbf{t}heta)G_2+
$$
$$
+5\left[\sin^2(\mathbf{t}heta)\cos(\mathbf{t}heta)(\Gamma_{11}^2+\Gamma_{12}^1+\Gamma_{21}^1)-\sin(\mathbf{t}heta)\cos^2(\mathbf{t}heta)(\Gamma_{22}^1+\Gamma_{12}^2+\Gamma_{21}^2) \right]
$$
$$
-3\left[\sin^2(\mathbf{t}heta)\cos(\mathbf{t}heta)(\Gamma_{22}^2-\Gamma_{12}^1-\Gamma_{21}^1)+\sin(\mathbf{t}heta)\cos^2(\mathbf{t}heta)(\Gamma_{21}^2+\Gamma_{12}^2-\Gamma_{11}^1) \right].
$$
Using now equations \mathbf{e}qref{eq:AffineBundle1} and \mathbf{e}qref{eq:AffineBundle2} we obtain
\begin{equation*}
\overline{G}_1=\cos(\mathbf{t}heta)G_1-\sin(\mathbf{t}heta)G_2-3(\cos(\mathbf{t}heta)d\mathbf{t}heta(X_1)+\sin(\mathbf{t}heta)d\mathbf{t}heta(X_2))
\mathbf{e}nd{equation*}
Similar calculations leads to
\begin{equation*}
\overline{G}_2=\sin(\mathbf{t}heta)G_1+\cos(\mathbf{t}heta)G_2+3(-\sin(\mathbf{t}heta)d\mathbf{t}heta(X_1)+\cos(\mathbf{t}heta)d\mathbf{t}heta(X_2)),
\mathbf{e}nd{equation*}
thus proving the lemma.
\mathbf{e}nd{proof}
\begin{corollary}
We have that
\begin{equation}
\left[
\begin{array}{c}
d\overline\mathbf{e}ta(Y_1)+\mathbf{e}psilon \overline{G}_1\\
d\overline\mathbf{e}ta(Y_2)-\overline{G}_2
\mathbf{e}nd{array}
\right]
=R_{\mathbf{e}psilon}(\mathbf{t}heta)\cdot\left[
\begin{array}{c}
d\mathbf{e}ta(X_1) +\mathbf{e}psilon G_1 \\
d\mathbf{e}ta(X_2) -G_2
\mathbf{e}nd{array}
\right].
\mathbf{e}nd{equation}
\mathbf{e}nd{corollary}
\begin{proof}
By lemma \ref{lemma:rankH}, $\overline{\mathbf{e}ta}=\mathbf{e}ta+3\mathbf{t}heta$. Thus, if $\mathbf{e}psilon=1$,
\begin{equation*}
\begin{array}{c}
d\overline{\mathbf{e}ta}(Y_1)=\cos(\mathbf{t}heta)d\mathbf{e}ta(X_1)+\sin(\mathbf{t}heta)d\mathbf{e}ta(X_2)+3(\cos(\mathbf{t}heta)d\mathbf{t}heta(X_1)+\sin(\mathbf{t}heta)d\mathbf{t}heta(X_2))\\
d\overline{\mathbf{e}ta}(Y_2)=-\sin(\mathbf{t}heta)d\mathbf{e}ta(X_1)+\cos(\mathbf{t}heta)d\mathbf{e}ta(X_2)+3(-\sin(\mathbf{t}heta)d\mathbf{t}heta(X_1)+\cos(\mathbf{t}heta)d\mathbf{t}heta(X_2)),
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
which implies that
\begin{equation*}
\begin{array}{c}
d\overline{\mathbf{e}ta}(Y_1)+\overline{G}_1=\cos(\mathbf{t}heta)(d\mathbf{e}ta(X_1)+G_1)+\sin(\mathbf{t}heta)(d\mathbf{e}ta(X_2)-G_2)\\
d\overline{\mathbf{e}ta}(Y_2)-\overline{G}_2=-\sin(\mathbf{t}heta)(d\mathbf{e}ta(X_1)+G_1)+\cos(\mathbf{t}heta)(d\mathbf{e}ta(X_2)-G_2).
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
If $\mathbf{e}psilon=-1$,
\begin{equation*}
\begin{array}{c}
d\overline{\mathbf{e}ta}(Y_1)=\cosh(\mathbf{t}heta)d\mathbf{e}ta(X_1)+\sinh(\mathbf{t}heta)d\mathbf{e}ta(X_2)+3(\cosh(\mathbf{t}heta)d\mathbf{t}heta(X_1)+\sinh(\mathbf{t}heta)d\mathbf{t}heta(X_2))\\
d\overline{\mathbf{e}ta}(Y_2)=\sinh(\mathbf{t}heta)d\mathbf{e}ta(X_1)+\cosh(\mathbf{t}heta)d\mathbf{e}ta(X_2)+3(\sinh(\mathbf{t}heta)d\mathbf{t}heta(X_1)+\cosh(\mathbf{t}heta)d\mathbf{t}heta(X_2)),
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
implying that
\begin{equation*}
\begin{array}{c}
d\overline{\mathbf{e}ta}(Y_1)-\overline{G}_1=\cosh(\mathbf{t}heta)(d\mathbf{e}ta(X_1)-G_1)+\sinh(\mathbf{t}heta)(d\mathbf{e}ta(X_2)-G_2)\\
d\overline{\mathbf{e}ta}(Y_2)-\overline{G}_2=\sinh(\mathbf{t}heta)(d\mathbf{e}ta(X_1)-G_1)+\cosh(\mathbf{t}heta)(d\mathbf{e}ta(X_2)-G_2),
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
thus proving the corollary.
\mathbf{e}nd{proof}
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\section{Proof of the Main Theorem}
We begin with the following lemma:
\begin{lemma}\label{lemma:Proof}
The system of equations
\begin{equation}\label{eq:Equiv1}
dA(X_1)=G_1B;\ dA(X_2)=-\mathbf{e}psilon G_2B;\ dB(X_1)=-\mathbf{e}psilon G_1A;\ dB(X_2)=G_2A;
\mathbf{e}nd{equation}
is equivalent to
\begin{equation}\label{eq:Equiv2}
A^2+\mathbf{e}psilon B^2=c;\ d\mathbf{e}ta(X_1)=-\mathbf{e}psilon G_1;\ d\mathbf{e}ta(X_2)=G_2,
\mathbf{e}nd{equation}
for some constant $c$, where $\mathbf{t}an(\mathbf{e}ta)=\frac{B}{A}$, if $\mathbf{e}psilon=1$, and $\mathbf{t}anh(\mathbf{e}ta)=\frac{B}{A}$, if $\mathbf{e}psilon=-1$.
\mathbf{e}nd{lemma}
\begin{proof}
If we assume that equations \mathbf{e}qref{eq:Equiv1} hold, then
\begin{equation*}
AdA(X_1)+\mathbf{e}psilon BdB(X_1)=0; \ \ AdA(X_2)+\mathbf{e}psilon BdB(X_2)=0,
\mathbf{e}nd{equation*}
which implies $A^2+\mathbf{e}psilon B^2=c$, for some costant $c\mathbf{n}eq 0$, and
\begin{equation*}
d\mathbf{e}ta(X_1)=-\mathbf{e}psilon G_1;\ \ d\mathbf{e}ta(X_2)=G_2.
\mathbf{e}nd{equation*}
On the other hand, if equations \mathbf{e}qref{eq:Equiv2} hold, then we can define
\begin{equation*}
\mathbf{t}ilde{G}_1=\frac{dA(X_1)}{B}=-\mathbf{e}psilon\frac{dB(X_1)}{A}
\mathbf{e}nd{equation*}
to obtain
\begin{equation}
d\mathbf{e}ta(X_1)=-\mathbf{e}psilon \mathbf{t}ilde{G}_1
\mathbf{e}nd{equation}
and conclude that $\mathbf{t}ilde{G}_1=G_1$. In a similar way we show that $dA(X_2)=-\mathbf{e}psilon G_2B;\ dB(X_2)=G_2A$,
which completes the proof of the lemma.
\mathbf{e}nd{proof}
\paragraph{Proof of theorem \ref{thm1}, part 1:}
Assume that $\Omega$ is a parallel symplectic form such that $S$ is $\Omega$-Lagrangian. Differentiating
\begin{equation}
\Omega(X_1,X_2)=0
\mathbf{e}nd{equation}
with respect to $X_1$ and $X_2$ we obtain
\begin{equation*}
\begin{array}{c}
\Omega(D_{X_1}X_1,X_2)+\Omega(X_1,D_{X_1}X_2)=0\\
\Omega(D_{X_2}X_1,X_2)+\Omega(X_1,D_{X_2}X_2)=0,
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
which is equivalent to
\begin{equation*}
\begin{array}{c}
\Omega(\mathbf{x}i_1,X_2)+\Omega(X_1,\mathbf{x}i_2)=0\\
\Omega(\mathbf{x}i_2,X_2)+\Omega(X_1,-\mathbf{e}psilon\mathbf{x}i_1)=0,
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
Write then
\begin{equation}
\Omega(X_1,\mathbf{x}i_2)=A,\ \Omega(X_2,\mathbf{x}i_1)=A; \ \ \Omega(X_1,\mathbf{x}i_1)=B,\ \Omega(X_2,\mathbf{x}i_2)=-\mathbf{e}psilon B;
\mathbf{e}nd{equation}
for some functions $A$ and $B$.
Differentiating $A$ with respect to $X_1$ in the first two equations we obtain
\begin{equation}\label{eq:DA1}
\begin{array}{c}
dA(X_1)=\left( \Gamma_{11}^1+\mathbf{t}au_2^2(X_1) \right) A+ \left( -\mathbf{e}psilon\Gamma_{11}^2 +\mathbf{t}au_2^1(X_1) \right) B+\Omega(\mathbf{x}i_1,\mathbf{x}i_2); \\
dA(X_1)= \left( \Gamma_{12}^2+\mathbf{t}au_1^1(X_1) \right)A +\left( \Gamma_{12}^1-\mathbf{e}psilon \mathbf{t}au_1^2(X_1) \right)B-\Omega(\mathbf{x}i_1,\mathbf{x}i_2).
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
or equivalently
\begin{equation*}
\begin{array}{c}
(\Gamma_{11}^1+\mathbf{t}au_2^2(X_1)-\Gamma_{12}^2-\mathbf{t}au_1^1(X_1)) A +2\Omega(\mathbf{x}i_1,\mathbf{x}i_2)= (\Gamma_{12}^1-\mathbf{e}psilon\mathbf{t}au_1^2(X_1)+\mathbf{e}psilon\Gamma_{11}^2-\mathbf{t}au_2^1(X_1)) B\\
2dA(X_1)=(\Gamma_{11}^1+\Gamma_{12}^2+\mathbf{t}au_1^1(X_1)+\mathbf{t}au_2^2(X_1))A+(\Gamma_{12}^1-\mathbf{e}psilon\Gamma_{11}^2+\mathbf{t}au_2^1(X_1)-\mathbf{e}psilon\mathbf{t}au_1^2(X_1))B.
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
By using equations \mathbf{e}qref{eq:SymmetryC1}, \mathbf{e}qref{eq:SymmetryC2}, \mathbf{e}qref{eq:Det1}, \mathbf{e}qref{eq:AffineBundle1} and \mathbf{e}qref{eq:AffineBundle2}, we verify that these equations are equivalent to
\begin{equation}\label{eq:DA11}
F_{21}A+F_{22}B+4\Omega(\mathbf{x}i_1,\mathbf{x}i_2)=0
\mathbf{e}nd{equation}
and
\begin{equation}\label{eq:DA12}
dA(X_1)=G_1 B.
\mathbf{e}nd{equation}
Differentiating $A$ with respect to $X_2$ we obtain
\begin{equation}\label{eq:DA2}
\begin{array}{c}
dA(X_2)=\left( \Gamma_{21}^1+\mathbf{t}au_2^2(X_2) \right) A+ \left( -\mathbf{e}psilon\Gamma_{21}^2 +\mathbf{t}au_2^1(X_2) \right) B; \\
dA(X_2)= \left( \Gamma_{22}^2+\mathbf{t}au_1^1(X_2) \right)A +\left( \Gamma_{22}^1-\mathbf{e}psilon \mathbf{t}au_1^2(X_2) \right)B,
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
which are equivalent to
\begin{equation}\label{eq:DA21}
F_{11} A +F_{12} B=0
\mathbf{e}nd{equation}
and
\begin{equation}\label{eq:DA22}
dA(X_2)=-\mathbf{e}psilon G_2 B.
\mathbf{e}nd{equation}
Now differentiate $B$ with respect to $X_1$ to obtain
\begin{equation}\label{eq:DB1}
\begin{array}{c}
dB(X_1)=\left( \Gamma_{11}^2+\mathbf{t}au_1^2(X_1) \right) A+ \left( \Gamma_{11}^1 +\mathbf{t}au_1^1(X_1) \right) B; \\
-\mathbf{e}psilon dB(X_1)= \left( \Gamma_{12}^1+\mathbf{t}au_2^1(X_1) \right)A -\mathbf{e}psilon\left( \Gamma_{12}^2+ \mathbf{t}au_2^2(X_1) \right)B.
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
We can verify that these equations are equivalent to
\begin{equation}\label{eq:DB11}
F_{11} A + F_{12} B=0
\mathbf{e}nd{equation}
and
\begin{equation}\label{eq:DB12}
dB(X_1)=-\mathbf{e}psilon G_1 A.
\mathbf{e}nd{equation}
Differentiating $B$ with respect to $X_2$ we get
\begin{equation}\label{eq:DB2}
\begin{array}{c}
dB(X_2)=\left( \Gamma_{21}^2+\mathbf{t}au_1^2(X_2) \right) A+ \left( \Gamma_{21}^1 +\mathbf{t}au_1^1(X_2) \right) B-\Omega(\mathbf{x}i_1,\mathbf{x}i_2); \\
-\mathbf{e}psilon dB(X_2)= \left( \Gamma_{22}^1+\mathbf{t}au_2^1(X_2) \right)A -\mathbf{e}psilon\left( \Gamma_{22}^2+ \mathbf{t}au_2^2(X_2) \right)B-\mathbf{e}psilon\Omega(\mathbf{x}i_1,\mathbf{x}i_2),
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
which are equivalent to
\begin{equation}\label{eq:DB21}
F_{21} A +F_{22} B-4\Omega(\mathbf{x}i_1,\mathbf{x}i_2)=0
\mathbf{e}nd{equation}
and
\begin{equation}\label{eq:DB22}
dB(X_2)=G_2 A.
\mathbf{e}nd{equation}
From equations \mathbf{e}qref{eq:DA11} and \mathbf{e}qref{eq:DB21} we conclude that $\Omega(\mathbf{x}i_1,\mathbf{x}i_2)=0$.
It follows that equations \mathbf{e}qref{eq:DA11}, \mathbf{e}qref{eq:DA21}, \mathbf{e}qref{eq:DB11} and \mathbf{e}qref{eq:DB21} are reduced to
\begin{equation*}\label{eq:KernelF}
\begin{array}{c}
F_{11}A+F_{12}B=0\\
F_{21}A+F_{22}B=0,
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
which says that $[A,B]^t$ belongs to the kernel of $F$.
Differentiating $\Omega(\mathbf{x}i_1,\mathbf{x}i_2)=0$ we obtain
\begin{equation}
\begin{array}{c}
\lambda_{11}^1A-\lambda_{11}^2\mathbf{e}psilon B-\lambda_{21}^1B-\lambda_{21}^2A=0\\
\lambda_{12}^1A-\lambda_{12}^2\mathbf{e}psilon B-\lambda_{22}^1B-\lambda_{22}^2A=0,
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
which can be written as
\begin{equation*}\label{eq:KernelL}
\begin{array}{c}
L_{11}A+L_{12}B=0\\
L_{21}A+L_{22}B=0.
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
We conclude that $[A,B]^t$ belongs to the kernel of $L$ and hence $rank(H)<2$.
Finally, by lemma \ref{lemma:Proof}, equations \mathbf{e}qref{eq:DA12}, \mathbf{e}qref{eq:DA22}, \mathbf{e}qref{eq:DB12} and \mathbf{e}qref{eq:DB22} are equivalent to
$A^2+\mathbf{e}psilon B^2=c$, for some constant $c$ and to equations \mathbf{e}qref{eq:DerivEta}. Equation $A^2+\mathbf{e}psilon B^2=c$ implies $\Omega\mathbf{w}edge\Omega=c[\cdot,\cdot,\cdot,\cdot]$.
\paragraph{Proof of theorem \ref{thm1}, part 2:}
Assume that $rank(H)=1$ and equations \mathbf{e}qref{eq:DerivEta} hold. Denote by
$[A,B]^t$ a column-vector in $Ker(H)$ satisfying $A^2+\mathbf{e}psilon B^2=c$, for some constant $c\mathbf{n}eq 0$. Define the symplectic form $\Omega$ by the conditions
\begin{equation*}
\begin{array}{c}
\Omega(X_1,X_2)=\Omega(\mathbf{x}i_1,\mathbf{x}i_2)=0\\
\Omega(X_1,\mathbf{x}i_2)=\Omega(X_2,\mathbf{x}i_1)=A\\
\Omega(X_1,\mathbf{x}i_1)=-\mathbf{e}psilon \Omega(X_2,\mathbf{x}i_2)=B
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
We shall prove that the symplectic form $\Omega$ is parallel.
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From equations \mathbf{e}qref{eq:DA11} and \mathbf{e}qref{eq:DB21} we conclude that $\Omega(\mathbf{x}i_1,\mathbf{x}i_2)=0$.
It follows that equations \mathbf{e}qref{eq:DA11}, \mathbf{e}qref{eq:DA21}, \mathbf{e}qref{eq:DB11} and \mathbf{e}qref{eq:DB21} are reduced to
\begin{equation*}\label{eq:KernelF}
\begin{array}{c}
F_{11}A+F_{12}B=0\\
F_{21}A+F_{22}B=0,
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
which says that $[A,B]^t$ belongs to the kernel of $F$.
Differentiating $\Omega(\mathbf{x}i_1,\mathbf{x}i_2)=0$ we obtain
\begin{equation}
\begin{array}{c}
\lambda_{11}^1A-\lambda_{11}^2\mathbf{e}psilon B-\lambda_{21}^1B-\lambda_{21}^2A=0\\
\lambda_{12}^1A-\lambda_{12}^2\mathbf{e}psilon B-\lambda_{22}^1B-\lambda_{22}^2A=0,
\mathbf{e}nd{array}
\mathbf{e}nd{equation}
which can be written as
\begin{equation*}\label{eq:KernelL}
\begin{array}{c}
L_{11}A+L_{12}B=0\\
L_{21}A+L_{22}B=0.
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
We conclude that $[A,B]^t$ belongs to the kernel of $L$ and hence $rank(H)<2$.
Finally, by lemma \ref{lemma:Proof}, equations \mathbf{e}qref{eq:DA12}, \mathbf{e}qref{eq:DA22}, \mathbf{e}qref{eq:DB12} and \mathbf{e}qref{eq:DB22} are equivalent to
$A^2+\mathbf{e}psilon B^2=c$, for some constant $c$ and to equations \mathbf{e}qref{eq:DerivEta}. Equation $A^2+\mathbf{e}psilon B^2=c$ implies $\Omega\mathbf{w}edge\Omega=c[\cdot,\cdot,\cdot,\cdot]$.
\paragraph{Proof of theorem \ref{thm1}, part 2:}
Assume that $rank(H)=1$ and equations \mathbf{e}qref{eq:DerivEta} hold. Denote by
$[A,B]^t$ a column-vector in $Ker(H)$ satisfying $A^2+\mathbf{e}psilon B^2=c$, for some constant $c\mathbf{n}eq 0$. Define the symplectic form $\Omega$ by the conditions
\begin{equation*}
\begin{array}{c}
\Omega(X_1,X_2)=\Omega(\mathbf{x}i_1,\mathbf{x}i_2)=0\\
\Omega(X_1,\mathbf{x}i_2)=\Omega(X_2,\mathbf{x}i_1)=A\\
\Omega(X_1,\mathbf{x}i_1)=-\mathbf{e}psilon \Omega(X_2,\mathbf{x}i_2)=B
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
We shall prove that the symplectic form $\Omega$ is parallel.
Observe first that
\begin{equation*}
\begin{array}{c}
D_{X_1}\Omega(X_1,X_2)=-A+A=0\\
D_{X_2}\Omega(X_1,X_2)=-B+ B=0.
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
Moreover
\begin{equation*}
\begin{array}{c}
D_{X_1}\Omega(\mathbf{x}i_1,\mathbf{x}i_2)=L_{11}A+L_{12}B=0\\
D_{X_2}\Omega(\mathbf{x}i_1,\mathbf{x}i_2)=L_{21}A+L_{22}B=0.
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
We must prove now that $(D_{X_k}\Omega)(X_i,\mathbf{x}i_j)=0$, for any $i,j,k=1,2$. We shall prove for $(i,j,k)=(1,2,1)$ and $(i,j,k)=(2,1,1)$, the other cases being similar.
We have
\begin{equation*}
\begin{array}{c}
dA(X_1)-\Gamma_{11}^1A-\Gamma_{11}^2(-\mathbf{e}psilon B)-\mathbf{t}au_2^1(X_1)B-\mathbf{t}au_2^2(X_1)A=0\\
dA(X_1)-\Gamma_{12}^1B-\Gamma_{12}^2A-\mathbf{t}au_1^1(X_1)A-\mathbf{t}au_1^2(X_1)(-\mathbf{e}psilon B)=0.
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
But, as we have seen above, this pair of equations are equivalent to
\begin{equation*}
\begin{array}{c}
F_{21}A+F_{22}B=0\\
dA(X_1)-G_1 B=0,
\mathbf{e}nd{array}
\mathbf{e}nd{equation*}
which holds by lemma \ref{lemma:Proof}.
\begin{thebibliography}{99}
\bibitem{Burstin27} Burstin,C. and Mayer,W.: \mathbf{t}extit{ Die Geometrie zweifach ausgedehnter Mannigfaltigkeiten $F_2$ in affinen Raum $\mathbb{R}_4$}, Math.Z. 27, 373-407, 1927.
\bibitem{Morvan87} Chen,B.Y. and Morvan,J.M.: \mathbf{t}extit{ G\'eom\'etrie des surfaces Lagrangiennes de $\mathbb{C}^2$}, J.Math. Pures et Appl., 66, 321-335, 1987.
\bibitem{Craizer14} Craizer,M., Domitrz,W. and Rios,P.deM.: \mathbf{t}extit{Even dimensional improper affine spheres}, J. of Mathematical Analysis and Applications, 421, 1803-1826, 2015.
\bibitem{Dillen} Dillen, F., Mys, G., Verstraelen, L. and Vrancken, L.:\mathbf{t}extit{The affine mean curvature vector for surfaces in $\mathbb{R}^4$}, Math.Nachr., 166, 155-165, 1994.
\bibitem{Klingenberg51} Klingenberg,W.: \mathbf{t}extit{Zur affinen Differetialgeometrie, Teil II: Uber 2-dimesionale Fl\"achen im 4-dimensionalen Raum}, Math.Z., 54, 184-216, 1951.
\bibitem{Magid} Magid, M., Scharlach, C. and Vrancken, L.:\mathbf{t}extit{ Affine umbilical surfaces in $\mathbb{R}^4$}, Manuscripta Math., 88, 275-289, 1995.
\bibitem{Martinez05} Martinez,A.: \mathbf{t}extit{Improper affine maps}, Math.Z. 249, 755-766, 2005.
\bibitem{Nomizu93} Nomizu,K. and Vrancken,L.: \mathbf{t}extit{A new equiaffine theory for surfaces in $\mathbb{R}^4$}, Int. J. of Mathematics., 4(1), 127-165, 1993.
\bibitem{Verstraelen} Verstraelen, L., Vrancken, L. and Witowicz, P.:\mathbf{t}extit{Indefinite affine umbilical surfaces in $\mathbb{R}^4$}, Geom.Dedicata, 79, 109-119, 2000.
\bibitem{Vrancken} Vrancken, L.:\mathbf{t}extit{ Affine surfaces whose geodesics are planar curves}, Proc.Amer.Math.Soc., 123(12), 3851-3854, 1995.
\mathbf{e}nd{thebibliography}
\mathbf{e}nd{document}
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\begin{document}
\title{Sparsity of integral points on moduli spaces of varieties}
\begin{abstract}
Let $X$ be a quasi-projective variety over a number field,
admitting (after passage to $\mathbb{C}$) a geometric variation of Hodge structure whose period mapping
has zero-dimensional fibers.
Then the integral points of $X$ are {\em sparse}: the number of such points of height $\leq B$ grows slower than any positive power of $B$.
For example, homogeneous integral polynomials in a fixed number of variables and degree, with discriminant divisible only by a fixed set of primes, are sparse
when considered up to integral linear substitutions.
\end{abstract}
\author{Jordan S. Ellenberg, Brian Lawrence, and Akshay Venkatesh}
\maketitle
\begin{comment}
Following Koll\`{a}r~\cite[4.1]{koll:shaf}, we make the following definitions.
\begin{defn} If $X$ is a normal variety, a {\em
normal cycle} on $X$ is a morphism $w:W \to X$ such that $W$ is an
irreducible normal variety and $w$ is finite and birational onto its
image.
\end{defn}
\begin{defn} Let $X$ be a normal projective variety over an algebraically
closed field, and let $Z$ be a proper closed
subvariety of $X$. We say $X$ has {\em large algebraic fundamental group away
from $Z$} if, for every positive-dimensional normal cycle $W \to X$,
the induced map $\pi_1(W \backslash (W \cap Z)) \to \pi_1(X \backslash Z)$ has infinite image.
\end{defn}
If $X/k$ is a variety over an arbitrary field $k$ of characteristic
$0$, we say $X$ has large algebraic fundamental group away from $Z$ if
the criterion above holds for $X_{\overline{k}}$. Note that if $X$ has
large algebraic fundamental group, the same is true for $X \times_k K$
where $K$ is any uncountable algebraically closed field of
characteristic $0$. {\bf This needs further justification, it is not
just Lefschetz principle but I think it is standard.}
The main theorem of this paper is that varieties with large
fundamental group away from $Z$ have few $Z$-integral points.
\end{comment}
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\section{Introduction}
Let $K \subset \field{C}$ be a finite field extension of the rational numbers,
and $S$ a finite set of primes of $K$.
We will consider $S$-integral points on quasi-projective $K$-varieties $X^{\circ} \subset \mathbb{P}^m$.
More precisely:
in this situation we will write $X$ for the Zariski closure of $X^{\circ}$ in $\mathbb{P}^m$,
$\mathcal{L} = \mathcal{O}(1)$ the associated hyperplane bundle, and $Z$ for $X \backslash X^{\circ}$.
After choosing a good integral model of $X$ and $Z$ (see \S \ref{integralmodel} for details)
we obtain a notion of ``$S$-integral point of $X^{\circ}$,''
and the projective embedding allows us to refer to the ``height'' of such a point.
(By ``height'' we always mean the \emph{multiplicative} height.)
We will denote by $X_{\field{C}}$
the complex variety obtained by base extension via the fixed embedding $K \hookrightarrow \field{C}$,
by $X(\field{C})$ its complex points,
and by $X^{{\text{an}}}$ its complex analytification.
\begin{thm} \getsbel{thm1}
Let $X^{\circ} \subset \mathbb{P}^m$ be a quasi-projective variety over $K$
such that $X^{\circ}_{\field{C}}$ admits a geometric variation of Hodge structure,
whose associated period map (see \cite{Griffiths} for definitions) is locally finite-to-one, i.e. has zero-dimensional fibers. Then integral points on $X^{\circ}$ are sparse, in the sense that
\begin{equation}
\getsbel{th:main}\# \{x \text{ an $S$-integral point of $X^{\circ}$, of (multiplicative) height at most $B$} \} =O_{\epsilonsilon}(B^{\epsilonsilon}). \end{equation}
\end{thm}
By ``geometric variation of Hodge structure''
we mean a direct summand of a VHS arising from a smooth projective family over $X^{\circ}_{\field{C}}$; in particular the Hodge structures arising here are pure.
In the situation of the Theorem, the ``period map'' is understood to be the complex-analytic map $\widetilde{X^{\circ {\text{an}}}} \rightarrow D$
classifying the VHS, where $\widetilde{X^{\circ {\text{an}}}}$ is the universal cover of $X^{ \circ {\text{an}}}$ and $D$
is the period domain associated to the varying Hodge structures.
We note that the condition we are imposing on $X^{\circ}$ depends only on the complex algebraic variety $X^{\circ}_{\field{C}}$ and not on its rational form over $K$.
The notation $O_{\epsilonsilon}(B^{\epsilonsilon})$ is that of analytic number theory:
for each $\epsilonsilon > 0$ there exists $c_{\epsilonsilon}$ such that
the left hand side is at most $c_{\epsilonsilon} B^{\epsilonsilon}$.
Theorem \ref{thm1} will be deduced from the following more general theorem.
\begin{thm} \getsbel{thm2}
Let $\pi: \mathfrak{X} \rightarrow X^{\circ}$
be a projective smooth morphism of $K$-varieties,
and (for some $i \geq 0$) let $\field{P}hi$ be the period map
associated to the variation of Hodge structure
$\mathsf{V} = R^i \pi_* \field{Q}$ on $(X^{\circ})^{{\text{an}}}$.
Then
the $S$-integral points of $X^{\circ}$ with height at most $B$
are covered by $O_{\epsilonsilon}(B^{\epsilonsilon})$ geometrically irreducible
$K$-varieties, each lying in a single fiber of $\field{P}hi$.
\end{thm}
We have abused language in the statement: for an irreducible analytic subvariety $V \subset X^{\circ {\text{an}}}$ we shall say that $V$ ``lies in a fiber of $\field{P}hi$'' if some component (hence every component)
of the preimage of $V$ in the universal cover
lies in such a fiber. We will use similar language elsewhere in the paper -- if a property of the period map is local,
we will phrase it on $X^{\circ}$ rather than the universal cover.
The bound \eqref{th:main} applies to many natural moduli spaces of varieties.
For instance, one may take $X$ to be the projective space parametrizing hypersurfaces
of a given degree and dimension,
and $Z$ the locus of singular hypersurfaces, and obtain the following corollary, which we will derive from Theorem~\ref{thm1} in \S \ref{Corproof}.
\begin{cor} \getsbel{Corhyp} Fix $n \geq 2$ and $d \geq 3$ and a
finite set $S$ of rational primes.
Then the $S$-integral homogeneous degree $d$ polynomials $P= \sum a_{i_1\dots i_n} x_1^{i_1} \dots x_n^{i_n}$ in $n$ variables with $S$-integer discriminant and
$\max_{i, v \in S} |a_i|_v \leq B$
lie in at most $O_{S,d,n,\epsilonsilon}(B^{\epsilonsilon})$ orbits of the group of integral linear substitutions $\GL_n(\field{Z}[\frac{1}{S}])$.
Similarly the set of such integral polynomials (i.e. with $\field{Z}$ coefficients),
with discriminant exactly equal to a nonzero
integer $N \in \mathbb{Z}$, and $\max|a_i| \leq B$,
lie in at most $O_{N,d,n,\epsilonsilon}(B^{\epsilonsilon})$ orbits of the group of unimodular integral linear substitutions $\SL_n(\field{Z})$. \end{cor}
Recall that the {\em discriminant} of a homogeneous polynomial is a certain homogeneous polynomial in the coefficients of $P$
which vanishes precisely when the hypersurface defined by $P=0$ is singular; see \cite{PS} for more discussion of its significance.
In the paper \cite{LV18} of the second- and third- named author, it is proved by a much more complicated argument that
for ``large enough'' $n,d$ the
set of polynomials with $S$-integer discriminant is not Zariski dense in the ambient affine hypersurface (which of course neither implies nor is implied by what we prove here).
It is expected (\cite[Conj.\ 1.4]{JL}) that this set is {\em finite} as part of a ``Shafarevich-type'' statement
about hypersurfaces with good reduction;
in fact, if one assumes the Lang-Vojta conjecture,
then results on the hyperbolicity of period domains
imply finiteness of $S$-integral points in the general context of Theorem \ref{thm2}; see \cite{Zuo} and \cite[Thm.\ 1.5]{JL}.\footnote{
The theorem in loc.\ cit.\ is stated only for complete intersections,
but the proof applies in the generality of our Theorem \ref{thm2}.}
We note that the Shafarevich assertion can hold even in cases where there is no Torelli theorem; for instance, it holds for the case $(n,d) = (4,3)$ of cubic surfaces, by a result of Scholl~\cite{scholl:delpezzo}.
In addition to cubic surfaces, the Shafarevich assertion is also known for cubic and quartic threefolds \cite{JL}.
\subsection{Discussion of the proof}
The reduction of Theorem \ref{thm1} to Theorem \ref{thm2} is relatively formal, and so we will focus
on the latter Theorem here.
The idea of the proof goes back to ideas initiated in the work of Bombieri-Pila \cite{BP}, Heath-Brown \cite{HBR}
(and, independently and in a different context, Coppersmith \cite{Coppersmith}):
One can show that an excess of rational (or integral) points on any $X$ as above must lie on the zero-loci
of some auxiliary functions, i.e., must be covered by a certain collection of divisors.
Then we iterate the process, replacing $X$ by these divisors, and covering the points by codimension-$2$ subvarieties, etc.; since we have almost no control over what these subvarieties look like, it is crucial to have, as in Bombieri-Pila and Heath-Brown, that the basic bounds are {\em uniform} in the ambient variety. See Remark \ref{Brobremark}
for a quick review of this method.
Unfortunately, this process is quite lossy when executed in dimensions larger than $1$.
The problem is the usual one in studying rational points on higher dimensional varieties: Even if a variety
has high degree, it may contain subvarieties of much lower degree, such as hyperplanes.
For this reason, there are very few situations where one can achieve $B^{\epsilonsilon}$-type bounds (see e.g.\ \cite{PW} for an example
in a different context).
The key point of our strategy here is to use the fact that the bounds of \cite{Broberg} improve
under {\'e}tale covers, which are plentiful under the conditions on $\mathfrak{X} \rightarrow X$ we have imposed; these covers can be used to raise the degree of intermediate subvarieties. To deploy this in a way suitable for an iterative argument, we use
the global invariant cycle theorem of Hodge theory to construct a cover $\widetilde{X} \rightarrow X$
which remains nontrivial when restricted to any subvariety of $X$. In some sense, this strategy generalizes the approach of \cite{EV:2d}, which used \'etale covers to slightly improve on Heath-Brown's bounds in case $X$ is a non-rational curve (i.e. a curve with interesting \'etale covers); in the case $\dim X = 1$, of course, the issue of restriction of covers to subvarieties becomes trivial.
In the end, the core part of the proof is quite short -- the main idea is the induction on dimension laid out in \S \ref{conclude}, with the key induction step being provided by Lemma \ref{lem:induction}. The reader may want to
skip directly to these, and refer back as necessary. One of the reasons for the length of the text is
the technicalities
involved in making various results uniform over all subvarieties of $X$.
One way of thinking about the strategy executed here is in terms of ``profinite repulsion" between low-height integral points. For simplicity of exposition we take $K=\field{Q}$ for this paragraph. For any prime $p$, the embedding of $X(\field{Z}[1/S])$ into $X(\field{Q}_p)$ induces a topology on the former set, and the methods of Bombieri-Pila and Heath-Brown rely on an argument that low-height points tend to repel each other in this topology. In the cases considered in this paper, there is an alternate profinite topology on $X^{\circ}(\field{Z}[1/S])$ afforded by the map
\begin{displaymath}
X^{\circ}(\field{Z}[1/S]) \to H^1(G_\field{Q}, \pi_1(X^{\circ}_{\overline{\field{Q}}}))
\end{displaymath}
and our approach can be thought of as exploiting the fact that in this topology, too, the low-height $S$-integral points repel each other. The two profinite repulsion phenomena working in tandem is what allows us to get a better upper bound on the number of low-height points. To be more precise: to say that two points $P_1$ and $P_2$ of $X^{\circ}(\field{Z}[1/S])$ are close together in the alternate profinite topology is to say that there is some high-degree cover $\widetilde{X}^\circ \to X^\circ$, with $\widetilde{X}^\circ$ irreducible, such that $P_1$ and $P_2$ both lie in the image of $\widetilde{X}^\circ(\field{Q})$. Then the method of Heath-Brown is used to control the low-height points of $\widetilde{X}^\circ(\field{Q})$, which is aided by the fact that this new variety has very high degree.
In the course of writing this paper we learned of the paper \cite{Br} of Brunebarbe, which uses the same basic idea of increasing the positivity of line bundles by passing to {\'e}tale covers, in the context of a VHS with finite period map, in order to prove uniform algebro-geometric assertions about covers of varieties carrying such a VHS. Indeed
Brunebarbe gives a more analytic (and therefore effective) proof of a result (\cite[Theorem 1.13]{Br}) of the type of Lemma \ref{lem31}, whereas
our argument uses general finiteness statements of algebraic geometry. As in our paper, a crucial input to \cite{Br}
is the invariant cycle theorem of Hodge theory.
Finally, let us mention some potential refinements of the result.
\begin{itemize}
\item The reader will observe that the {\em geometricity} of the variation of Hodge structure is,
in fact, barely used at all in our proof. It permits us to phrase
the argument in quite an explicit way, but it should
be possible to entirely avoid it.
\item Indeed, the whole apparatus of variations of Hodge structures is used simply to provide examples of varieties $X^\circ$ with the property that the image of $\pi_1(W^{\circ {\text{an}}}) \to \pi_1(X^{\circ {\text{an}}})$ has infinite order for every subvariety $W^\circ$ of $X^\circ$. We note that this is essentially the same as Koll\'{a}r's condition of {\em large fundamental group} from \cite{kollar}.
For an $X^{\circ}$ which possessed this property for some reason unrelated to Hodge structures, the arguments here should work just as well to control its rational points. For example, abelian varieties do not admit variations of Hodge structure but they do have large fundamental group, so the arguments of this paper should in principle imply sparseness of rational points; but for abelian varieties, we know this already, by the Mordell-Weil theorem.
\item One might ask what happens for varieties $X^\circ$ carrying a variation of {\em mixed} Hodge structures; it seems plausible that a theorem like the one proved here will still hold.
\item It would be interesting to refine our statement $O_{\epsilonsilon}(B^{\epsilonsilon})$ to a more precise upper bound. To do this would, at the very least, require that the inexplicit dependence on degree in Broberg's bounds be made explicit. Questions in this vein have been the topic of much recent progress; see, e.g., the recent result of Castryck, Cluckers, Dittman, and Nguyen~\cite[Theorem 2]{CCDN}, which gives bounds with explicit polynomial dependence on degree.
The quantitative approach of \cite{Br} will also likely be needed in order to get effective lower bounds for the degrees of covers that arise in the argument.
\end{itemize}
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The key point of our strategy here is to use the fact that the bounds of \cite{Broberg} improve
under {\'e}tale covers, which are plentiful under the conditions on $\mathfrak{X} \rightarrow X$ we have imposed; these covers can be used to raise the degree of intermediate subvarieties. To deploy this in a way suitable for an iterative argument, we use
the global invariant cycle theorem of Hodge theory to construct a cover $\widetilde{X} \rightarrow X$
which remains nontrivial when restricted to any subvariety of $X$. In some sense, this strategy generalizes the approach of \cite{EV:2d}, which used \'etale covers to slightly improve on Heath-Brown's bounds in case $X$ is a non-rational curve (i.e. a curve with interesting \'etale covers); in the case $\dim X = 1$, of course, the issue of restriction of covers to subvarieties becomes trivial.
In the end, the core part of the proof is quite short -- the main idea is the induction on dimension laid out in \S \ref{conclude}, with the key induction step being provided by Lemma \ref{lem:induction}. The reader may want to
skip directly to these, and refer back as necessary. One of the reasons for the length of the text is
the technicalities
involved in making various results uniform over all subvarieties of $X$.
One way of thinking about the strategy executed here is in terms of ``profinite repulsion" between low-height integral points. For simplicity of exposition we take $K=\field{Q}$ for this paragraph. For any prime $p$, the embedding of $X(\field{Z}[1/S])$ into $X(\field{Q}_p)$ induces a topology on the former set, and the methods of Bombieri-Pila and Heath-Brown rely on an argument that low-height points tend to repel each other in this topology. In the cases considered in this paper, there is an alternate profinite topology on $X^{\circ}(\field{Z}[1/S])$ afforded by the map
\begin{displaymath}
X^{\circ}(\field{Z}[1/S]) \to H^1(G_\field{Q}, \pi_1(X^{\circ}_{\overline{\field{Q}}}))
\end{displaymath}
and our approach can be thought of as exploiting the fact that in this topology, too, the low-height $S$-integral points repel each other. The two profinite repulsion phenomena working in tandem is what allows us to get a better upper bound on the number of low-height points. To be more precise: to say that two points $P_1$ and $P_2$ of $X^{\circ}(\field{Z}[1/S])$ are close together in the alternate profinite topology is to say that there is some high-degree cover $\widetilde{X}^\circ \to X^\circ$, with $\widetilde{X}^\circ$ irreducible, such that $P_1$ and $P_2$ both lie in the image of $\widetilde{X}^\circ(\field{Q})$. Then the method of Heath-Brown is used to control the low-height points of $\widetilde{X}^\circ(\field{Q})$, which is aided by the fact that this new variety has very high degree.
In the course of writing this paper we learned of the paper \cite{Br} of Brunebarbe, which uses the same basic idea of increasing the positivity of line bundles by passing to {\'e}tale covers, in the context of a VHS with finite period map, in order to prove uniform algebro-geometric assertions about covers of varieties carrying such a VHS. Indeed
Brunebarbe gives a more analytic (and therefore effective) proof of a result (\cite[Theorem 1.13]{Br}) of the type of Lemma \ref{lem31}, whereas
our argument uses general finiteness statements of algebraic geometry. As in our paper, a crucial input to \cite{Br}
is the invariant cycle theorem of Hodge theory.
Finally, let us mention some potential refinements of the result.
\begin{itemize}
\item The reader will observe that the {\em geometricity} of the variation of Hodge structure is,
in fact, barely used at all in our proof. It permits us to phrase
the argument in quite an explicit way, but it should
be possible to entirely avoid it.
\item Indeed, the whole apparatus of variations of Hodge structures is used simply to provide examples of varieties $X^\circ$ with the property that the image of $\pi_1(W^{\circ {\text{an}}}) \to \pi_1(X^{\circ {\text{an}}})$ has infinite order for every subvariety $W^\circ$ of $X^\circ$. We note that this is essentially the same as Koll\'{a}r's condition of {\em large fundamental group} from \cite{kollar}.
For an $X^{\circ}$ which possessed this property for some reason unrelated to Hodge structures, the arguments here should work just as well to control its rational points. For example, abelian varieties do not admit variations of Hodge structure but they do have large fundamental group, so the arguments of this paper should in principle imply sparseness of rational points; but for abelian varieties, we know this already, by the Mordell-Weil theorem.
\item One might ask what happens for varieties $X^\circ$ carrying a variation of {\em mixed} Hodge structures; it seems plausible that a theorem like the one proved here will still hold.
\item It would be interesting to refine our statement $O_{\epsilonsilon}(B^{\epsilonsilon})$ to a more precise upper bound. To do this would, at the very least, require that the inexplicit dependence on degree in Broberg's bounds be made explicit. Questions in this vein have been the topic of much recent progress; see, e.g., the recent result of Castryck, Cluckers, Dittman, and Nguyen~\cite[Theorem 2]{CCDN}, which gives bounds with explicit polynomial dependence on degree.
The quantitative approach of \cite{Br} will also likely be needed in order to get effective lower bounds for the degrees of covers that arise in the argument.
\end{itemize}
\subsection{Notation and integral models} \getsbel{integralmodel}
As above, $K$ is a number field embedded in $\field{C}$;
let $\overline{K}$ be the algebraic closure of $K$ in $\field{C}$ (so $\overline{K}$ is just another name for $\overline{\mathbb{Q}}$).
By a variety $V$ (over $K$), or $K$-variety for short, we mean, as usual, an integral, separated, finite-type scheme over $K$.
In this situation,
$V$ will always denote the scheme over $K$,
and $V_{\field{C}}$ the base extension of $V$ to a complex scheme via $K \hookrightarrow \field{C}$.
(Note that a $K$-variety is not required to be geometrically irreducible; thus, $V_{\field{C}}$ may have multiple components.)
We will sometimes use notation such as $Y_{\field{C}}$ even when $Y$ is not a base-change from $K$,
simply to emphasize that $Y_{\field{C}}$ is a complex variety.
We will write $V^{{\text{an}}}$ for the (complex) analytification of $V_{\field{C}}$.
As above let $S$ be a finite set of primes of $K$.
Let $\mathfrak{o}_S$ be the ring of $S$-integers of $K$,
i.e.\ the set of those elements of $K$ that are integral outside $S$.
\begin{defn} \getsbel{goodmodel}
Let $X^{\circ} \subset \mathbb{P}^m$
be a locally closed subvariety, and $X$
its Zariski closure.
Let $Z$ be the complement $X \backslash X^{\circ}$, a Zariski-closed subset of $\mathbb{P}^m$.
We will call a \emph{good integral model} for $(X, Z)$ a choice of a
projective flat $\mathfrak{o}_S$-scheme $X_S \subset \mathbb{P}^N_S$
extending $X$ on the generic fiber.
Given a good integral model, we define $Z_S$ to be the Zariski closure of $Z$ inside $X_S$, and take $X^{\circ}_S = X_S - Z_S$.
If we are given a projective smooth morphism $f: \mathfrak{X} \rightarrow X^{\circ}$
and we have chosen a good integral model for $(X,Z)$, a good integral model for $f$ will be
a projective smooth morphism
$f_S: \mathfrak{X}_{S} \rightarrow X^{\circ}_S$.
\end{defn}
Good integral models exist after possibly increasing $S$, by standard spreading arguments
(a nice reference here is Theorem 3.2.1 and Appendix C, Table 1 of \cite{PoonenRP}).
A $K$-point of $X$ will be said to be {\em $S$-integral}
if it extends to a morphism $\mathrm{Spec} (\mathfrak{o}_S) \rightarrow X_S-Z_S$.
Explicitly, assuming $\mathfrak{o}_S$ to have class number one for simplicity,
any $K$-point, represented by a point
with relatively prime homogeneous coordinates $\mathbf{x} = [x_0: \dots: x_N] \in \mathfrak{o}_S^{N+1}$,
is \emph{integral} at a prime $p \notin S$ if there exists
a homogenous function $f \in \mathfrak{o}_S[x_0, \dots, x_N]$
in the ideal of $Z_S$ such that $f(\mathbf{x})$
is nonzero mod $p$; it is $S$-integral if it is integral at all primes $p \notin S$.
If we choose two different choices of good integral model as above, there exists a finite set $T$
such that any $S$-integral point for the first model is $S \coprod T$-integral
for the second model, so the choice of good integral model is irrelevant to the statement of the theorem.
Finally, we fix some other notation to be used:
For any field $F$ and any
$F$-variety $Y$ mapping to $X$,
we denote by $Y^{\circ}$ the preimage of $X^{\circ}$ inside $Y$.
If the map $f:Y \rightarrow X$ is finite, the ``degree'' of $Y$ will
be the degree of the pullback $f^* \mathcal{L}$ of the hyperplane bundle $\mathcal{L}$, characterized by the asymptotic formula
\begin{equation} \getsbel{growth} \dim_F \Gamma(Y, f^* \mathcal{L}^{\otimes k}) \sim \frac{ \mathrm{deg} Y}{(\dim Y)!} k^{\dim Y} \end{equation}
for large $k$.
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\section{Some results used in the proof}
\getsbel{res}
We collect results we will use in the proof.
These
are a few lemmas from algebraic geometry, and a crucial
bound on rational points due to Broberg.
We emphasize again that, for $F$ a field, a variety over $F$
is assumed reduced and irreducible, but not geometrically irreducible.
\begin{lem} \getsbel{KS}
Suppose that $\pi: \mathfrak{X} \rightarrow X$
is a projective smooth morphism of algebraic varieties
over a subfield $\kappa \subset \field{C}$, with
$X$ smooth,
and let $\mathcal{T}$ be the tangent bundle of $X$.
Let
$\field{P}hi$ be the period map
associated to the variation of Hodge structure
$\mathsf{V} = R^i \pi_* \field{Q}$ on $(X)^{{\text{an}}}$,
for some $i>0$.
Then there is a morphism
\begin{equation}
\getsbel{gdef} g \colon \mathcal{H}_1 \otimes \mathcal{T} \rightarrow \mathcal{H}_2\end{equation}
of vector bundles
over $X$ (everything defined over $\kappa$) such that
the derivative of the period map along a tangent vector $t$ at $x$ vanishes if and only if
$g_x( -, t)$
is the zero map from $\mathcal{H}_1$ to $\mathcal{H}_2$.
\end{lem}
\proof
This is a consequence of standard facts in Hodge theory, in particular the algebraicity of the Gauss--Manin connection (see \cite{Katz_Oda} and \cite[\S 1]{Katz_nilp}).
The $i$th relative algebraic de Rham cohomology of $\mathfrak{X}/X$ gives a variation of Hodge structure $H_{\text{dR}}$ on $X$.
As a variation of Hodge structure, $H_{\text{dR}}$ is a vector bundle,
equipped with a flat connection and a filtration by algebraic subbundles $F^p H_{\text{dR}}$.
These data determine the period map $\widetilde{X^{\circ {\text{an}}}} \rightarrow D$,
where $D$ is a flag variety classifying filtrations
of a fixed vector space of dimension $\operatorname{rank} H_{\text{dR}}$
by subspaces of dimensions $\operatorname{rank} F^p H_{\text{dR}}$.
For each step $F^p H_{\text{dR}}$ of the filtration, the connection on $H_{\text{dR}}$ defines an $\mathcal{O}_X$-linear map
\[ g_p \colon F^p H_{\text{dR}} \otimes \mathcal{T} \rightarrow H_{\text{dR}} / F^p H_{\text{dR}}, \]
which describes how the subspace $F^p H_{\text{dR}, x}$ varies in $x \in X$.
Combining the maps $g_p$ over all $p$, we obtain
\[ g \colon \bigoplus_p F^p H_{\text{dR}} \otimes \mathcal{T} \rightarrow \bigoplus_p H_{\text{dR}} / F^p H_{\text{dR}}. \]
This map $g$ determines the differential of the period map.
More precisely, at every point, the tangent space to $D$ is identified with a subspace of
\[ \operatorname{Hom} \left ( \bigoplus_p F^p H_{\text{dR}}, \bigoplus_p H_{\text{dR}} / F^p H_{\text{dR}} \right ), \]
and with this identification, $g$ gives the differential of the period map.
Finally, we note that $g$ is defined over $\kappa$ by \cite{Katz_Oda} and \cite[\S 1]{Katz_nilp}.
\qed
\begin{lem} \getsbel{degree}
Let $F$ be a field of characteristic zero.
Suppose that $Y$ is a proper $F$-variety equipped with an ample line bundle $\mathcal{L}$,
and let $g: \widetilde{Y} \rightarrow Y$ be finite, with $\dim(\widetilde{Y}) = \dim(Y)$.
Writing $\deg g$ for the degree of $g$ at the generic point, we have
\begin{equation} \getsbel{degfinite} \deg_{g^* \mathcal{L}}(\widetilde{Y}) = (\deg g) \deg_{\mathcal{L}}(Y).\end{equation}
\end{lem}
We will apply this only when $\widetilde{Y}$ is a variety, but the argument does not require that, taking
\eqref{growth} as the definition of degree.
\proof
By the projection formula we have
$$\Gamma(\widetilde{Y}, g^* \mathcal{L}^{\otimes k}) = \Gamma(Y, g_* \mathcal{O} \otimes \mathcal{L}^{\otimes k}).$$
Now $g_* \mathcal{O}$ is isomorphic to $\mathcal{O}^{\oplus (\mathrm{deg} g)}$
away from a set of positive codimension on $Y$. Therefore
$\Gamma(\widetilde{Y}, g^* \mathcal{L}^{\otimes k})$
coincides with
$(\mathrm{deg} \ g) \frac{ \mathrm{deg}_{\mathcal{L}} Y}{(\dim Y)!} k^{\dim Y}$
for large $k$, up to terms $O(k^{\dim Y-1})$. We conclude using $\dim \widetilde{Y} = \dim Y$.
\qed
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\begin{lem} \getsbel{degree}
Let $F$ be a field of characteristic zero.
Suppose that $Y$ is a proper $F$-variety equipped with an ample line bundle $\mathcal{L}$,
and let $g: \widetilde{Y} \rightarrow Y$ be finite, with $\dim(\widetilde{Y}) = \dim(Y)$.
Writing $\deg g$ for the degree of $g$ at the generic point, we have
\begin{equation} \getsbel{degfinite} \deg_{g^* \mathcal{L}}(\widetilde{Y}) = (\deg g) \deg_{\mathcal{L}}(Y).\end{equation}
\end{lem}
We will apply this only when $\widetilde{Y}$ is a variety, but the argument does not require that, taking
\eqref{growth} as the definition of degree.
\proof
By the projection formula we have
$$\Gamma(\widetilde{Y}, g^* \mathcal{L}^{\otimes k}) = \Gamma(Y, g_* \mathcal{O} \otimes \mathcal{L}^{\otimes k}).$$
Now $g_* \mathcal{O}$ is isomorphic to $\mathcal{O}^{\oplus (\mathrm{deg} g)}$
away from a set of positive codimension on $Y$. Therefore
$\Gamma(\widetilde{Y}, g^* \mathcal{L}^{\otimes k})$
coincides with
$(\mathrm{deg} \ g) \frac{ \mathrm{deg}_{\mathcal{L}} Y}{(\dim Y)!} k^{\dim Y}$
for large $k$, up to terms $O(k^{\dim Y-1})$. We conclude using $\dim \widetilde{Y} = \dim Y$.
\qed
The next Lemma
is \cite[Expos{\'e} I, Corollaire 10.8]{SGA1}.
\begin{lem} \getsbel{disjointness}
Suppose that $X$ is an irreducible normal noetherian scheme
and $f: Y \rightarrow X$ a finite {\'e}tale cover. Then the irreducible
components of $Y$ are disjoint.
\end{lem}
The following result asserts, essentially, ``boundedness''
of the set of irreducible varieties in a fixed projective space and bounded degree.
Results of this type are also stated and used in work of Salberger \cite[Lemma 1.4, Thm.\ 3.2]{Salberger}
in a similar context; in the interest of self-containedness
we give a proof of precisely what we use.
\begin{lem} \getsbel{Kleiman}
Let $F$ be a field of characteristic zero.
\begin{itemize}
\item[(a)] Suppose that $V \subset \mathbb{P}^m_F$ is a closed subvariety (irreducible, reduced closed subscheme) of degree $d$.
Then there are bounds, depending only on $m,d$ (not on $F$) for each coefficient of the Hilbert polynomial of $V$,
and in particular the homogeneous ideal $I(V)$ of $V$ is generated in degree $O_{m,d}(1)$.
\item[(b)] Suppose given integers $m, n, d, R$. Then there exist bounds $D$ and $N$ with the following property:
Suppose $V_1, \ldots, V_r$ (with $r \leq R$) is a collection of closed subvarieties of $\mathbb{P}^m_F$,
with each $V_i$ of dimension $\leq n$ and degree $\leq d$.
Let $Z$ be the intersection
\begin{equation} \getsbel{Zdef} Z = \bigcap_{i=1}^{r} V_i. \end{equation}
Then the number of irreducible components of $Z$ is at most $N$,
and the degree of each such component (endowed with the reduced scheme structure) is at most $D$.
Furthermore, the bounds $D$ and $N$ are independent of the field $F$.
\item[(c)] Suppose that $V \subset \mathbb{P}^m_F$ is a
closed variety of dimension $n$ and degree $d$ which is {\em not} geometrically irreducible.
Then there exist finitely many subvarieties $V_1, \ldots, V_N \subseteq V$, defined over $F$, such that:
\begin{itemize}
\item Each $V_i$ is irreducible of dimension $\leq n-1$,
\item $V(F) = \bigcup V_i(F)$, and
\item the number $N$ of $V_i$'s, and degree of each $V_i$ can be bounded in terms of $n, m, d$ (but independently of the field $F$ and the variety $V$).
\end{itemize}
\item[(d)] Suppose that $V \subset \mathbb{P}^m_F$ is a closed variety of dimension $n$ and degree $d$. Then
the set of points in $V(F)$ which are singular on $V$ can again be covered by varieties $V_1, \dots, V_N$
with the same properties as (c).
\end{itemize}
\end{lem}
\begin{comment}
\begin{rem}
In statement (c) we understand $\bigcap V_i$ to be endowed with its scheme
structure as a fiber product; it need not be reduced, and neither
do its irreducible components, which are considered as schemes via
a schematic closures of their generic points. However, the statement immediately implies
the same result if we endow $Z$ instead with its reduced scheme structure.
\end{rem}
\end{comment}
\begin{exmp}
As an example of ``bad'' examples for part (a): take
a $d$-dimensional variety, and adjoin to it a large set of disjoint points;
this modification does not affect the degree, and shows the need for irreducibility or at least
equidimensionality. Similarly, consideration of
embedded points shows that ``reduced'' is also important.
As an example of the situation in part (c), consider the plane curve defined by $x^2 + y^2 = 0$.
This is irreducible over $\mathbb{Q}$ but not geometrically irreducible;
it only has one rational point $(0, 0)$, which is contained in a zero-dimensional, geometrically irreducible subvariety.
More generally, suppose $F$ is a number field, and consider ``the affine line over $F$, with the origin reduced to a $\mathbb{Q}$-point'' -- that is, $\operatorname{Spec} ( \mathbb{Q} + T F[T])$.
If $F \neq \mathbb{Q}$, this scheme is irreducible, but geometrically reducible, and its only $\mathbb{Q}$-point is the origin.
(Taking $F = \mathbb{Q}[i]$ recovers the original example.)
\end{exmp}
\proof
The first assertion of (a) follows from \cite[Expos\'e XIII, Corollary 6.11(a)]{SGA6}.
That assertion applies as formulated to ``special positive cycles'' over an algebraically closed field;
in our situation $V_{\overline{F}} \subset \mathbb{P}^m_{\overline{F}}$
is reduced equidimensional and its decomposition into irreducible components give the closed
subschemes appearing in {\em loc. cit.} D{\'e}finition 6.9.
The consequence
on bounded generation of $I(V)$ follows, because such a bound on generation of
the defining ideal is valid in any finite type subscheme
of the Hilbert scheme.
For (b), we may as well consider one fixed $r$.
Let $\mathcal{P}$ be the finite set of polynomials arising from (a),
and let $\operatorname{Hilb}$ be the Hilbert scheme parametrizing closed subschemes of $\mathbb{P}^m$
with Hilbert polynomial in $\mathcal{P}$. This is a finite-type $\mathbb{Q}$-scheme by (a).
Now tuples $(V_1, \ldots, V_r)$ are classified by suitable $K$-points of $\operatorname{Hilb}^r$, which is again of finite type;
and the result now follows from standard results on families over a finite-type base.
Specifically, we have the universal schemes $\mathcal{V}_1, \ldots, \mathcal{V}_r$ over $\operatorname{Hilb}^r$;
let $\mathcal{Z} \subseteq \mathbb{P}^m \times \operatorname{Hilb}^r$ be their fiber product over $\mathbb{P}^m$,
so fiberwise $\mathcal{Z}$ gives the intersection of the $\mathcal{V}$s (although
with a possibly non-reduced schematic structure). We will work by Noetherian induction.
Let $\mathrm{et}a$ be the generic point of a closed irreducible subscheme $H \subseteq \operatorname{Hilb}^r$.
We will show that there exists a relatively open subset $U \subseteq H$ (that is, open in $H$, but not necessarily in $\operatorname{Hilb}^r$)
such that the number, dimensions, and degrees of the irreducible components of fibers $\mathcal{Z}_h$, for $h \in U$, are bounded.
Here, as in the statement, ``degree'' is taken with reference to the reduced scheme structure.
The number of irreducible components of any geometric fiber of $\mathcal{Z}$ is bounded by \cite[9.7.9]{EGAIV3}.
This bounds the number of geometric components, and so also the number of irreducible components,
of any $Z$ as in \eqref{Zdef}.
Now we turn to the degree.
For each $j$ with $0 \leq j \leq n$,
let $Z_j$ be the closure in $\mathcal{Z}$ of the union of $j$-dimensional components of $\mathcal{Z}_{\mathrm{et}a}$ (thought of, for now, merely as a closed subset of $\mathcal{Z}$ in the Zariski topology; we will revisit the issue of scheme structure shortly.)
In particular, the fiber over $\mathrm{et}a$ of each $Z_j$ is equidimensional of dimension $j$.
By \cite[9.5.1, 9.5.5]{EGAIV3}, we can restrict to an open $U \subseteq H$, on which:
\begin{itemize}
\item[(i)] For each $s \in U$, the fiber $(Z_j)_s$ is equidimensional of dimension $j$,
\item[(ii)] For each $s \in U$, the fiber $\mathcal{Z}_s$ is set-theoretically covered by the various $(Z_j)_s$, and
\item[(iii)] For each $s \in U$, and for all $j' < j$, the intersection $(Z_j)_s \cap (Z_{j'})_s$ has all components of dimension strictly less than $j'$.
\end{itemize}
Take $s \in U$ and let $K$ be any irreducible component of $\mathcal{Z}_s$, say of dimension $q$;
by (ii) it is contained in some irreducible component of some $(Z_j)_s$.
Since $K$ is maximal among irreducible subsets, we must have equality here, i.e.\
$K$ coincides with an irreducible component of this $(Z_j)_s$, and
since $(Z_j)_s$ is equidimensional of dimension $j$
we must have $j=q$.
Now let us endow $Z_j$ (so far merely a closed set) with its reduced scheme structure.
But now by generic flatness (\cite[6.9.1]{EGAIV2}),
we further shrink $U$ so that each $Z_j$ is flat over $U$.
Then for each $j$, the degree of each $(Z_j)_s$ is independent of $s$.
Now, the scheme structure on $(Z_j)_s$ need not be reduced, but
nonetheless
there is an inequality of degrees $\mathrm{deg} \ (Z_j)_s \geq \mathrm{deg} \ (Z_j)_s^{\mathrm{red}}$
(where $(Z_j)_s^{\mathrm{red}}$ denotes the fiber taken with the reduced scheme structure)
and the degree of $(Z_q)_s^{\mathrm{red}}$ bounds from above
the degree of $K$ taken with its reduced scheme structure.
Now we turn to (c).
Consider the geometrically irreducible components $W_1, \dots, W_h$ inside the base change $V_{\overline{F}}$ of $V$
to an algebraic closure.
Note that $h$ is bounded by the degree of $V$, which we have assumed bounded,
and similarly the degree of each $W_i$ is bounded by the degree of $V$.
The intersection $\cap_{i=1}^h W_i$ (with its reduced structure)
is a Galois-stable closed subscheme of $V_{\overline{F}}$ and thereby descends to a reduced $F$-subscheme $W \subset V$.
Since $V$ is not geometrically irreducible, we know that $\operatorname{dim} W \leq \operatorname{dim} V - 1$.
Also (b), applied with $F$ replaced by $\overline{F}$, implies that $W_{\bar{F}}$ has a bounded number of irreducible components, each of bounded degree.
The same is then true for the $F$-scheme $W$.
We claim, further, that $W(F) = V(F)$.
To see this, note that any $F$-point of $V$ must be Galois-invariant;
since Galois permutes the geometric components of $V$ transitively, the associated element of $V(\overline{F})$
belongs to all $W_i$.
(See \cite[Lemma 0G69]{Stacks} for a similar argument.)
The argument for (d) is similar to that for (b). Again we can parameterize all such $V$ by a suitable finite type Hilbert scheme $\mathrm{Hilb}$
and, writing $\mathcal{H} \rightarrow \mathrm{Hilb}$ for the universal subscheme, the smooth locus of $\pi$ coincides
with the set of points of $\mathcal{H}$ that are smooth in their fiber over $\mathrm{Hilb}$
(see e.g.\ \cite[17.5.1]{EGAIV4}).
Let $\mathcal{Z}$
be the complement of this smooth locus, endowed with the reduced structure. Then proceed as in (b).
\qed
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Finally, the following theorem of Broberg \cite{Broberg}
builds on fundamental ideas of Heath-Brown \cite{HBR} and Bombieri-Pila \cite{BP}:
\begin{thm}[Broberg, 2004] \getsbel{Brobs}
Let $V \subset \mathbb{P}^M_{K}$ be an irreducible closed subvariety of dimension $n$ and degree $d$.
Then the points
of $V(K)$ whose naive height is at most $H$ are
contained in a set of $K$-rational divisors of cardinality $\ll_{\epsilonsilon, M} H^{\frac{n+1 +\epsilonsilon}{d^{1/n}} }$,
and each of which has degree $O_{\epsilonsilon, M}(1)$.
\end{thm}
We note that, as stated in \cite{Broberg}, Broberg requires a bound on the generation of the ideal of $V$. This bound is however automatic from Lemma \ref{Kleiman} part (a).
Also ``divisor'' in the statement means ``effective Cartier divisor.''
\begin{rem} \getsbel{Brobremark}
Because Theorem \ref{Brobs} is so crucial, particularly its dependence on degree,
we briefly outline
where it comes from, taking $K=\field{Q}$ to simplify notation; this is not needed in the remainder of the paper.
One chooses a large integer $k$
and embeds $V \hookrightarrow \mathbb{P}^{e-1}$
via a basis of sections of $\Gamma(V, \mathcal{O}(1)^{\otimes k})$. In fact, we can and do choose from $\Gamma(\field{P}^M, \mathcal{O}(1)^{\otimes k})$ a set of monomials $f_1, \ldots, f_e$ of degree $k$ in the $M+1$-variables which freely span $\Gamma(V, \mathcal{O}(1)^{\otimes k})$.
Choose a ``good'' prime $p$ and examine a collection of $e$ points $P_i \in V(\field{Q})$
which all reduce to the same point modulo $p$ of $\field{P}^M$, and whose height is at most $H$. Expressing each $P_i$ in coprime integer coordinates,
we can speak of the evaluation $f_i(P_j) \in \mathbb{\field{Z}}$.
Consider
$$ \Delta := \det \left[ f_i(P_j) \right]_{1 \leq i,j \leq e} \in \mathbb{Z}$$
which measures the volume of the $e$-simplex in $\mathbb{Z}^e$ spanned by the $P_i$ and the origin.
On the one hand,
$\Delta$ is bounded by a constant multiple of $H^{ke}$.
On the other hand, $\Delta$ is highly divisible by $p$,
because the values of $f_j$ modulo power of $p$ are highly constrained,
and therefore there are many relations (mod $p^k$) between rows of $\Delta$.
To see this more formally, fix $r \geq 1$ and let
denote by $V(\field{Z}_p)_0$ the subset of $V(\field{Z}_p)$
consisting of points with a given reduction modulo $p$. Set
$$M_r := \{ \mbox{functions: $V(\field{Z}_p)_0 \rightarrow \field{Z}/p^r$}\}.$$
Each $f_j$ gives an element of $M_r$, by evaluation and reduction modulo $p^r$.
For $r=1$, all these functions (for varying $j$) lie in a $\field{Z}/p$-module of rank one: the constant functions.
For $r=2$, these functions
depend only on the ``constant term'' and ``derivative'' of $f_j$,
and thereby
lie in a $\mathbb{Z}/p^2$-submodule of $M_2$ of rank $n+1$.
For $r=3$ we get $\mathbb{Z}/p^3$-submodule of $M_3$ of rank $(n+1)(n+2)/2$,
where the number comes from counting possible
Taylor expansions of $f_j$ up to degree two.
Each such statement gives linear constraints on the rows of $f_i(P_j)$,
and therefore leads to divisibility for $\Delta$.
Computing with this we find
$$v_p(\Delta) \gtrsim k e \cdot \frac{ d^{1/n} }{1+1/n},$$
where $d$ arises on the right-hand side
eventually through the asymptotic behaviour of $e = \dim \Gamma(V, \mathcal{O}(k))$,
cf.\ \eqref{growth}.
Choosing $p$ so that
$p^{v_p(\Delta)}$ is larger than the size bound $H^{ke}$ then forces $\Delta=0$;
so the points $P_i$ lie on a hyperplane of $\mathbb{P}^{e-1}$, i.e.
all the points of $V$ with a fixed mod $p$ reduction lie on a divisor.
So we produce $\approx p^{n}$
divisors covering the points of height $\leq H$. The argument above is so flexible -- in particular, using freedom to choose $p$ --
that one can achieve bounds that are uniform in $V$.
Notice the crucial point: as the degree of $\mathcal{O}(1)$ on $V$ increases, $\mathcal{O}(1)^{\otimes k}$ has more sections for a fixed $k$, giving
stronger divisibility for $\Delta$ and thus stronger bounds.
\end{rem}
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\section{Reduction to Theorem \ref{thm2}}
We describe how Corollary \ref{Corhyp} is reduced to Theorem \ref{thm2}.
\getsbel{Main1}
\subsection*{Deduction of Corollary \ref{Corhyp} from Theorem \ref{thm1}} \getsbel{Corproof}
We will focus on the first statement of the Corollary, with $S$-integral points,
and remark at the end of the proof on the only modification needed to handle
the statement about fixed discriminant.
For $n=2$ (binary forms of degree three and above) one in fact knows finiteness (Birch--Merriman \cite{BM}).
The same is true for the case $n=4, d=3$ of cubic surfaces (Scholl \cite{scholl:delpezzo}).
We may now restrict to the remaining cases $n \geq 3, d \geq 3$ and $(n,d) \neq (4,3)$.
We will apply Theorem \ref{thm2} taking $X=\mathbb{P}^M$ the projective space
parameterizing polynomials of degree $d$ in $n$ variables up to scaling,
with $Z$ the zero-locus of the discriminant, and taking the geometric variation of Hodge structure to arise
from the middle cohomology of the universal family of hypersurfaces over $X^{\circ}$.
Here the infinitesimal Torelli theorem is known, see \cite[Thm.\ 9.8(b)]{Griffiths}; that is, every fiber of the period map is locally contained in an orbit of $\mathrm{PGL}_n(\field{C})$; so,
noting that the Weil height of $[a_0: \dots : a_M]$ is given by $\prod_{v} \max_{i} |a_{i,v}|$ and is thereby bounded
by a power of $B$ in the situation of the Corollary,
Theorem \ref{thm2} shows that the integral points in question are covered by $O_{\epsilonsilon}(B^{\epsilonsilon})$ orbits of $\field{P}GL_n(\field{C})$
or equivalently $\GL_n(\field{C})$.
We must replace $\GL_n(\field{C})$ by $\GL_n(\field{Z}[S^{-1}])$.
This is not difficult
but one must take care because a hypersurface could have automorphisms in characteristic $p$ that do not lift to characteristic zero.
Fix $P_0$ as in the Corollary. We will show that the number of $\GL_n(\field{Z}[S^{-1}])$-orbits on
$S$-integral polynomials $P \in \GL_n(\field{C}) P_0$ with $S$-integral discriminant is bounded in terms of $n,d,S$. Let $h$ be the degree of the discriminant polynomial. For any $S$-integral $P \in \GL_n(\field{C}) P_0$ with $S$-integral discriminant, there exists a rescaling of $P$ by an $S$-unit
whose discriminant has $p$-valuation between $0$ and $h$.
It suffices, then, to show that for any integer $N = \prod_{p \in S} p^{a_p}$ (with $0\leq a_p < h$)
the set of $S$-integral polynomials in $\GL_n(\field{C}) P_0$ with discriminant $N$
lie in a union of $O_{n,d,S}(1)$ orbits of $\SL_n(\field{Z}[S^{-1}])$.
Now, write $Y$ for the affine hypersurface defined by $\mathrm{disc}(P)=N$,
which we can regard as an affine scheme over $\field{Q}$ (and even over $\field{Z}$).
It is equipped
with an action of the $\field{Q}$-algebraic group $G=\mathrm{SL}_n$.
We must show that the intersection of $Y(\field{Z}[S^{-1}])$ with any $G(\field{C})$-orbit
is covered by $O_{n,d,S}(1)$ orbits of $G(\field{Z}[S^{-1}])$. We will need:
\begin{quote} {\em Claim 1:} The action morphism
$G \times Y \rightarrow Y \times Y$
is a {\em finite} morphism of algebraic varieties over $\field{Q}$.
\end{quote}
This follows essentially from the theorem of Matsumura and Monsky \cite{MM} that each stabilizer
$G_y$ for $y \in Y(\field{C})$ is finite.
The deduction can be carried out using results of geometric invariant theory
to show that the stack of smooth hypersurfaces is separated; see
\cite{JLmoduli}.
We give a self-contained argument using similar ideas.
\proof (of {\em Claim 1.})
It is sufficient
to prove
this over $\field{C}$, and, since the morphism is quasi-finite, it is sufficient to prove that it is proper.
Using the singular value decomposition
one reduces to checking that the action of the diagonal subgroup $T = \{ (t_1, \dots, t_n) \text{ with }
t_i \in \field{R}_+ \}$ on $Y(\field{C})$ is proper, which can be checked for the analytic topology \cite[XII, Prop 3.2]{SGA1}. In other words, one must show that for any compact regions $\Omega_1$ and $\Omega_2$ in $Y(\field{C})$, the set $\{g \in T: g\Omega_1 \cap \Omega_2 \mbox{ nonempty} \}$ is bounded. It suffices to show that, if $P = \sum a_{i_i \dots i_n} x_1^{i_1} \dots x^{i_n}$ and $Q = \sum b_{i_1\dotsi_n} x_1^{i_1} \dots x^{i_n}$ and $(t_1, \ldots, t_n) \cdot P = Q$ then the absolute values of the $t_j$ can be bounded in terms of the coefficients of $P$ and $Q$.
Write $\Sigma$ for the set of $I = (i_1, \ldots, i_n)$ such that $a_I \neq 0$. For each $t = (t_1, \ldots, t_n)$, write $t^I$ for $\prod_j t_j^{i_j} = \exp(\sum i_j \log t_j)$. Then the maximal absolute value of a coefficient of $(t_1, \ldots, t_n) \cdot P$ is $\max_{I \in \Sigma} |a_I| t^I$, so the condition that $(t_1, \ldots, t_n) \cdot P = Q$ provides upper bounds on $|a_I| t^I$ for all $I \in \Sigma$.
An upper bound on $|a_I| t^I$ restricts $(\log t_1, \ldots \log t_n)$ to a half-space; we are done once we show that the intersection of these half-spaces over all $I \in \Sigma$ is a compact region in $T$.
This is the case exactly when the region
\begin{equation} \getsbel{badset} \{ t \in T: \sum i_j \log t_j > 0 \mbox{ for all $I \in \Sigma$}\}\end{equation}
is {\em empty}. Suppose otherwise; then there exists $t$ in this region
of the form $t_0^{m_1}, \ldots, t_0^{m_n}$ for some (whence any) $t_0 \in \field{R}_+$ and with $m_j \in \field{Z}$.
By assumption on $t$, the limit in the analytic topology $P_0 := \lim_{t_0 \rightarrow 0} (t_0^{m_1}, \ldots, t_0^{m_n}) \cdot P$ exists; explicitly,
$P_0 = \sum_{I \in \Sigma_0} a_I t^I$
where
$\Sigma_0$ is that subset of $\Sigma$ consisting of those $I$ with $\sum i_j m_j = 0$.
Then $\mathrm{disc}(P_0) = \lim_{t_0 \to 0} \mathrm{disc} (t \cdot P) = \mathrm{disc}(P)$ is nonzero, so $P_0$ cuts out a smooth affine hypersurface.
But the identity $
\sum m_j X_j \frac{d}{dX_j} P_0 = 0$
means that the $n$ conditions $\frac{dP}{dX_j} = 0$ cutting out the nonsmooth locus are dependent,
contradicting smoothness of $P_0=0$
(cf. proof in \cite{MM}).
Thus region \eqref{badset} is empty, so the action of $T$ is proper, so the action of $G$ is proper as well.
This concludes the proof of {\em Claim 1.} \qed
Take $y_1, y_2 \in Y(\field{Z}[S^{-1}])$. We will now prove
\begin{quote}
{\em Claim 2 :} The action of $\Gal(\overline{\field{Q}}/\field{Q})$
on the stabilizers $G_{y_i}(\overline{\field{Q}})$ and also on the set $G_{12} := \{g \in G(\overline{\field{Q}}): g y_1 = y_2\}$
is unramified outside a set of primes $\mathcal{P}$ depending only on $n,d,S$.
\end{quote}
\proof (of {\em Claim 2}).
It is enough to prove the claim for $G_{12}$; take $y_1=y_2$ to get the claim about stabilizers.
From finiteness of the action map we see that the matrix entries of
$g^{\pm 1}$ for $g \in G_{12}$ satisfy monic polynomials whose coefficients are rational polynomials in the coordinates of $y_i$. Take $A_1(y_1,y_2), \ldots, A_M(y_1,y_2)$ to be the finite collection of all coefficients arising in this way.
Now, for any extension of the $p$-valuation on $\field{Q}$ to $\overline{\field{Q}}$, and for every matrix entry $g^{\pm 1}_{ij}$ of $g^{\pm 1}$ we have
\begin{equation} \getsbel{cb} |g^{\pm 1}_{ij}|_p \leq \max_k |A_k(y_1, y_2)|_p. \end{equation}
Take $P$, larger than any prime in $S$, such that the coefficients of each $A_i$ are $p$-integral for all $p > P$; it now follows from \eqref{cb} that, for all $p > P$, any element $g$ which lies in $G_{12}$ for any $y_1,y_2 \in Y(\field{Z}[S^{-1}])$ has for entries roots of monic polynomial with $p$-integral coefficients; in other words, it is $p$-integral.
Thus, for all such $p$ there is an induced map
$$ G_{12} \rightarrow \SL_n(\overline{\field{Z}}_p) \rightarrow \SL_n(\overline{\mathbb{F}}_p)$$
which is necessarily injective if we also require $p$ to be larger
than the order of $G_{12}$, since any torsion element of the kernel of $\SL_n(\overline{\field{Z}}_p) \rightarrow \SL_n(\overline{\mathbb{F}}_p)$
must have $p$-power order. For general constructibility reasons (similar to Lemma \ref{Kleiman})
the order of $G_{12}$ is bounded above; choose $P$ to be larger than this bound.
Now for any $\sigma$ belonging to the inertia group at $p >P$, and any $g \in G_{12}$,
$g^{\sigma}$ and $g$ have the same image in $\SL_n(\overline{\mathbb{F}_p})$, so must coincide. This is precisely to say that the Galois action on $G_{12}$ is unramified at $p$, so we have proved {\em Claim 2}
with $\mathcal{P}$ the set of primes less than $P$.
\qed
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Take $y_1, y_2 \in Y(\field{Z}[S^{-1}])$. We will now prove
\begin{quote}
{\em Claim 2 :} The action of $\Gal(\overline{\field{Q}}/\field{Q})$
on the stabilizers $G_{y_i}(\overline{\field{Q}})$ and also on the set $G_{12} := \{g \in G(\overline{\field{Q}}): g y_1 = y_2\}$
is unramified outside a set of primes $\mathcal{P}$ depending only on $n,d,S$.
\end{quote}
\proof (of {\em Claim 2}).
It is enough to prove the claim for $G_{12}$; take $y_1=y_2$ to get the claim about stabilizers.
From finiteness of the action map we see that the matrix entries of
$g^{\pm 1}$ for $g \in G_{12}$ satisfy monic polynomials whose coefficients are rational polynomials in the coordinates of $y_i$. Take $A_1(y_1,y_2), \ldots, A_M(y_1,y_2)$ to be the finite collection of all coefficients arising in this way.
Now, for any extension of the $p$-valuation on $\field{Q}$ to $\overline{\field{Q}}$, and for every matrix entry $g^{\pm 1}_{ij}$ of $g^{\pm 1}$ we have
\begin{equation} \getsbel{cb} |g^{\pm 1}_{ij}|_p \leq \max_k |A_k(y_1, y_2)|_p. \end{equation}
Take $P$, larger than any prime in $S$, such that the coefficients of each $A_i$ are $p$-integral for all $p > P$; it now follows from \eqref{cb} that, for all $p > P$, any element $g$ which lies in $G_{12}$ for any $y_1,y_2 \in Y(\field{Z}[S^{-1}])$ has for entries roots of monic polynomial with $p$-integral coefficients; in other words, it is $p$-integral.
Thus, for all such $p$ there is an induced map
$$ G_{12} \rightarrow \SL_n(\overline{\field{Z}}_p) \rightarrow \SL_n(\overline{\mathbb{F}}_p)$$
which is necessarily injective if we also require $p$ to be larger
than the order of $G_{12}$, since any torsion element of the kernel of $\SL_n(\overline{\field{Z}}_p) \rightarrow \SL_n(\overline{\mathbb{F}}_p)$
must have $p$-power order. For general constructibility reasons (similar to Lemma \ref{Kleiman})
the order of $G_{12}$ is bounded above; choose $P$ to be larger than this bound.
Now for any $\sigma$ belonging to the inertia group at $p >P$, and any $g \in G_{12}$,
$g^{\sigma}$ and $g$ have the same image in $\SL_n(\overline{\mathbb{F}_p})$, so must coincide. This is precisely to say that the Galois action on $G_{12}$ is unramified at $p$, so we have proved {\em Claim 2}
with $\mathcal{P}$ the set of primes less than $P$.
\qed
Now, with $y \in Y(\field{Z}[S^{-1}])$, there is an injection
$$ \mbox{$G(\field{Q})$-orbits on $Gy \cap Y(\field{Q})$} \hookrightarrow \mbox{$G_{y}$-torsors over $\field{Q}$}$$
sending $y' \in Y(\field{Q})$ to the right torsor $\{g: gy=y'\}$. The {\em Claim} means that,
under this map, elements of $Y(\field{Z}[S^{-1}])$ are sent to unramified-away-from-$\mathcal{P}$
torsors for the unramified-away-from-$\mathcal{P}$-group $G_y$, whose order is moreover bounded. Hermite-Minkowski gives an upper bound on the size of unramified Galois $H^1$ in this setting, and we conclude that
the {\em $S$-integral} points lying in $Gy \cap Y(\field{Q})$ lie in a collection of $G(\field{Q})$-orbits
whose cardinality is bounded in terms of $d,n,S$.
Finally, we pass to $G(\field{Z}[S^{-1}])$-orbits using \eqref{cb}.
This proves the first statement of the Corollary.
In the last sentence of the argument we used \eqref{cb} only at $p \notin S$;
but using it also for $p \in S$ allows one to conclude similarly that
\begin{equation} \getsbel{wwp2} \{y \in Y(\mathbb{Z}): \mathrm{disc}(y) =N, \mbox{ht.}(y) \leq B\}\end{equation}
is covered by $O_{\epsilonsilon}(B^{\epsilonsilon})$ orbits
of $\SL_n(\field{Z})$. Here $\mathrm{ht}(y)$ just refers to the largest coefficient of $y$, rather than a Weil height. This is the second statement of the Corollary.
\subsection*{Reduction of Theorem \ref{thm1} to Theorem \ref{thm2}}
Suppose that $X^{\circ}$ is quasi-projective and $X^{\circ}_{\field{C}}$
admits a geometric variation of Hodge structure, i.e.\ there is a morphism
$$ \pi_{\field{C}}: \mathfrak{X}_{\field{C}} \rightarrow X_{\field{C}}^{\circ}$$
such that the variation of Hodge structure
in the theorem statement is a direct summand of $R^i \pi_{\field{C}*} \field{C}$,
for some $i \geq 0$. (That such $\mathfrak{X}$ exists is understood to be the content of the word ``geometric.'')
Theorem \ref{thm1} assumes that the period morphism associated to $\mathsf{V}$
is locally finite-to-one.
In this case, the period morphism associated to $R^i \pi_{\field{C}*} \field{C}$ has
the same property.
The issue
to be handled is that $\mathfrak{X}_{\field{C}}$ is defined only over $\field{C}$
whereas Theorem \ref{thm2} requires a $K$-morphism as input.
To verify sparsity it will suffice, by descending
Noetherian induction and extending the base field, to produce a proper
Zariski-closed subset $E \subset X_{\overline{K}}$
with the property that integral points on $X^{\circ}$ that
do not lie inside $E$ are sparse. (Warning: this is not the same as studying
``integral points on the quasi-projective variety $X^{\circ} \backslash E$.'')
By a standard
spreading out technique we may extend
$\pi_{\field{C}}: \mathfrak{X}_{\field{C}} \rightarrow X_{\field{C}}$
over the spectrum of a subring $R \supset K$ of $\field{C}$, finitely generated over $K$:
$$ \pi: \mathfrak{X}_R \rightarrow X_R = X \times_{\mathrm{Spec} \ K} S,$$
where $S$ is the spectrum of $R$. This recovers $\pi_{\field{C}}$
upon taking the pullback via the map
$s : \Spec \field{C} \rightarrow S$
associated to $R \rightarrow \field{C}$.
The image of
$s$ is the generic point $\mathrm{et}a_S$ of $S$.
Since $S$ is the spectrum of the finitely generated integral domain $R$,
$S \rightarrow \operatorname{Spec} K$ is smooth at $\mathrm{et}a_S$.
Then, deleting the nonsmooth locus, we may suppose that $S$ is
a smooth $K$-variety.
Consider the morphism of vector bundles supplied by Lemma \ref{KS}, which, applied to the morphism $\mathfrak{X}_R \rightarrow X \times S$, gives
a morphism of locally free sheaves \begin{equation} \getsbel{g2} g: T_{X \times S} \otimes \mathcal{H}_1 \rightarrow \mathcal{H}_2\end{equation}
over $X \times S$.
Let $T_{X} \subset T_{X \times S}$ be the sub-bundle defined by the pullback of the tangent bundle of $T_X$.
Since $\pi_{\field{C}}$ has finite-to-one period map, the specialization $g_{x,s}|_{T_X}$ is injective for each $x \in X(\field{C})$;
the same is then true for a Zariski-open neighbourhood
$U$ of $X \times \{\mathrm{et}a_S\}$ inside $X \times S$.
Choose $s' \in S(\overline{K})$
such that $X(\overline{K}) \times s'$ meets $U(\overline{K})$.
The fiber of $\mathfrak{X}_R \rightarrow X_R$ over
$s'$ gives a proper smooth morphism of $\bar{K}$-varieties
$\mathfrak{X}_{s'} \rightarrow X_{\bar{K}}$.
The associated period map has generically injective derivative; let $E \subset X_{\bar{K}}$ be the locus where its
derivative has a nontrivial kernel, i.e., where $g_{x,s'}|T_X$ fails to be injective.
$E$ is a proper Zariski closed subset of $X$ defined over some finite extension $K_1 \supset K$, and
Theorem \ref{thm2} (applied after passage to $K_1$) implies that integral points on $X^{\circ}$ that lie on the complement of $E$ are sparse.
This establishes the inductive step, and therefore
concludes the reduction of Theorem \ref{thm1} to Theorem \ref{thm2}.
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\section{Proof of Theorem \ref{thm2}}
The remainder of the paper is devoted to the proof of Theorem \ref{thm2}. We use notation as in the statement.
We will fix throughout a good integral model for both $(X,Z)$ and the morphism $\mathfrak{X} \rightarrow X^{\circ}$,
as in Definition \ref{goodmodel}.
As we have noted, the proof involves proving that integral points of $X$ lie on
various collections of subvarieties, whose dimension will be steadily reduced until they
are either points or fibers of the period map.
The key inductive statement used to reduce the dimension of the subvarieties is
Lemma \ref{lem:induction}.
It may be helpful to note, in advance, that we will not need to keep track of any {\em integral} structure on our subvariety. The notion
of ``integral point on a subvariety'' will simply mean a $K$-rational point of the subvariety that is integral
as a rational point on $X$.
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\subsection{Large fundamental group.} \getsbel{large}
Enlarging $S$ if necessary, we fix a prime
$p \in S$ for which the integral cohomology of complex fibers of $\mathfrak{X}$ over $X^{\circ}$
is $p$-torsion-free, say of rank $r$ over $\field{Z}$.
Fixing $x \in X^{\circ}(\field{C})$
we get a monodromy representation of the topological fundamental group
\begin{equation} \getsbel{Gmdef} \pi_1(X^{\circ, {\text{an}}}, x) \longrightarrow G_n := \mathrm{Aut}(H^i(\mathfrak{X}_x, \field{Z}/p^n)) \simeq \GL_r(\field{Z}/p^n \field{Z}).\end{equation}
In this setting, the {\em global invariant cycle theorem} (\cite[Corollaire 4.1.2 and 4.1.3.3]{Deligne_Hodge2}) implies the following statement: For any complex irreducible subvariety $\iota: V \hookrightarrow X_{\field{C}}$,
not contained in $Z_{\field{C}}$ and with $V^{\circ}$ not contained in a fiber of the period map,
the image of the monodromy representation
\begin{equation} \getsbel{i-inf} \pi_1(V^{\circ,{\text{an}}}) \rightarrow G_n \end{equation}
(topological $\pi_1$, taken for an arbitrary choice of basepoint) has size that grows without bound as $n \rightarrow \infty$.
Indeed \cite[Corollaire 4.1.2 and 4.1.3.3]{Deligne_Hodge2}
applies, after passing from $V$ to the smooth part $V'$ of its intersection
with $X^{\circ}$, to show that the image of
$$ \pi_1({V'}^{{\text{an}}}) \longrightarrow \mathrm{Aut} \ H^i(\mathfrak{X}_x, \field{Q})$$
is infinite.
In particular, the image of the monodromy representation of $\pi_1((V^{\circ})^{{\text{an}}})$
on $H^i(\mathfrak{X}_x, \field{Z})$ is infinite,
so the size of the image of monodromy on $H^i(\mathfrak{X}_x, \field{Z})/p^n$
grows without bound as $n \rightarrow \infty$.
These results transpose, as usual, to the {\'e}tale topology.
Indeed, $R^i \pi^{\mathrm{et}}_*(\field{Z}/p^n \field{Z})$ defines
a locally constant {\'e}tale sheaf of $\field{Z}/p^n$-modules on $X^{\circ}$,
which, by the local constancy of direct images for a smooth proper morphism (\cite[Theorem 5.3.1]{SGA4.5}),
extends to a locally constant {\'e}tale sheaf on the good integral model $X^{\circ}_S$ over $\mathfrak{o}_S$.
For $V$ as above, standard comparison theorems show that the homomorphism in \eqref{i-inf} factors as
\[ \pi_1(V^{\circ,{\text{an}}}) \rightarrow \pi_1^{\mathrm{et}}(V^{\circ}) \rightarrow G_n, \]
so
the image of {\'e}tale $\pi_1$
\begin{equation} \getsbel{fff} \pi_1^{\mathrm{et}}(V^{\circ}) \rightarrow G_n\end{equation}
(again, with an arbitrary geometric basepoint in $(V^{\circ})$)
has size that grows without bound as $n \rightarrow \infty$.
Next, let $E \supset K$
be an arbitrary algebraically closed field, with base change $X_E := X \times_{K} E$;
let $i: V \hookrightarrow X_E$ be a closed $E$-subvariety
not contained in $Z_E$ or a fiber of the period map.
(Note that we can make sense of the latter condition without reference to $\field{C}$ by using Lemma \ref{KS}:
by ``$V$ is contained in a fiber of the period map'' we mean that the associated morphism of vector bundles is zero on the smooth locus of $V$.)
We get by base change $\pi_E: \mathfrak{X}_E \rightarrow X_E$ and an {\'e}tale local
system $R^i \pi^{\mathrm{et}}_{E*} (\field{Z}/p^n \field{Z})$ on $X_E^{\circ}$; and
the same conclusion as above holds, i.e.\
the monodromy representation \eqref{fff} for $V$ on $R^i \pi^{\mathrm{et}}_* (\field{Z}/p^n \field{Z})$ has ``large image''
in the sense specified above.
We will use this only in the case when $E$ is the algebraic closure
of a finitely generated field; we may then choose a $K$-embedding $\sigma: E \rightarrow \field{C}$,
and thus also $V_{\field{C}} \subset X_{\field{C}}$ compatibly with $V \subset X_E$.
The local system $R^i \pi^{\mathrm{et}}_{E*} (\field{Z}/p^n \field{Z})$ on $V_E$
pulls back to the similarly defined system on $V_{\field{C}}$,
so ``large monodromy'' for $V$ follows from the same statement for $V_{\field{C}}$.
\subsection*{Construction of a suitable cover of $X$} \getsbel{bigcover}
Our proof will involve an induction over higher and higher-codimension subvarieties of $X$ about which we know almost nothing apart from their degree.
It is thus crucial to have at hand covers of $X$ whose monodromy is uniformly bounded below on restriction to {\em every} subvariety of bounded dimension and degree.
\begin{lem}\getsbel{lem31} Fix $d, n, D \geq 1$,
and let $H$ be any (locally closed, finite type) complex subvariety of the Hilbert scheme of
subschemes of $X_{\field{C}}$ of degree $\leq d$ and dimension $n$.
There are a finite group $G$ and a
finite morphism $f: \widetilde{X} \to X$ of $K$-varieties,
equipped with an injection $G \hookrightarrow \mathrm{Aut}(\widetilde{X} / X)$, \
such that:
\begin{itemize}
\item[(a)] $f|_{X^{\circ}}$ is finite {\'e}tale Galois with deck group $G$,
and, moreover, extends to a finite {\'e}tale cover of the good integral model $X^{\circ}_S$.
\item[(b)]
Let $U \subset X$ be any $n$-dimensional irreducible closed complex subvariety
of degree $\leq d$.
Suppose that:
\begin{itemize}
\item The point of the Hilbert scheme classifying $U$ lies in $H(\field{C})$, and
\item $U$ is not contained in $Z$ and $U^{\circ {\text{an}}}$ is not contained in a single fiber of the period map $\field{P}hi$.
\end{itemize}
Let $Q$ be any irreducible component of $f^{-1} U$, endowed with the reduced structure,
and such that the induced finite map $f:Q \rightarrow U$ is dominant
(note that it is automatically \'etale over $U^{\circ}$).
Then the degree of $f:Q \rightarrow U$ at the generic point is $\geq D$,
i.e., the induced map of function fields has degree $\geq D$.
\end{itemize}
\getsbel{lem:bigcover}
\end{lem}
\begin{proof}
Through the rest of this proof, $U$ will represent a single subvariety of $X$, classified by a point of the Hilbert scheme,
and we will use $\mathcal{U}$ for the universal family.
We are going to find a cover $f: \widetilde{X} \to X$
and a proper Zariski-closed subset $H_1 \subseteq H$
such that the conclusion of (b) holds for any $U=U_h$,
satisfying the assumptions of (b), and
with $h \in (H-H_1)(\field{C})$. The result will follow by Noetherian induction.
In particular, removing the singular locus of $H$ at the start,
we may suppose that $H$ is smooth.
Take a geometric generic point $\mathrm{et}a \rightarrow H$.
Let $U_{\mathrm{et}a} \subset X_{\mathrm{et}a}$ be the
corresponding generic subscheme.
We may assume without loss of generality that:
\begin{quote}
{\em Situation:}
\begin{itemize}
\item[(a)] Every geometric fiber of $\mathcal{U} \rightarrow H$ is integral;
\item[(b)] Every fiber of $\mathcal{U} \rightarrow H$ meets $X^{\circ}$;
\item[(c)] On each fiber $\mathcal{U}_h$
for $h \in H(\field{C})$
the period map is not locally constant.
\end{itemize}
\end{quote}
For (a), note that
the locus of points with geometrically integral fiber by \cite[12.2.1(x)]{EGAIV3}
is open on the base, so if there exists one $h \in H(\field{C})$
for which the fiber $U_h$ is integral, then
(after shrinking $H$ to a suitable nonempty open neighbourhood) we can suppose it is true for all $h$. If there is no such $h$, then the conclusion of the theorem holds for $H$ vacuously.
For (b), note that the set of $h$ for which $U_h$ meets $X^{\circ}$ is constructible.
To see this, first note that $U_h$ is reduced for every $h$.
Now note that $U_h$ meets $X^{\circ}$
if and only if $(U_h \cap Z)_s \rightarrow U_s$ is not surjective, and apply \cite[9.6.1(i)]{EGAIV3}.
Thus, restricting to an open subset of $H$,
we can assume that $U_h$ meets $X^{\circ}$ either for no $h$ or for all $h$.
In the former case, the statement is vacuously true;
so we can assume that (b) holds for all $h$.
For (c) let $\mathcal{U}'$ be the smooth locus of the morphism $\mathcal{U} \rightarrow H$,
which, by flatness of the morphism, coincides with the locus of points which are smooth points of their fibers (\cite[17.5.1]{EGAIV4}).
Note that:
\begin{itemize}
\item $\mathcal{U}'$ is itself smooth over $\mathrm{Spec} \ K$, since it is smooth over $H$ and $H$ was assumed smooth.
\item $\mathcal{U}'$ contains an open dense subset of every fiber, since these fibers are all integral.
\item $\mathcal{U}'$ is a $K$-variety: this follows from the previous conditions.
It is reduced by smoothness,
and since $\mathcal{U}' \rightarrow H$ is flat (\cite[Tag 01VF]{Stacks}), $H$ is irreducible, and the fibers
are irreducible, it readily follows (\cite[Tag 004Z]{Stacks}) that $\mathcal{U}'$ is itself irreducible.
\end{itemize}
The tangent bundle $T_{\mathcal{U}'}$ has a sub-bundle $T_{\mathcal{U}'/H}$
made up of`vertical'' vector fields. Restricting the morphism of Lemma \ref{KS} to this sub-bundle we get
$$ g: \mathcal{H}_1 \otimes T_{\mathcal{U}'/H} \rightarrow \mathcal{H}_2$$
Now, we may certainly assume there is some $h \in H(\field{C})$
such that $\mathcal{U}_h$ satisfies the conditions of (b) in the statement of the Lemma, or else the Lemma once again holds vacuously.
In particular, there exists a point $u \in \mathcal{U}_h(\field{C})$, smooth in the fiber $\mathcal{U}_h$,
such that $g_u$ is nonzero. It follows that $g_u$ is nonzero on a nonempty Zariski-open
subset of $\mathcal{U}'$; the image of this Zariski-open
by the dominant morphism $\mathcal{U}' \rightarrow H$ contains an nonempty open subset of $H$,
and we replace $H$ by this open to obtain the second part of the {\em Situation.}
So we proceed assuming ourselves to be in the {\em Situation} above.
We continue to write $\mathcal{U}'$ for the smooth locus of $\mathcal{U}/H$
and $\mathcal{U}'^{\circ}$ for the preimage of $X^{\circ}$ in $\mathcal{U}'$. Recall that our assumptions
guarantee that $\mathcal{U}'^{\circ}$ is fiberwise dense in $\mathcal{U}'$.
By \S \ref{large} we can find
an $m$ for which the image of the {\em geometric} monodromy representation of $\pi_1(U_{\mathrm{et}a}^{'\circ})$
in $G_m$ has size at least $D$ (same notation as in \S \ref{large}).
This choice of $m$ determines a finite \'etale Galois cover of $X^{\circ}$ with Galois group $G_m$, which extends to a finite {\'e}tale Galois
cover of the $\mathfrak{o}_S$-model $X^{\circ}_S$.
Let $f: \widetilde{X} \rightarrow X$ be the normalization of $X$ in this $G_m$ cover;
then $\widetilde{X}$ is a normal $K$-variety and the morphism $f$ is finite (although not necessarily flat).
The action of $G_m$ by deck transformations above $X^{\circ}$ extends uniquely
to a $G_m$-action on the morphism $f$.
The morphism $f$ gives of course a morphism $f: \widetilde{X}_{\mathrm{et}a} \rightarrow X_{\mathrm{et}a}$
after base-change from $\operatorname{Spec} \ K$ to $\mathrm{et}a$. The restricted map
\begin{equation} \getsbel{fU0} f_{\mathrm{et}a}^{-1} \mathcal{U}_{\mathrm{et}a}'^{\circ} \rightarrow U_{\mathrm{et}a}'^{\circ} \end{equation}
is finite {\'e}tale and has degree $\geq D$ restricted to
each geometric component of the source
by choice of $f$.
Now this morphism is the geometric generic fiber of a
finite {\'e}tale morphism of smooth $H$-schemes:
\begin{equation} \getsbel{fU} f^{-1} \mathcal{U}'^{\circ} \rightarrow \mathcal{U}'^{\circ}\end{equation}
and we want to draw the same conclusion about degrees
for the fibers of \eqref{fU} over
a nonempty open subset of $H$.
This will imply the desired conclusion, for -- with $Q$ as in the statement --
the assumed dominance implies that $Q \cap f^{-1} U'^{\circ}$
is an open nonempty subset of $Q$.
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For (b), note that the set of $h$ for which $U_h$ meets $X^{\circ}$ is constructible.
To see this, first note that $U_h$ is reduced for every $h$.
Now note that $U_h$ meets $X^{\circ}$
if and only if $(U_h \cap Z)_s \rightarrow U_s$ is not surjective, and apply \cite[9.6.1(i)]{EGAIV3}.
Thus, restricting to an open subset of $H$,
we can assume that $U_h$ meets $X^{\circ}$ either for no $h$ or for all $h$.
In the former case, the statement is vacuously true;
so we can assume that (b) holds for all $h$.
For (c) let $\mathcal{U}'$ be the smooth locus of the morphism $\mathcal{U} \rightarrow H$,
which, by flatness of the morphism, coincides with the locus of points which are smooth points of their fibers (\cite[17.5.1]{EGAIV4}).
Note that:
\begin{itemize}
\item $\mathcal{U}'$ is itself smooth over $\mathrm{Spec} \ K$, since it is smooth over $H$ and $H$ was assumed smooth.
\item $\mathcal{U}'$ contains an open dense subset of every fiber, since these fibers are all integral.
\item $\mathcal{U}'$ is a $K$-variety: this follows from the previous conditions.
It is reduced by smoothness,
and since $\mathcal{U}' \rightarrow H$ is flat (\cite[Tag 01VF]{Stacks}), $H$ is irreducible, and the fibers
are irreducible, it readily follows (\cite[Tag 004Z]{Stacks}) that $\mathcal{U}'$ is itself irreducible.
\end{itemize}
The tangent bundle $T_{\mathcal{U}'}$ has a sub-bundle $T_{\mathcal{U}'/H}$
made up of`vertical'' vector fields. Restricting the morphism of Lemma \ref{KS} to this sub-bundle we get
$$ g: \mathcal{H}_1 \otimes T_{\mathcal{U}'/H} \rightarrow \mathcal{H}_2$$
Now, we may certainly assume there is some $h \in H(\field{C})$
such that $\mathcal{U}_h$ satisfies the conditions of (b) in the statement of the Lemma, or else the Lemma once again holds vacuously.
In particular, there exists a point $u \in \mathcal{U}_h(\field{C})$, smooth in the fiber $\mathcal{U}_h$,
such that $g_u$ is nonzero. It follows that $g_u$ is nonzero on a nonempty Zariski-open
subset of $\mathcal{U}'$; the image of this Zariski-open
by the dominant morphism $\mathcal{U}' \rightarrow H$ contains an nonempty open subset of $H$,
and we replace $H$ by this open to obtain the second part of the {\em Situation.}
So we proceed assuming ourselves to be in the {\em Situation} above.
We continue to write $\mathcal{U}'$ for the smooth locus of $\mathcal{U}/H$
and $\mathcal{U}'^{\circ}$ for the preimage of $X^{\circ}$ in $\mathcal{U}'$. Recall that our assumptions
guarantee that $\mathcal{U}'^{\circ}$ is fiberwise dense in $\mathcal{U}'$.
By \S \ref{large} we can find
an $m$ for which the image of the {\em geometric} monodromy representation of $\pi_1(U_{\mathrm{et}a}^{'\circ})$
in $G_m$ has size at least $D$ (same notation as in \S \ref{large}).
This choice of $m$ determines a finite \'etale Galois cover of $X^{\circ}$ with Galois group $G_m$, which extends to a finite {\'e}tale Galois
cover of the $\mathfrak{o}_S$-model $X^{\circ}_S$.
Let $f: \widetilde{X} \rightarrow X$ be the normalization of $X$ in this $G_m$ cover;
then $\widetilde{X}$ is a normal $K$-variety and the morphism $f$ is finite (although not necessarily flat).
The action of $G_m$ by deck transformations above $X^{\circ}$ extends uniquely
to a $G_m$-action on the morphism $f$.
The morphism $f$ gives of course a morphism $f: \widetilde{X}_{\mathrm{et}a} \rightarrow X_{\mathrm{et}a}$
after base-change from $\operatorname{Spec} \ K$ to $\mathrm{et}a$. The restricted map
\begin{equation} \getsbel{fU0} f_{\mathrm{et}a}^{-1} \mathcal{U}_{\mathrm{et}a}'^{\circ} \rightarrow U_{\mathrm{et}a}'^{\circ} \end{equation}
is finite {\'e}tale and has degree $\geq D$ restricted to
each geometric component of the source
by choice of $f$.
Now this morphism is the geometric generic fiber of a
finite {\'e}tale morphism of smooth $H$-schemes:
\begin{equation} \getsbel{fU} f^{-1} \mathcal{U}'^{\circ} \rightarrow \mathcal{U}'^{\circ}\end{equation}
and we want to draw the same conclusion about degrees
for the fibers of \eqref{fU} over
a nonempty open subset of $H$.
This will imply the desired conclusion, for -- with $Q$ as in the statement --
the assumed dominance implies that $Q \cap f^{-1} U'^{\circ}$
is an open nonempty subset of $Q$.
We now use \cite[Lemma 055A]{Stacks}
(see also \cite[Prop.\ 9.7.8]{EGAIV3} and references therein).
which guarantees the existence of a morphism $g: H' \rightarrow H$ (which in fact factors as a finite {\'e}tale
surjection followed by an open immersion) such that, after base change of
$f^{-1} \mathcal{U}'^{\circ} \rightarrow \mathcal{U}'^{\circ}$
by $g$ -- i.e.\ replacing $\mathcal{U}'^{\circ}$ by $\mathcal{U}'^{\circ} \times_{H} H'$
and similarly for $f^{-1} \mathcal{U}'^{\circ}$ --
the following assertions hold:
\begin{itemize}
\item[(a)] Each irreducible component of the generic fiber
$(f^{-1} \mathcal{U}'^{\circ})_{\mathrm{et}a'}$ (with $\mathrm{et}a'$ the generic point of $H'$ -- not a geometric generic point here) is in fact
a geometrically irreducible component of that generic fiber.
\item[(b)] Let $\overline{Z_1}, \dots, \overline{Z_r}$ be the Zariski closures of these generic irreducible components $Z_1, \dots, Z_r$
inside $f^{-1} \mathcal{U}'^{\circ}$.
These $\overline{Z_i}$ give, upon intersection with the fiber $(f^{-1} \mathcal{U}'^{\circ})_{h'}$ above any $h' \in H'$,
the decomposition of that fiber
into irreducible components, and indeed each of these irreducible components are geometrically irreducible. \end{itemize}
In the decomposition of (b)
$$ (f^{-1} \mathcal{U}'^{\circ}) = \coprod \overline{Z_{\alpha}}$$
the sets $Z_{\alpha}$ are disjoint.
Indeed, upon restriction to each fiber, this decomposition recovers the decomposition of $f^{-1} \mathcal{U}'^{\circ}_{h'}$;
however, this is finite {\'e}tale over $\mathcal{U}'^{\circ}_{h'}$ and Lemma \ref{disjointness} implies the disjointness.
In particular, the $\overline{Z_{\alpha}}$ are
both closed and open, and in particular inherit a scheme structure as open sets in $f^{-1} \mathcal{U}'^{\circ}$.
The restriction of the map $f$ to each $\overline{Z_{\alpha}}$ is then a finite {\'e}tale map
$f_{\alpha}:\overline{Z_{\alpha}} \rightarrow \mathcal{U}'^{\circ}$.
The degree of such a map is locally constant on the base, and here, by assumption,
that degree is $\geq D$
everywhere on the generic fiber $\mathcal{U}'^{\circ}_{\mathrm{et}a'}$.
That generic fiber is dense because it contains every generic point of
of $\mathcal{U}'^{\circ}$, and, consequently the degree of $f_{\alpha}$ is everywhere $\geq D$.
Restricting to a single fiber $\mathcal{U}'^{\circ}_{h'}$
for $h' \in H'$ gives the desired conclusion --
that is, the bound stated in (b) of the Theorem holds for all fibers $U_h$
for all $h$ in a nonenmpty open subset of $H$, explicitly, the image of $H' \rightarrow H$.
Finally, we conclude by Noetherian induction.
We have shown that, given $H$, there is a Zariski-closed $H_1 \subseteq H$
and an $m > 0$, such that the image of geometric monodromy in $G_m$
gives an \'etale cover for $X$, that satisfies the required properties for
any $h \in (H - H_1)(\mathbb{C})$.
There is no harm in replacing $G_m$ by $G_{m'}$, for $m' \geq m$.
Thus we can apply Noetherian induction to find one $m$ that works for all $h \in H(\mathbb{C})$.
\qed
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\subsection*{Bounding rational points on a subvariety} \getsbel{ss:induction}
The following result is the key inductive step.
We note that {\em all constants appearing in this discussion are permitted to
depend on the variety $X$, and indeed on the integral model
chosen in \S \ref{integralmodel}, without explicit mention.}
\begin{lem}
\getsbel{lem:induction}
Let $V$ be a geometrically irreducible closed subvariety of $X$ defined over $K$,
of dimension $n$ and degree $d$, such that $V_{\field{C}}$
is not contained in $Z$ and $V^{\circ {\text{an}}}$ is not
contained in a fiber of $\field{P}hi$.
Then all integral points of $X^\circ$ of height $\leq B$
that lie on $V$ can be covered by
$O_{d,\epsilonsilon} (B^{\epsilonsilon})$ irreducible (but not necessarily geometrically irreducible) subvarieties, all defined over $K$,
with dimension $\leq n-1$ and degree $O_{d, \epsilonsilon}(1)$. \end{lem}
\proof
Choose
$D$ so that $\frac{n+1}{D^{1/n}} < \epsilonsilon$.
Let $\mathcal{P}$ be the
(finite, by
Lemma \ref{Kleiman}) set of Hilbert polynomials that arise
from irreducible subvarieties of $X$ of dimension $n$ and degree $d$,
and let $H_{\mathcal{P}}$ be the associated Hilbert scheme. Write $H^{red}_{\mathcal{P}}$ for the reduced induced closed subscheme of $H_{\mathcal{P}}$.
For this choice of $n,d,D, H=H^{red}_{\mathcal{P}}$
take a finite group $G$ and a finite morphism $f: \widetilde{X} \rightarrow X$ as provided by Lemma~\ref{lem:bigcover}. We note that $H_{\mathcal{P}}(\field{C}) = H^{red}_{\mathcal{P}}(\field{C})$, so the passage to the reduced subscheme structure is irrelevant for the statements on complex subvarieties proved in Lemma~\ref{lem:bigcover}.
For every $x \in X^{\circ}(K)$, the action of $G$ on the fiber of $\widetilde{X}$ over $x$
defines a class in the Galois cohomology group $H^1(\Gal(\overline{K}/K), G)$.
Concretely, since the Galois action on $G$ is trivial,
$H^1(\Gal(\overline{K}/K), G)$ classifies
homomorphisms $\rho: \Gal(\overline{K}/K) \rightarrow G$ up to conjugacy.
Choosing a point $\widetilde{x}$ above $x$,
we define a homomorphism $\rho$ by the rule
\begin{equation} \getsbel{tww} \sigma(\widetilde{x}) = \rho(\sigma) \cdot \widetilde{x}\end{equation}
for $\sigma$ in the Galois group.
If $x$ is $S$-integral, this homomorphism
$\rho$ is in fact unramified outside $S$.
Such a $\rho$ can also be used to twist $\widetilde{X} \rightarrow X$, namely,
one modifies the Galois action on $\widetilde{X}$ through $\rho$;
and then \eqref{tww} means precisely that $x$ will lift to a $K$-rational point on the
twist of $\widetilde{X}$ indexed by $\rho$.
(See $\S$ 4.5 and Thm.\ 8.4.1 of \cite{PoonenRP} for further discussion.)
There are only finitely many homomorphisms $\Gal(\overline{K}/K) \rightarrow G$, unramified outside $S$;
call them $\rho_1, \rho_2, \dots, \rho_R$. This list does not depend on $B$.
Each such $\rho_j$
can be used to twist $f$ to a map $f_j:\widetilde{X}_j \rightarrow X$.
Our previous discussion now shows that any
integral point of $X^\circ$ lifts along some $f_j$ to a point of $\widetilde{X}_j(K)$.
For a sufficiently large integer $e$ the pullback $(f_j^* \mathcal{L})^{\otimes e}$ is
very ample and defines, after fixing a basis of sections, a projective embedding $\widetilde{X}_j \hookrightarrow \mathbb{P}^{M_j}$.
Now the data of the diagram of $K$-varieties and line bundles
\begin{equation} \getsbel{diag} (X, \mathcal{L}^{\otimes e}) \stackrel{f_j}{\longleftarrow}(\widetilde{X}_j, (f_j^* \mathcal{L})^{\otimes e}) \hookrightarrow (\mathbb{P}^{M_j}, \mathcal{O}(1))\end{equation}
depends on various choices,
but these choices can (and will) be made once and for all depending only on $d,\epsilonsilon$.
Then for $P \in \widetilde{X}_j(K)$ we get
\begin{equation} \getsbel{imp}
H_\mathcal{L}(f_j(P))^e \asymp H_{f_j^* \mathcal{L}}(P)^e \asymp H_{\mathbb{P}^{M_j}}(P)
\end{equation}
where the symbol $\asymp$ means that
the ratio is bounded above and below by constants that may depend on $f_j$.
Since there are only finitely many $f_j$, and their coefficients are bounded in terms of $d$ and $\epsilonsilonilon$ (and, as always, $X$ and $S$) but don't depend on $B$,
these constants depend only on $d$ and $\epsilonsilonilon$.
Therefore, we have shown that the integral points of $X^\circ$ with height $\leq B$
belonging to $V$ all have the form $f_j(P)$, where $P$ is a $K$-rational point of $f_j^{-1}(V)$ with $H_{\mathbb{P}^{M_j}}(P) \leq c_{d,\epsilonsilonilon}B^e$.
It will suffice to prove the conclusion for those $P$ for which $f_j(P)$ is a {\em smooth} point of $V$,
simply by including each irreducible component of the singular locus of $V$ in the list of subvarieties (see Lemma \ref{Kleiman} part (d) for the necessary bounds).
Let $V' \subset V$ be the (open) smooth locus.
Consider those geometric components $Q^{\circ} \subset
(f_j^{-1} V')^{\circ}$ that have a $K$-rational point.
Because $ (f_j^{-1} V')^{\circ}$ is a finite {\'e}tale cover
of the geometrically irreducible smooth $K$-variety $V'^{\circ}$, its
geometric components are pairwise disjoint (Lemma \ref{disjointness})
and permuted by the Galois group; so
any such $Q^{\circ}$ is defined over $K$ and the number of such $Q^{\circ}$ is bounded in number
by the size of the group $G$.
The Zariski closure $Q$ of any $Q^{\circ}$ is again geometrically irreducible and defined over $K$;
we understand it to be endowed with its reduced scheme structure.
The map $f_j: \widetilde{X}_j \rightarrow X$ induces a compatible map $f_j: Q \rightarrow V$,
which is dominant since, by construction of $Q$, the image contains a nonempty open set of $V'^{\circ}$.
Indeed, $f_j: Q \rightarrow V$ is {\'e}tale over $V^{\circ}$,
with degree between $D$ and $(\# G)$; the lower bound comes from (b) of Lemma \ref{lem31},
using also the fact that $f_j$ is a twist of $f$.
The degree of $V$ with respect to $\mathcal{L}^{\otimes e}$ is $d e^n$, and therefore,
by Lemma \ref{degree}
the degree of $Q$, considered as a closed subvariety of $\mathbb{P}^{M_i}$ via \eqref{diag}, satisfies
$$ D d e^n \leq \mathrm{deg} Q \leq (\# G) d e^n.$$
We apply Theorem \ref{Brobs} to
each $Q$ that arises in the above fashion, i.e.\
to the
Zariski closure of any irreducible geometric component of $(f_j^{-1} V')^{\circ}$ that has a $K$-point.
Theorem \ref{Brobs} and our choice of $D$ implies that the
set of rational points of $Q$ of height $\leq c B^e$
are supported on a set
of proper closed subvarieties of $Q$ of degree $O_{d,\epsilonsilon}(1)$ with cardinality $\ll_{d,\epsilonsilonilon} B^{2\epsilonsilonilon}$. These subvarieties are defined over $K$ and need not be geometrically irreducible.
For any such $Q$ and any such proper subvariety $Y \subset Q$,
the scheme-theoretic image $f_j(Y)$ under the finite map $f_j$ is a proper subvariety $f_j(Y) \subset V$, in particular, of dimension $\leq n-1$.
Moreover, $f_j$ restricts to a finite map $Y \rightarrow f_j(Y)$.
By Lemma \ref{degree} the $\mathcal{L}$-degree of $f_j(Y)$
is no larger than the $f_j^* \mathcal{L}$-degree of $Y$, in particular, $O_{d,\epsilonsilon}(1)$.
The number of maps $f_j$ depends only on $d,\epsilonsilon$,
and the number of $Q$ arising is then at most
the number of $f_j$
multiplied by the order of $G$, which is again $O_{d,\epsilonsilon}(1)$.
Consequently,
the number of $Y$ arising as in the prior paragraph
is $O_{d,\epsilonsilonilon}( B^{2\epsilonsilonilon})$, concluding the proof (after the obvious
scaling $\epsilonsilonilon \leftarrow \epsilonsilonilon/2$.)
\end{proof}
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\subsection{Conclusion of the proof of Theorem \ref{thm2}}
\getsbel{conclude}
\begin{proof} Fix $\epsilonsilon > 0$.
We use descending induction via Lemma \ref{lem:induction}. The inductive statement is the following:
\begin{quote}
$(\star)_n$: For every $n$ with $0 \leq n \leq \dim X$, there exists an integer $d_n$ with the following property: for all $B > 0$,
the $S$-integral points of $X^\circ$ are covered by a collection of
\[ O_{\epsilonsilon}(B^{(\dim X - n) \epsilonsilon}) \]
irreducible subvarieties of $X$, all defined over $K$, each of which is either
\begin{itemize}
\item[--]$\textrm{(a)}_n$: a subvariety of dimension $\leq n$ and degree $\leq d_n$, or
\item[--]$\textrm{(b)}$: a geometrically irreducible subvariety that is contained in a single fiber of the period map.
\end{itemize}
\end{quote}
The base case is given by $n = \dim X$, in which case, of course, the single subvariety $X \subseteq X$ suffices.
The implication $(\star)_n \implies (\star)_{n-1}$ follows from Lemma~\ref{lem:induction}:
Let $\mathcal{V}_n$ be the collection of $n$-dimensional varieties in the statement of $(\star)_n$.
For each $V \in \mathcal{V}_n$, we will
construct a set of varieties covering all the integral points of $X^{\circ}$ lying on $V$. We subdivide into cases:
\begin{itemize}
\item $V$ is not geometrically irreducible.
In this case,
we take the set $\{V_i\}$ of subvarieties given by
part (c) of Lemma \ref{Kleiman}.
These varieties number at most
$O_{n,d_n}(1)$ and they have
dimension $\leq n-1$ and degree $O_{n,d_n}(1)$.
\item $V$ is geometrically irreducible but $V^{\circ {\text{an}}}$ is contained in a fiber of $\field{P}hi$: then
we take the singleton set $\{V\}$.
\item $V$ is contained in $Z$; in this case
we can take the empty set $\emptyset$.
\item $V$ is geometrically irreducible and not contained in $Z$, and $V^{\circ{\text{an}}}$ is not contained in a fiber of $\field{P}hi$; then we may apply Lemma \ref{lem:induction} to show
that integral points of height $\leq B$ on $V$ are covered by $O_{d_n,\epsilonsilon}(B^{\epsilonsilon})$ irreducible $K$-varieties of dimension $\leq n-1$ and degree $O_{d_n,n,\epsilonsilonilon}(1)$.
\end{itemize}
We take $d_{n-1}$ to be the largest of the implicit constants $O_{n,d_n}(1)$ and $O_{n, d_{n}, \epsilonsilonilon}(1)$
appearing in the above proof. Then, to sum up,
by $(\star)_n$ we know that the $S$-integral points of $X^\circ$ of height at most $B$ are covered by $O_{\epsilonsilon}(B^{(\dim X - n) \epsilonsilon})$ subvarieties $V$
satisfying either $\textrm{(a)}_n$ or (b),
and we know that for each of those $V$, the subset of those points lying on $V$ is covered by $O_{\epsilonsilon}(B^{\epsilonsilon})$ subvarieties
satisfying either $\textrm{(a)}_{n-1}$ or (b);
together, these facts yield $(\star)_{n-1}$.
We emphasize that this is the point in the argument where the uniformity in Broberg's result is crucial. We have no control of the heights of the varieties making up the collection $\mathcal{V}_n$, and indeed these heights will grow with $B$; but since the implicit constants in Lemma~\ref{lem:induction} depend only on $X$ and $\epsilonsilon$, not on $V$, this lack of control does not present a problem.
The case $n = 0$ gives the Theorem.
\end{proof}
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\section{Author affiliations}
Jordan S.\ Ellenberg, University of Wisconsin
Brian Lawrence, University of California, Los Angeles; \url{[email protected]}
Akshay Venkatesh, Institute for Advanced Study
\end{document}
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\begin{document}
\title{Geometric phase gates in dissipative quantum dynamics}
\author{Kai Müller}
\author{Kimmo Luoma}
\email{[email protected]}
\author{Walter T. Strunz}
\affiliation{Institut f{\"u}r Theoretische Physik, Technische Universit{\"a}t Dresden,
D-01062,Dresden, Germany}
\date{\today}
\begin{abstract}
Trapped ions are among the most promising candidates
for performing quantum information processing tasks. Recently,
it was demonstrated how the properties of geometric phases
can be used to implement an entangling
two qubit phase gate with significantly reduced operation time
while having a built-in resistance against certain types of errors
{(Palmero et. al., Phys. Rev. A {\bf 95}, 022328 (2017))}.
{In this article, we investigate the influence of both quantum and thermal
fluctuations on the geometric phase in the Markov regime.}
We show that additional
environmentally induced phases as well as a loss of coherence result from
the non-unitary evolution, {even at zero temperature}. We
connect these effects to the associated dynamical and geometrical phases. This suggests
a strategy to compensate the detrimental environmental influences and restore
some of the properties of the ideal implementation.
Our main result is a strategy for zero temperature to construct forces for the geometric phase gate
which compensate the
dissipative effects and leave the produced phase as well as the
final motional state identical to the isolated case. We show that the same strategy
helps also at finite temperatures.
Furthermore, we
examine the effects of dissipation on the fidelity and the robustness
of a two qubit phase gate against certain error types.
\end{abstract}
\maketitle
\section{Introduction}
During the last decades there has been an increasing effort to develop
reliable, large scale quantum information processors. Since such a device
could utilize quantum properties like superpositions and entanglement, its
computing power could potentially surpass every conceivable
classical device for certain problems \cite{Feynman1982,Nielson&Chuang} with potential applications
in various fields of science and technology.
At the moment there are several physical realizations
developed in parallel, each with their own benefits and drawbacks.
One of the most advanced platforms for quantum information processing is based
on {trapped ions} \cite{Linke3305}, where many elementary operations have already been
experimentally demonstrated with high precision \cite{PhysRevLett.113.220501,PhysRevLett.117.060504,PhysRevLett.117.060505}.
Up to date there are, however, still various
difficulties to overcome \cite{Wineland1998}. One of the most severe issues
is dissipation and decoherence resulting from the interaction of the quantum system
with the environment leading to detrimental effects on the quantum
resources and to quantum gate errors.
Although there
exist quantum error correction schemes that can compensate small
errors of the quantum gates these only allow error rates of
roughly 1\% and come at the cost of a high computational overhead
\cite{errorrates}. This means that in order to construct an efficient
quantum information processor it is necessary to reduce the error rates of the
individual quantum gates as much as possible. It is therefore
important to have a good understanding of the environmental
effects and how to compensate for them.
In this work we want to
specifically focus on two-qubit phase gates,
which perform the following operation
\begin{align}
\label{eq:phaseGateOperation}
&\ket{00} \to \ket{00},\hspace{20pt} \ket{11} \to \ket{11},\mathbb Notag\\
&\ket{01} \to e^{i\mathsf{P}hi}\ket{01}, \hspace{6pt}\ket{10} \to e^{i\mathsf{P}hi} \ket{10}.
\end{align}
Two-qubit phase gates are important since
they can be used to convert the separable state $ 1/2(\ket{11} +
\ket{10} + \ket{01} + \ket{00} ) $ into a maximally entangled state
$ 1/2(\ket{11} + i\ket{10} + i\ket{01} + \ket{00} ) $.
First experimental implementations were
realized over a decade ago
\cite{Sackett2000,experimentalDemonstration}, based on theoretical proposals in \cite{PhysRevLett.82.1971,PhysRevA.59.R2539,MSJ}.
However due to recent efforts in theory \cite{PhysRevA.95.022328,Steane_2014,PhysRevA.71.062309,PhysRevLett.91.157901}
and experimental techniques \cite{Schafer2018} the operation times and error rates have significantly reduced. These realizations
leverage the idea of geometric phases first introduced by Berry \cite{Berry45,PhysRevLett.58.1593}
where the cyclic evolution of a quantum state results in the acquisition of a phase.
{The aim of this article is to investigate the effects of quantum and thermal
fluctuations on the geometric phase gate given by Eq.~({\rm Re}f{eq:phaseGateOperation}).
We show how to extend the ideal (fluctuation free) implementation of the gate
given in~\cite{PhysRevA.95.022328} in order to compensate the
detrimental environmental effects.}
{The outline of the remainder of the article is the following.}
In Sec.~{\rm Re}f{sec:model} we first review the ideal isolated case, introduce
our notation and then present our open system model in the context
of a single trapped ion. In Sec.~{\rm Re}f{sec:geometricPhases} we then show
how dissipation leads to additional phases
and in which way they can be connected to the conventional geometrical and
dynamical phases. Furthermore we show which conditions the
experimental protocol must satisfy in order to implement a phase gate and how the
sensitivity of the gate against small experimental errors is
altered compared to the case where the system is perfectly
isolated from its environment. In {\rm Re}f{sec:twoQubit} we then apply our
results to the two-qubit phase gate protocol proposed in~\cite{PhysRevA.95.022328}
and examine the impact on the fidelity.
{In Sec.~{\rm Re}f{sec:finiteTemperatureEffects} we use the results of the
previous sections to draw conclusions for the finite temperature case.}
We close the {main part of the article} with a summary and an outlook.
{Lastly, some of the analytic computations are presented in the Appendix.}
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\section{Model}
\label{sec:model}
In this section we will first introduce a model {for} an isolated
phase gate. Then we
expand the model to account for
{detrimental environmental effects.}
\subsection{Isolated system}
We consider the ion trap as a quantum harmonic oscillator with mass $m$
and frequency $\omega$ which is driven by an
external force leading to the Hamiltonian
\begin{equation}
\label{eq:modelIsolated}
H_{{\mathrm{isol}}}(t) = \hbar\omega a^{\dagger}a + V(t).
\end{equation}
Here $a$ $(a^\dagger)$ is the annihilation (creation) operator for the vibrational
mode satisfying bosonic commutation relations $[a,a^\dagger]=1$. The potential $ V(t) $ arises
from the externally applied force $ F(t) $. Since we want to implement the operation
described in Eq.~\eqref{eq:phaseGateOperation} we need to introduce
state-dependent forces $ F_{1} $ and $ F_{0} $ that depend on an
internal (e.g. spin-) state of the ion in order to distinguish these states. An ion in the internal state
$ \ket{1} $ will only experience $ F_{1} $ and vice versa. In
the following we will use the notation $ F_{j} $ with $j = 0,1$ labeling
the internal state of the ion. Furthermore, the external forces $ F_{j}(t) $ are assumed to be
homogeneous over the extent of the motional state. This can be assumed, for example,
for forces realized by lasers if
the wavelength of the laser is much greater than the amplitude of
oscillation. Under
these circumstances the Hamiltonian can be written in the following
form
\begin{align}
\label{eq:Hschroedinger}
H(t) &= \hbar\omega(a^{\dagger}a) + \kb{0}{0}\otimes V_{0}(t) + \kb{1}{1}\otimes V_{1}(t), \\
V_{j}(t) &= F_{j}(t) x = f_j(t)(a+a^{\dagger}),
\end{align}
with $ f_j(t) = \dfrac{\hbar}{2m\omega}F_j(t) $.
Before we determine the evolution of a quantum state in this model we will
simplify the equations more by switching to an interaction picture with
respect to $\hbar\omega a^\dagger a$ leading to a simpler Hamiltonian
\begin{align}
\label{eq:coupledPotential}
\widetilde{H}(t) &= \kb{0}{0}\otimes \widetilde{V}_{0}(t) + \kb{1}{1}\otimes \widetilde{V}_{1}(t),\\
\widetilde{V}_{j}(t) &= \widetilde{f}_j^{*}(t)a+\widetilde{f}_j(t)a^{\dagger}, \mathbb Notag
\end{align}
where $\widetilde{f}_{j} = e^{i\omega t} f_{j}$.
The equation of motion in the interaction picture
for a quantum state represented by the density operator $ \rho $
is the von-Neumann equation
\begin{equation}
\label{eq:VNisolated}
\dot{\rho} = -i\frac{1}{\hbar} [\widetilde{H}(t),\rho].
\end{equation}
This equation can be solved by inserting
{an ansatz $\rho=\kb{\mathsf{P}si_t}{\mathsf{P}si_t}$}, where
\begin{align}
\label{eq:ansatzisol}
{\ket{\mathsf{P}si_t}} =& {\sum_{j=0}^1a_je^{i\textrm{Var}phi_j(t)}\ket{j,z_j(t)}},
\end{align}
where $ j $ represents the internal state and
{$\ket{z_j(t)}=e^{-\abs{z_j(t)}^2/2}\sum_{n=0}^\infty ((z_j(t))^n/\sqrt{n!})\ket{n}$ is a coherent state
for the motional degree of freedom of the ion when the internal state is $\ket{j}$, ($j=0,1$)
and $a_j$ is a constant determined from the initial conditions.}
Inserting this ansatz into Eq.~({\rm Re}f{eq:VNisolated}) leads to the following equation
for the coherent state label $z_j(t)$
\begin{equation}
\label{eq:z(t)isol}
\dot{z}_{j} = \dfrac{1}{i\hbar}\widetilde{f}_{j}(t).
\end{equation}
{In the {context of implementing} a phase gate{,} we want $ f_j(t) $ to be part of some protocol
which is switched on at a certain time and is completed some time $ T $ later.
Therefore $ f_j $ shall only be non-zero in the interval $ \left[0,T\right] $ and it shall be such
that the motional state undergoes a cyclic evolution $ z_{j}(0) = z_{j}(T) $
whereas the internal degrees of freedom acquire a phase according to Eqs.~({\rm Re}f{eq:phaseGateOperation}).}
It is known that such a cyclic quantum evolution
leads to the acquisition of a phase $\mathcal{P(H)}i_j = \mathcal{P(H)}i_{\mathrm{g},j}+\mathcal{P(H)}i_{\mathrm{d},j}$, where the dynamical and
geometrical phases satisfy $\dot\mathcal{P(H)}i_{\mathrm{d},j}=-(1/\hbar)\bra{j,z_j(t)}H(t)\ket{j,z_j(t)}$
and $\dot\mathcal{P(H)}i_{\mathrm{g},j}=i\bra{j,z_j(t)}\partial_t\ket{j,z_j(t)}$, respectively.
The total phase acquired is equal to twice the area enclosed by the trajectory $ z_j(t) $
in the interaction picture (see Eq.~({\rm Re}f{eq:phiA}))~\cite{experimentalDemonstration,PhysRevA.95.022328}.
In the following we will expand this model to include
{quantum and thermal fluctuations}.
\subsection{Open system}
\label{ssec:dissipativeCase}
Effects of the ion coupling
to some external environment
are modeled phenomenologically by a
Gorini-Kossakowski-Sudarshan-Lindblad (GKSL)
{master} equation \cite{Lindblad1976,doi:10.1063/1.522979} of the following
form
{\begin{align}
\label{eq:model}
\dot{\rho} &= -i\frac{1}{\hbar} [\widetilde{H}(t),\rho] + \gamma(\bar{n} +1) (2a\rho a^{\dagger} - a^{\dagger}a\rho - \rho a^{\dagger}a)\mathbb Notag\\
& + \gamma\bar{n} (2a^{\dagger}\rho a - aa^{\dagger}\rho - \rho aa^{\dagger}),
\end{align}}{in the interaction picture}.
{The model ({\rm Re}f{eq:model}) describes dissipation
and thermal excitation of the motional state of the ion with rates
$\gamma(\bar{n} +1)$ and $\gamma\bar{n}$, respectively. $\bar{n}$ models the
average occupation number of the bosonic heat bath modes at a relevant system
frequency at finite temperature. At zero temperature, $\bar{n}=0$ and only quantum fluctuations
and damping with rate $\gamma$ are present.
}
{{Motional} coherence times $(\gamma\bar{n})^{-1}$ for the trapped ion systems are
of the order of {1 - 100 milliseconds~\cite{Wineland1998, lucas2007longlived}}. Typical frequency
for the harmonic motion of the ion around the minimum of the trap is in the
MHz range, whereas operation times for the two qubit phase gate investigated
later in the article is in the $\mu$s range~\cite{PhysRevA.95.022328}.}
{
As shown in \cite{PhysRevA.74.022102,Strunz2005},
finite temperature effects can be incorporated in the zero temperature
model by adding a fluctuating force $\sqrt{2\gamma\bar{n}}\hbar\mathbf{C}i(t)$
to the Hamiltonian.
Here $\mathbf{C}i(t)$ is a Gaussian white noise process with
$\langle\mathbf{C}i(t)\mathrm{ran}\,gle = \langle \mathbf{C}i(t)\mathbf{C}i(t')\mathrm{ran}\,gle = 0 $ and
$\langle\mathbf{C}i(t)\mathbf{C}i^*(t')\mathrm{ran}\,gle = \delta(t-t') $.
Thus the Hamiltonian reads
\begin{align}
\label{eq:H_xi}
\widetilde{H_{\mathbf{C}i}}(t) &= \kb{0}{0}\otimes \widetilde{V}_{0,\mathbf{C}i}(t) + \kb{1}{1}\otimes \widetilde{V}_{1,\mathbf{C}i}(t),\\
\widetilde{V}_{j}(t) &= (\widetilde{f}_j^{*}(t) + \sqrt{2\gamma\bar{n}}\hbar\mathbf{C}i^*(t))a\\
&+(\widetilde{f}_j(t)+ \sqrt{2\gamma\bar{n}}\hbar\mathbf{C}i(t))a^{\dagger}. \mathbb Notag
\end{align}
Note that the noise does not depend on the internal state $j$.
The finite temperature model~({\rm Re}f{eq:model}) is thus equivalent to
an ensemble of zero temperature models with stochastic Hamiltonian
\begin{align}
\label{eq:rho_xi}
\dot{\rho_{\mathbf{C}i}} = -i&\frac{1}{\hbar} [\widetilde{H_{\mathbf{C}i}}(t),\rho_{\mathbf{C}i}] +
\gamma (2a\rho_{\mathbf{C}i} a^{\dagger} - a^{\dagger}a\rho_{\mathbf{C}i} - \rho_{\mathbf{C}i} a^{\dagger}a).
\end{align}
The evolution of the density operator can be recovered by taking an average over
$\mathbf{C}i(t)$
\begin{equation}
\label{eq:rhoAverage}
\rho(t) = \langle\rho_{\mathbf{C}i}(t)\mathrm{ran}\,gle,
\end{equation}
and the average state $\rho(t)$ satisfies Eq.~({\rm Re}f{eq:model}).}
{Remarkably, Eq.~({\rm Re}f{eq:rho_xi}) can still be solved
by a coherent state ansatz, such as Eq.~({\rm Re}f{eq:ansatzisol}),
although it contains the effects of thermal and quantum fluctuations.
Inserting the ansatz from Eq.~\eqref{eq:ansatzisol}
for a particle in the internal state $\ket{j}$
into Eq.~({\rm Re}f{eq:model}) leads to the following equation
for the coherent state label $z_j(t)$
\begin{equation}
\label{eq:z(t)}
\dot{z}_{j} + \gamma z_{j} = \dfrac{1}{i\hbar}\left(\widetilde{f}_{j}(t) + \sqrt{2\gamma\bar{n}}\hbar\mathbf{C}i(t)\right).
\end{equation}}
{For current ion-traps the motional coherence time $1/(\gamma\bar{n})$ is much longer
than the operation time~\cite{Wineland1998, lucas2007longlived}} so that the motion
of the ion is dominated by the deterministic force.\\
In the following sections we will first investigate the zero temperature case
for an arbitrary force. In section {\rm Re}f{sec:finiteTemperatureEffects} we can apply
these results to a noisy force and average over the noise.
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\section{Consequences of quantum fluctuations}
\label{sec:geometricPhases}
In this section we investigate the consequences of coupling the
trapped ion to a zero temperature bath. The effect of thermal
fluctuations will be considered in Sec.~{\rm Re}f{sec:finiteTemperatureEffects}.
\subsection{Consequences for the phase}
\label{ssec:consequncesPhase}
In the following we want to investigate how the Lindblad terms in the
time evolution equation affect the phase. We therefore consider a
model which is in principle identical to ({\rm Re}f{eq:model})
{at zero temperature} but slightly
more general. The result can then be applied to the phase gate model.
The Hamiltonian shall be of the form
\begin{equation*}
H(t) = \kb{0}{0}\otimes H_0(t) + \kb{1}{1}\otimes H_1(t),
\end{equation*}
where $ \ket{0} $, $ \ket{1} $ represent the internal states of
the ion and $ H_0(t) $ and $ H_1(t) $ act on the motional degree of freedom.
We assume that the dissipation and decoherence is well described by a
general GKSL master equation and thus arrive at the following model
\begin{align}
\label{eq:time evolution}
\dot{\rho} &= \frac{-i}{\hbar}[H(t),\rho] + \mathcal{L}\left[\rho\right],\\
\mathcal{L}[\rho] &= \sum_{l=1}^{N} L_l\rho L_l^{\dagger} - \dfrac{1}{2}\left(L_l^{\dagger}L_l\rho + \rho L_l^{\dagger}L_l\right).\mathbb Notag
\end{align}
Furthermore we assume that this model has a pure state solution
{$\kb{\mathsf{P}hi_t}{\mathsf{P}hi_t}$,
where $\ket{\mathsf{P}hi_t}=\sum_j\ket{j,\mathsf{P}si_j(t)}$ and the}
internal state $ j $ remains unchanged during the evolution. Although these are very limiting
assumptions we will see that the results can nevertheless be applied to
the phase gate scenario mentioned before.\\
Under these assumptions, we can repeat the argument proposed in~\cite{PhysRevLett.58.1593}
for a cyclic quantum evolution governed by a
time-dependent Schrödinger equation to determine the relative phase that arises if the initial state is in a superposition of internal states. In our case, however, the cyclic quantum
evolution of a pure state is modified by a damping term. The details
of the computation can be found in the appendix~{\rm Re}f{sec:Anhangphidissipative}.
We then get a new complex valued term $\xi$ in addition to the dynamical and geometrical phase
\begin{flalign}
\label{eq:phi}
\mathcal{P(H)}i &= \mathcal{P(H)}i_{\mathrm{g}} + \mathcal{P(H)}i_{\mathrm{d}} + \xi,\\
\xi &= \mathcal{P(H)}i_{\mathrm{L}} + i\eta,
\end{flalign}
where the individual terms are defined as follows:
\begin{align}
\dot{\mathcal{P(H)}i}_{\mathrm{g}} &= i\left(\bra{\mathsf{P}si_0}\partial_t\ket{\mathsf{P}si_0} - \bra{\mathsf{P}si_1}\partial_t\ket{\mathsf{P}si_1}\right), \mathbb Notag\\
\dot{\mathcal{P(H)}i}_{\mathrm{d}} &= -\frac{1}{\hbar}(\langle H_0(t)\mathrm{ran}\,gle - \langle H_1(t)\mathrm{ran}\,gle),\\
\dot{\xi} &= -i \sum_{l=1}^{N}\bra{\mathsf{P}si_0}L_l\ket{\mathsf{P}si_0}\bra{\mathsf{P}si_1}L_l^{\dagger}\ket{\mathsf{P}si_1} \mathbb Notag\\
&- \dfrac{1}{2}\left(\bra{\mathsf{P}si_0}L_l^{\dagger}L_l\ket{\mathsf{P}si_0} + \bra{\mathsf{P}si_1}L_l^{\dagger}L_l\ket{\mathsf{P}si_1}\right). \mathbb Notag
\end{align}
The first two terms are identical to the unitary case mentioned before and
also found in~\cite{PhysRevLett.58.1593}.
Therefore, they correspond to dynamical and
geometrical phases which arise during a cyclic evolution of a quantum
system. Since we have constructed relative phases they are expressed
as the difference between the dynamical/geometrical phases of
particles in the internal states $0$ and $1$.
The last sum cannot be expressed in such a way and
contains dissipative effects. In general it leads to real terms in the
exponent which result in a loss of coherence. We can apply this equation to the
damped harmonic oscillator if we set $ L = \sqrt{\gamma}a $ and $
\ket{\mathsf{P}si_j} = \ket{j,z_j} $. Furthermore we can identify $H_j(t)$, which
corresponds to the Hamiltonian seen by a particle in the internal state
$\ket{j}$ as $\hbar\omega a^{\dagger}a + V_j(t)$ (see
Eq.~({\rm Re}f{eq:Hschroedinger})). This means we can calculate the dynamical
phase for a particle in the internal state $j$ with $\ket{j,z_j}$ in
the interaction picture by using Eqs.~({\rm Re}f{eq:z(t)}) and~\eqref{eq:coupledPotential} as
\begin{align}
\mathcal{P(H)}i_{\mathrm{d},j} &= -\frac{1}{\hbar}\langle H_j(t)\mathrm{ran}\,gle \mathbb Notag\\
&= \frac{-1}{\hbar}\int\limits_{0}^{T}\bra{j,z_j(t)} e^{-i\omega t a^{\dagger}a} H(t) e^{i\omega t a^{\dagger}a}\ket{j,z_j(t)} {\rm d} t\mathbb Notag\\
&= \frac{-1}{\hbar}\int\limits_{0}^{T}\bra{j,z_j(t)}\hbar\omega a^{\dagger}a + \widetilde{V}_j(t)\ket{j,z_j(t)} {\rm d} t\mathbb Notag\\
&= \int\limits_{0}^{T}2\operatorname{Im}\left(\dot{z}_j(t) z_j^{*}(t)\right) - \omega\abs{z_j(t)}^2 {\rm d}{t}.
\end{align}
For the geometric phase we arrive at
\begin{flalign}
\mathcal{P(H)}i_{\mathrm{g},j} &= i\left(\bra{z_j(t)}e^{-i\omega t a^{\dagger}a}\partial_te^{i\omega t a^{\dagger}a}\ket{z_j(t)}\right) \mathbb Notag\\
&= \int\limits_{0}^{T}-\operatorname{Im}(\dot{z}_j(t)z_j^{*}(t)) + \omega\abs{z_j(t)}^2 {\rm d}{t},\\
\end{flalign}
again with $\ket{j, z_j}$ in the interaction picture. As we can see
the dynamical and geometrical phase are remarkably similar for the
harmonic oscillator. Furthermore we can combine these two phases for
the total phase in the isolated ($\gamma = 0$) case $\mathcal{P(H)}i_{{\mathrm{isol}}}$:
\begin{flalign}
\mathcal{P(H)}i_{{\mathrm{isol}}} &= (\mathcal{P(H)}i_{d,0} - \mathcal{P(H)}i_{d,1}) + (\mathcal{P(H)}i_{g,0} -\mathcal{P(H)}i_{g,1}) \mathbb Notag\\
\label{eq:phaseAsIm}
&= \int\limits_{0}^{T}\operatorname{Im}\left((\dot{z}_0(t)z_0^{*}(t)\right) - \dot{z}_1(t)z_1^{*}(t)) {\rm d}{t}.
\end{flalign}
For a cyclic evolution this reduces to the known result
\cite{PhysRevA.95.022328,experimentalDemonstration}
\begin{equation}
\label{eq:phiA}
\mathcal{P(H)}i_{{\mathrm{isol}}} = 2 (A_0 - A_1),
\end{equation}
where $A_j$ is the area enclosed by the cyclic evolution of
$z_j$. This is shown in Fig.~{\rm Re}f{fig:phasespacearea}. From now on,
we do not always write the time dependence of the coherent state labels
explicitly in order to shorten the notation.
\begin{figure}
\caption{(color online) In the isolated case the relative phase between two cyclic
evolutions is proportional to the difference of the swept phase space
area in the interaction picture.}
\label{fig:phasespacearea}
\end{figure}
The influence of the dissipation is {contained}
in the term $ \xi $ with
{
\begin{align*}
\label{eq:phiL}
i\dot{\xi} &= \gamma\left(2z_0z_1^{*} - \left(\abs{z_0}^2 + \abs{z_1}^2\right)\right)\\
&= -\gamma \abs{z_{1}-z_{0}}^2 + i \gamma\abs{z_{1}}\abs{z_{0}}\sin(\theta_{1} - \theta_{0}),
\end{align*}
}where the phases $\theta_j$ are defined by $z_j=\abs{z_j}e^{i\theta_j}$.
{Note that $ \xi $ consists of a real as well as of an imaginary part.\\
In summary, an initial state which is in a superposition of spin states
{\begin{equation*}
\rho(0) =
\begin{pmatrix}
\abs{a}^{2} && ab^{*} \\
a^{*}b && \abs{b}^{2}
\end{pmatrix} \otimes \kb{z(0)}{z(0)},
\end{equation*}}
will be transformed into the following state after the cyclic evolution:
{\begin{equation*}
\rho(T) =
\begin{pmatrix}
\abs{a}^{2} && ab^{*} e^{i\mathcal{P(H)}i_{{\mathrm{isol}}} + i\xi} \\
a^{*}b e^{-i\mathcal{P(H)}i_{{\mathrm{isol}}} - i\xi^{*}} && \abs{b}^{2}
\end{pmatrix} \otimes \kb{z(0)}{z(0)}.
\end{equation*}}
This shows that for $ \gamma \mathbb Neq 0 $ the damping results in an
additional real term $ \eta = \gamma\int_{0}^{T} \abs{z_{1}-z_{0}}^2 {\rm d} t $ in the exponent which does only depend on the
damping strength $ \gamma $ and the difference of the amplitudes of the paths. This real
term leads to a dephasing of the spin state by diminishing the off-diagonal elements of the density matrix.
We will therefore refer to it as dephasing term from now on. }
We can also see a new phase term
$\mathcal{P(H)}i_{\mathrm{L}}= \gamma\int\limits_{0}^{T}\abs{z_{1}}\abs{z_{0}}\sin(\theta_{1} - \theta_{0}) {\rm d} t$
which depends on the absolute values of $ z_0 $
and $ z_1 $. The integral over this term can vanish for sufficiently
symmetrical $ z_j(t) $ (e.g. if $ z_j(t) = z_j(T-t) $) with $
j\in\{0,1\} $ or if $ z_{1} $ or $ z_{0} $ is in the ground state
during the entire operation. In Sec.~{\rm Re}f{sec:twoQubit}
we will see that the 2-qubit phase gates proposed in \cite{PhysRevA.95.022328,PhysRevA.71.062309}
{and realized in \cite{experimentalDemonstration}} do indeed
have the latter property which means that even with damping the
phases produced by those phase gates are still only determined by the
respective areas. The dephasing term can, however, only vanish if $
f_{1} = f_{0} $ which implies that there is no relative phase as
well. We can also conclude that the dephasing is stronger for higher
energies of the particle which means it is especially relevant for
short operation times as we will see in Sec.~{\rm Re}f{ssec:fidelity}.
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\subsection{Consequences for the path}
\label{ssec:force}
We have seen in the previous section how the damping results in
additional phase terms. However, from Eq.~({\rm Re}f{eq:z(t)}) it is clear that
the damping alters the path as well. Therefore, the paths which are
closed in the isolated case are no longer closed in the damped case.
It is a natural question to ask which forces $ \widetilde{f}_j(t) =
f_j(t)e^{i\omega t} $ can be used to achieve the cyclical
evolution ($z_j(0) = z_j(T) $) in the damped case and whether some of
those forces should be used preferably because they minimize the
dephasing term. First we note that it is not possible to completely
compensate the effects of the damping by applying some sophisticated
force $f_{\mathrm{c}}$. This can be seen from Eq.~({\rm Re}f{eq:z(t)}), since the isolated
dynamics of a coherent state is described by $\dot{z}_j = 0$, such a
force would need to satisfy $\widetilde{f} = e^{i\omega t}f_{\mathrm{c}}(t) = \mathrm{const} $,
which is impossible for real $f_{\mathrm{c}}(t)$.
To determine
the effects of a force $ f_j(t) $ on the path $ z_j(t) $ we have to
solve Eq.~({\rm Re}f{eq:z(t)}).
{This leads to the solution}
\begin{flalign}
z_j(t) &= z_{j,\mathrm{hom}} + z_{j,\mathrm{inhom}} \mathbb Notag\\
\label{eq:alpha(t)}
&= z_j(0) e^{-\gamma t} + \int_{0}^{t} \dfrac{-i}{\hbar} \widetilde{f}_j e^{-\gamma (t-\tau)} {\rm d}{\tau}.
\end{flalign}
We can therefore conclude that in order to achieve the cyclic dynamics
$ z_j(0) = z_j(T) $ we need the forces to satisfy
\begin{equation}
\label{eq:conditionf}
z_j(0) \left(e^{\gamma T} - 1\right) = \int_{0}^{T}f_j(\tau)e^{i\omega\tau}e^{\gamma\tau} {\rm d}{\tau}.
\end{equation}
The equation shows explicitly that for $ \gamma \mathbb Neq 0 $ the condition
depends on the initial state $ z_j(0) $. This means that in contrast to
the undamped case where $ f_j $ would always lead to closed trajectories it now
only works for a specific initial condition $z_j(0)$.
The fault tolerance of a quantum phase gate
towards the initial motional state is therefore lost in the damped
case. \\
An interesting observation at this point is that if we
consider $z_j(0) = 0$, we can derive forces $ f_{\mathrm{d}} $ which
return $ z_j $ to the ground state after time $ T $ in the damped
case from the forces $ f_{\mathrm{nd}} $ which accomplish this in
the undamped case by using the formula
\begin{equation}
\label{eq:relationFDamping}
f_{\mathrm{d}} = f_{\mathrm{nd}}\cdot e^{-\gamma t}.
\end{equation}
We will use this link between the damped and the undamped scenarios in the
next section to generalize an already existing protocol for 2-qubit
phase gate to account for dissipative effects.
For an experimental realization
it is desirable to minimize the dephasing term for a given relative
phase. To examine how this can be done we want to consider the case
$f_{0}(t) = 0 $ and $ \ket{z_j(0)} = \ket{0} $ for the sake of
simplicity. This means that $ \ket{z_0} $ is the ground
state at all times and we only need to discuss the dynamics
of $ \ket{z_1} $. These simplifications are well justified because in
an experimental setup the ground state can be prepared initially and
an additional force $ f_{0} $ does not bring any benefits but just
makes the computations more complex. We can then derive simple
expressions for the phase and dephasing terms after time $ T $
from Eq. \eqref{eq:phiA} if we write $ z_1 = r(t) e^{i\theta(t)} $
{(in the following equations we have omitted the time dependence of
$ r $ and $ \theta $)}
\begin{flalign*}
\int_{0}^{T} z_1^{*}\dot{z_1} {\rm d}{t} &= \int_{0}^{T} re^{-i\theta}\left(\dot{r} e^{i\theta} + ir\dot{\theta} e^{i\theta}\right){\rm d} t \\
&= \int_{0}^{T} r\dot{r} {\rm d}{t} + i\int_{0}^{T} \dot{\theta}r^2 {\rm d}{t}.
\end{flalign*}
Since $ z_1(0) = z_1(T) $ we can see that the first integral vanishes
by integrating by parts and we are left with the following expression
for the phase:
\begin{equation}
\label{eq:comparisonPhase}
\mathcal{P(H)}i_{\mathrm{isol}}= \int_{0}^{T} \dot{\theta}r^2 {\rm d}{t},\quad \mathcal{P(H)}i_{\mathrm{L}}=0.
\end{equation}
In this case, $\mathcal{P(H)}i_{\mathrm{L}}=0$ because $z_0=0$ at all times.
For the dephasing term we find
\begin{equation}
\label{eq:comparisonDephasing}
\eta=\gamma \int_{0}^{T} r^2 {\rm d}{t}.
\end{equation}
We can see that the easiest way to minimize
the dephasing for a given relative phase is to make $ \dot{\theta} $
as large as possible. This result also makes intuitive sense since if
the path goes around the origin multiple times (large $ \dot{\theta} $)
it needs a smaller amplitude (which results in less dephasing) in
order to sweep over the same area. Since $\theta$ corresponds to the
interaction picture it is affected by the frequency $\omega$ of
the harmonic oscillator.
Fig.~{\rm Re}f{fig:paths1} displays the paths
$z_1(t)$ which are generated by forces of the form $f_1 =
e^{-\gamma t} \sin(\Omega t)$ and $f_{0} =0$
(see Eq.~({\rm Re}f{eq:z(t)})).
\begin{figure}
\caption{(color online){Path of $ z_1 $ in the complex plane
{when using the force $ f(t) = e^{-\gamma t}
\label{fig:paths1}
\end{figure}
The paths (a) and (b) correspond to the isolated case with $\gamma =
0$ whereas $\gamma > 0$ for the paths (c) and (d).
{The exact parameters used in this numerical simulation were
$ \Omega = 2\,\mathrm{MHz}$ for the frequency of the driving force in all trajectories,
$ \omega = 2\Omega $ for the frequency of the harmonic trap in
trajectories (a) and (c) and $ \omega = 4\Omega $ for (b) and (d).
The damping constant for paths (c) and (d) was set to $\gamma/\Omega = 0.2$.}
We can see how
the upper two paths are symmetrical with respect to the imaginary
axis. As shown in~\cite{PhysRevA.95.022328} this implies that the
phase does not change (in first order) if the force is subjected to a homogeneous,
small constant offset $ f \mapsto f + \delta f $ in the $ \gamma = 0 $
case. In contrast, the paths for $ \gamma \mathbb Neq 0 $ are no longer
symmetrical.
However whether the path is symmetric or not depends on the
force. In the next section we therefore want to investigate how
the robustness can be maintained in the damped case by
constructing forces differently compared to Eq.~\eqref{eq:relationFDamping}.
\subsection{Consequences for the robustness}
\label{sec:consequenceRobustness}
At first we want to show how the condition for the robustness against
small constant offsets of the force $ f \mapsto f+\delta f $ reads in the
dissipative case studied here. According to Eq.~({\rm Re}f{eq:phaseAsIm}) with $
z_0 = 0 $,
\begin{flalign*}
\mathcal{P(H)}i_{\mathrm{isol}} &= \int_{0}^{T}\operatorname{Im}(\dot{z}_1 z_1^{*}) {\rm d} t \mathbb Notag\\
&= \dfrac{-1}{\hbar}\int_{0}^{T}\operatorname{Re}(z_{1}^{S} f_1) {\rm d} t,
\end{flalign*}
where we used Eq.~\eqref{eq:z(t)} and $ z_{1}^{S} = e^{-i\omega t} z_1 $ is
the path in the Schrödinger picture. Therefore the offset to the phase
in first order becomes
\begin{equation*}
\delta \mathcal{P(H)}i_{\mathrm{isol}} = \dfrac{-\delta f}{\hbar} \int_{0}^{T}\operatorname{Re}(z_{1}^{S}) {\rm d} t \overset{!}{=} 0.
\end{equation*}
This result is identical to the one in the isolated case found in
\cite{PhysRevA.95.022328}. By inserting Eq.~\eqref{eq:alpha(t)}, assuming $
z_{1}(0) = 0 $ and integrating by parts we can express this as a
condition for the force
\begin{equation}
\label{eq:forceResisanceCondition}
0 = \int_{0}^{T}f(t){\rm d} t.
\end{equation}
Together with the condition for a cyclic evolution, Eq.~\eqref{eq:conditionf},
we therefore have a set of conditions that for $ z_j(0) = 0 $ may be
seen as orthogonality conditions for $ f(t) $
\begin{equation}
\label{eq:orthogonal}
f(t) \perp \{e^{\gamma t} \sin(\omega t), e^{\gamma t} C_0(\Omega)s(\omega t), 1\} =: \mathcal{C}.
\end{equation}
This means that we can construct forces to suit our needs by a
Gram-Schmidt procedure. It is useful to orthogonalize the set $
\mathcal{C}$ and then do one more orthogonalizing step for the
arbitrary function $g$ which will then become orthogonal
to the set $ \mathcal{C} $. This method makes it possible to construct
a plethora of forces which will leave the phase unchanged under a
small constant offset $ \delta f $ and produce a cyclic evolution. By
superposing many of such forces one can then ensure to meet
further demands like e.g. $ f(0) = f(T) = 0 $.
We can conclude that
it is possible to maintain the robustness of the phase gate against
small constant offsets of the force $ f \mapsto f +\delta f $ in the
damped case. However as we have seen in the previous section (Eq.~\eqref{eq:conditionf}) the gate
loses its resistance against fluctuations in the initial motional
state.
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\section{Application to 2-qubit phase gates}
\label{sec:twoQubit}
In this section we want to show how the relations we found in the
previous sections apply to two-qubit phase gates which have been realized in
\cite{experimentalDemonstration,Schafer2018}.
These two-qubit gates consist of two
ions in a harmonic trap potential which experience a force that
depends on the internal state ($ \ket{\uparrow} $ or $
\ket{\downarrow} $) of the ion. As shown in \cite{PhysRevA.95.022328},
the Hamiltonian of such a system can be written as
\begin{align}
H_{\mathrm{tot}} &= H_{+} + H_{-},\mathbb Notag\\
\label{eq:H2Qubit}
H_{\pm} &= \dfrac{p_{\pm}^2}{2} + \dfrac{1}{2}\Omega_{\pm}^2 x_{\pm}^2 + f_{\pm} x_{\pm}.
\end{align}
Here, $ H_{+} $ describes an oscillation of a stretch mode where the
displacement from the equilibrium position of the two ions are equal but
in opposite directions and $ H_{-} $ describes an oscillation of the
center-of-mass mode where the displacement of the ions is
identical. {$ x_{\pm} $ are mass weighted normal mode
coordinates which for equal mass ions take the form
\begin{equation*}
x_{\pm} = \sqrt{m}((x_{1}-x_{1}^{(0)}) \mp (x_{2}-x_{2}^{(0)})).
\end{equation*}
Here, $ x_{i} $ is the position operator for ion $ i $, $ x_{i}^{(0)} $ is the equilibrium
position of ion $i$ and the canonically conjugate momentum operators
are $p_{\pm}=-i\hbar\partial/\partial x_{\pm} $.}
Note that
here we ignored a term in $ H_{\mathrm{tot}} $ which is proportional to the
difference of the forces experienced by the two ions. This (purely time dependent)
term will therefore lead to additional phases for certain configurations. Similar to Eq.~\eqref{eq:orthogonal} we
will however later (see Eq.({\rm Re}f{eq:orthogonality2qubit})) present a way to
construct forces which satisfy $ \int_{0}^{T} f {\rm d} t = 0 $ so that this phase will vanish.
A more detailed derivation of the Hamiltonian and discussion of the purely time dependent
term can be found in \cite{PhysRevA.95.022328}.
{If the forces on the two ions take the form $ F_j = F(t)
\sigma_{j}^{z} $ one can derive the following values for the force in the interaction picture $ \widetilde{f}_{\pm} $
\begin{flalign}
\widetilde{f}_{+}(P) &= \widetilde{f}_{-}(A) = 0, \mathbb Notag\\
\widetilde{f}_{-}(\uparrow\uparrow) &= -2F/\sqrt{2m}e^{i\Omega_- t},\mathbb Notag\\
\widetilde{f}_{+}(\uparrow\downarrow) &= -2F/\sqrt{2m}e^{i\Omega_+ t},\mathbb Notag\\
\widetilde{f}_{-}(\downarrow\downarrow) &= 2F/\sqrt{2m} e^{i\Omega_- t}, \mathbb Notag\\
\label{eq:2Qubitf}
\widetilde{f}_{+}(\downarrow\uparrow) &= 2F/\sqrt{2m} e^{i\Omega_+ t},
\end{flalign}
where $ P \in \{\uparrow\uparrow, \downarrow\downarrow\} $ denotes
parallel and $ A\in \{\uparrow\downarrow, \downarrow\uparrow\} $
anti-parallel spin combinations. This type of force can be realized
by off resonant lasers in the Lamb-Dicke regime~\cite{schleich2011quantum}.
For the optimization of the functional form of the
force see \cite{PhysRevA.95.022328}. Frequencies $\Omega_\pm$ are
defined as $\Omega_+ = \sqrt{3}\omega$ and $\Omega_-=\omega$~\cite{PhysRevA.95.022328}.} We can bring the Hamiltonian in the
same form as in the previous section by introducing
creation and annihilation operators for the stretch and center-of-mass
mode $ a_{\pm}, a_{\pm}^{\dagger} $ and switching to an interaction
picture
\begin{flalign}
\ket{\mathsf{P}si_{I}} &= e^{-iH_{0}t/\hbar}\ket{\mathsf{P}si}, \mathbb Notag\\
\label{eq:2QubitInteractionPicture}
H_{0} &= \hbar\Omega_{+}(a_{+}^{\dagger}a_{+} + \dfrac{1}{2}) + \hbar\Omega_{-}(a_{-}^{\dagger}a_{-} + \dfrac{1}{2}).
\end{flalign}
The Hamiltonian then reduces to
\begin{flalign}
\label{eq:2QubitInteractionHamiltonian}
\widetilde{V} &= \widetilde{f}_{+}^{*}a_{+} + \widetilde{f}_{+}a_{+}^{\dagger} + \widetilde{f}_{-}^{*}a_{-} + \widetilde{f}_{-}a_{-}^{\dagger} \mathbb Notag\\
&= \widetilde{V}_{+} + \widetilde{V}_{-}.
\end{flalign}
To model the damping we can introduce two Lindblad terms similar to
Eq. \eqref{eq:model}.
We assume identical damping rates $\gamma$ since all degrees of freedom couple
to the same bath which we assume to have a flat spectral density on the
relevant frequency scale. We then find in the interaction picture
\begin{align}
\label{eq:2QubitTimeEvolution}
\dot{\rho} &= \mathcal{L}_{+}[\rho] + \mathcal{L}_{-}[\rho] ,\\
\mathcal{L}_{\pm} &:= \frac{-i}{\hbar}[\widetilde{V}_{\pm},{\rho}]
+ \gamma\left(2a_{\pm}\rho a_{\pm}^{\dagger} - a_{\pm}^{\dagger}a_{\pm}\rho - \rho a_{\pm}^{\dagger}a_{\pm}\right). \mathbb Notag
\end{align}
We can see that the $ \mathcal{L}_{\pm} $ only act on one of the two
modes and are of the same form as the right hand side of Eq.~({\rm Re}f{eq:model}).
Therefore the two modes $+,-$ are not coupled by Eq.~\eqref{eq:2QubitTimeEvolution} and we can reuse the results from the previous Sec.~{\rm Re}f{sec:geometricPhases} to determine the evolution of an initial state that is in a superposition of spin states and in a coherent motional state
\begin{equation*}
\rho(0) =
\begin{pmatrix}
\abs{a}^{2} && ab^{*} \\
a^{*}b && \abs{b}^{2}
\end{pmatrix} \otimes \kb{z_{+}(0)}{z_+(0)}\otimes\kb{z_{-}(0)}{z_{-}(0)}.
\end{equation*}
The coherent motional state evolves according to
\begin{equation}
\label{eq:2Qubitalphat}
\dot{z}_{\pm}^{j}+\gamma z_{\pm}^{j} = \dfrac{1}{i\hbar}\widetilde{f}_{\pm}(j),
\end{equation}
where the internal degrees of freedom are given by $j$ and oscillation
in the stretch and center-of-mass mode is described by
$z_{+}^{j}$ and $z_{-}^{j}$ respectively. The forces $\widetilde{f}_{\pm}(j)$ are given in Eq.~\eqref{eq:2Qubitf}.
Furthermore, we can observe that Eq.~\eqref{eq:2QubitTimeEvolution} is of the
form of Eq.~\eqref{eq:time evolution} which means that we can apply the
results from Sec.~{\rm Re}f{ssec:consequncesPhase} to calculate the
phase and dephasing which arise during a cyclic evolution:
\begin{equation}
\label{eq:2QubitA}
\mathcal{P(H)}i(T) = \mathcal{P(H)}i_{{\mathrm{isol}}} + \xi,
\end{equation}
where
\begin{align}
\mathcal{P(H)}i_{{\mathrm{isol}}} &= 2(A_{+}^{1} + A_{-}^{1} - A_{+}^{0} - A_{-}^{0}),\mathbb Notag\\
\xi &= \gamma \int_{0}^{T} d_{+}(\tau) + d_{-}(\tau) {\rm d}{\tau}, \mathbb Notag\\
d_{\pm} &= i\abs{z_{\pm}^{1}-z_{\pm}^{0}}^2 + \abs{z_{\pm}^0}\abs{z_{\pm}^1}\sin(\theta_{\pm}^1 - \theta_{\pm}^0).\mathbb Notag
\end{align}
We see that $\mathcal{P(H)}i_{{\mathrm{isol}}} = \mathcal{P(H)}i_{\mathrm{g}} + \mathcal{P(H)}i_{\mathrm{d}}$ is proportional to the areas
swept in the stretch and center-of-mass modes of oscillation $A_{+}$ and
$A_{-}$, respectively, and it is identical to the phase of the isolated evolution
($\gamma =0$). Because of the symmetries of the forces described in
Eq.~\eqref{eq:2Qubitf}, $\mathcal{P(H)}i_{{\mathrm{isol}}}$ is only nonzero if $ 0 $ is a parallel and $
1 $ an anti-parallel spin combination or vice versa. The $\xi$ term
originates from the Lindblad operators and since $\operatorname{Im}(\xi) \mathbb Neq 0$
it will result in dephasing equivalently to the one ion case discussed
in Sec.~{\rm Re}f{ssec:consequncesPhase}. However, if the evolution starts in the
ground state either $z_{\pm}^{0}$ or $z_{\pm}^{1}$ will
remain in the ground state for the entire operation because of
Eq.~\eqref{eq:2Qubitf}. Therefore $\operatorname{Re}(\xi)$ will always be zero which
means that the phase depends only on the difference of the swept phase
space areas like in the undamped case studied in
\cite{PhysRevA.95.022328}.
Since the equations \eqref{eq:2Qubitalphat} for the evolution of $ z $
are identical to the one ion case,
Eq. \eqref{eq:relationFDamping} still holds true and we can easily
generalize the force $ F_{\mathrm{nd}}$ constructed in
\cite{PhysRevA.95.022328} to the damped oscillator:
\begin{align}
\label{eq:f_damped}
F(t) = &\kappa e^{-\gamma t}\cdot F_{\mathrm{nd}}.
\end{align}
According to Eq.~({\rm Re}f{eq:alpha(t)}) this means for the damped path
\begin{equation*}
{z_{\mathrm{d}}=\kappa e^{-\gamma t}z_{\mathrm{nd}}.}
\end{equation*}
Here we introduced two correction factors $ \kappa $ and
${e^{-\gamma t}}$ to the original force for the undamped case. $
\kappa $ is a constant to compensate for the smaller area due to the
damping and it therefore ensures that the phase (which corresponds to
the area) stays the same. The exponential factor ensures that $ z
$ returns to the ground state after time $ T $.
It would also be
possible to construct forces via the Gram-Schmidt process described in
Sec.~{\rm Re}f{sec:consequenceRobustness} to maintain the resistance against small
constant offsets of the force. Since these forces now have to produce
closed paths in both modes there are more orthogonality conditions
\begin{align}
\label{eq:orthogonality2qubit}
f \perp& \{e^{\gamma t} \sin(\Omega_+ t), e^{\gamma t} C_0(\Omega)s(\Omega_+ t), \mathbb Notag\\
&\quad e^{\gamma t} \sin(\Omega_{-} t), e^{\gamma t} C_0(\Omega)s(\Omega_{-} t), 1\},
\end{align}
but the method of constructing $ f $ stays the same. For the next
section we will nevertheless stick to Eq.~({\rm Re}f{eq:f_damped}) since the
resulting forces are less complex and sufficient for discussing the
impact on the fidelity. \\
The paths of the resulting $ z_{\pm} $ with
and without damping and under the influence of different forces are
shown in Fig.~{\rm Re}f{fig:2QubitAlphaPath}. The plots (a), (b) and (c) are in
the interaction picture whereas the plot (d) is in the Schrödinger
picture. The parameters for trajectory (a) were chosen identically to
\cite{PhysRevA.95.022328}: $ T = {0.8}$ $\mu$s,
$\omega/2\pi = {2}$ MHz and $ \gamma/\omega = 0 $. The
trajectory (b) corresponds to the same force and parameters as
trajectory (a) but now with a damping $ \gamma/\omega =
0.1 $. We can see how the path is no longer closed and that the area
decreased as well. In the plot (c) we used the same damping and
parameters as in (b) but with the adjusted force
\eqref{eq:f_damped}. Now the paths are closed again and the area
difference is identical to (a). The plot (d) shows the trajectory for
a shorter operation time $ T
= {0.3}{\mu}$s in the Schrödinger picture. It is
important to note that the ticks are different for this plot because
the trajectory has a much greater amplitude. The intuitive reason for
this is that the particle needs a large momentum to complete the loop
in a shorter time, the trajectory is therefore stretched out in $ p
\propto \operatorname{Im}(z^{S}) $ direction. This illustrates that
shorter operation times come with the trade-off of more dephasing (see
Sec.~{\rm Re}f{ssec:fidelity} and Fig.~{\rm Re}f{fig:Infidelity}).
\begin{figure}
\caption{(color online) Figure (a), (b) and (c) show paths of $
z_{+}
\label{fig:2QubitAlphaPath}
\end{figure}
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\subsection{Influence of damping on the fidelity}
\label{ssec:fidelity}
The fidelity measures the overlap of the final state $ \rho_{\mathrm{f}} $ with the desired state $ \ket{\mathsf{P}si_{\mathrm{d}}} $
\begin{equation*}
\mathcal{F} = \bra{\mathsf{P}si_{\mathrm{d}}}\rho_{\mathrm{f}}\ket{\mathsf{P}si_{\mathrm{d}}}.
\end{equation*}
{Since we want to implement a} two qubit phase gate our desired state $ \ket{\mathsf{P}si_{\mathrm{d}}} $ is
\begin{equation*}
\ket{\mathsf{P}si_{\mathrm{d}}} = ae^{i\mathcal{P(H)}i_{{\mathrm{isol}}}}\ket{P} + b\ket{A},
\end{equation*}
with $ \abs{a}^2 + \abs{b}^2 = 1 $. $P$ and $A$ denote an
arbitrary parallel and anti-parallel spin
combination(e.g. $ P =\uparrow\uparrow $ and $ A=\uparrow\downarrow $). The final (spin-) state (in the basis $ \{ \ket{P}, \ket{A}\} $)
after the cyclic evolution is
\begin{equation*}
\rho_{\mathrm{f}} = \begin{pmatrix}
\abs{a}^2 & ab^{*} e^{i\mathcal{P(H)}i_{{\mathrm{isol}}} - \Gamma} \\
a^{*}b e^{-i\mathcal{P(H)}i_{{\mathrm{isol}}} -\Gamma} & \abs{b}^2
\end{pmatrix},
\end{equation*}
where
\begin{equation}
\label{eq:Gamma}
\Gamma = \gamma\int_{0}^{T} \abs{z_{+}(A)}^2 + \abs{z_{-}(P)}^2 {\rm d}{t}.
\end{equation}
We can calculate the fidelity as
\begin{equation}
\label{eq:fidelity}
\mathcal{F} \geq \dfrac{1 + e^{-\Gamma}}{2}.
\end{equation}
The lower bound of the inequality above is reached if the prefactors satisfy $a=b=1/\sqrt{2}$.\\
Figure {\rm Re}f{fig:Infidelity} shows the maximal infidelity $ 1 -
\mathcal{F} $ and phase difference $ \Delta\mathcal{P(H)}i = 2(\mathcal{P(H)}i(A) - \mathcal{P(H)}i(P))
$ for different $ \gamma $ and $ T $. In (a) and (b) the operation
time was chosen as constant $ T={0.8}{\mu}$s whereas $
\gamma/\omega $ varied and in (c) and (d) we chose constant $ \gamma/\omega =
10^{-4} $ for varying $ T $. The plots (b) and (d) show
the phase difference for the force given in Eq.~\eqref{eq:f_damped} which accounts
for the damping (solid blue line) and for the original force (dashed
red line). We can see that the force we constructed (blue line)
correctly compensates for the phase whereas lack of compensation would lead
to a drastic phase deviation for larger damping strengths. Different
operation times do not affect the phase significantly for both forces
because for $ \gamma/\omega = 10^{-4} $ the lost area is still
marginal. For large $ \gamma $ the curve in (a) goes towards $ 1/2 $
apart from that the infidelity is roughly in the same order of
magnitude as $ \gamma/\omega $. In the plot (c) it is demonstrated
that short operation times $ T $ lead to a higher infidelity, because
the forces needed to achieve the desired phase result in a higher
amplitude $ \abs{z} $ and therefore in a larger $ \Gamma $ (see
Fig.~{\rm Re}f{fig:2QubitAlphaPath} (d)). The bump in figure (c) comes from our
choice of the path and has no other physical meaning. We can compare these
infidelities which are roughly of the order of $ \gamma/\omega $ to
infidelities from different sources studied by
\cite{PhysRevA.95.022328}. The infidelity caused by the anharmonic
Coulomb repulsion is below $ 10^{-4} $ whereas the infidelity caused
by considering the correct sinusoidal form of the force $ F(x,t) =
F(t)\cdot\sin(kx) $ is in between $ 10^{-5} $ and $ 0.1 $ depending on
the operation time.
\begin{figure}
\caption{(color online) The upper two plots show the maximum infidelity (a) and
phase (b) $ \Delta\mathcal{P(H)}
\label{fig:Infidelity}
\end{figure}
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\section{Finite temperature effects}
\label{sec:finiteTemperatureEffects}
In section {\rm Re}f{ssec:dissipativeCase} we outlined how the effects of
finite temperature can be taken into account by using a noisy force equal to
$ \widetilde{f}_{j}(t) + \sqrt{2\gamma\bar{n}}\hbar \mathbf{C}i(t) $, where $ \mathbf{C}i(t) $
is a complex valued Gaussian white noise process.
In this section we show how the previous results can be extended to include
finite temperature effects and what impact these effects have on the fidelity
of a two-qubit phase gate.\\
We begin with a stochastic GKSL-type master equation that describes the influence of a
finite temperature heat bath coupled to the trapped ions implementing
the two qubit phase gate. We model the effect of the heat bath by equal strength but independent
damping of both vibration modes ($a_\pm$) and by independent thermal
noise processes affecting both modes ($\mathbf{C}i_\pm(t)$)
\begin{align}
\label{eq:2QubitTimeEvolutionFiniteT}
\dot{\rho_{\mathbf{C}i}} =& \mathcal{L}_{\mathbf{C}i,+}[\rho_\mathbf{C}i] + \mathcal{L}_{\mathbf{C}i,-}[\rho_\mathbf{C}i] ,\\
\mathcal{L}_{\mathbf{C}i,\pm}[\rho] =& \frac{-i}{\hbar}[\widetilde{V}_{\mathbf{C}i,\pm},{\rho}]
+ \gamma\left(2a_{\pm}\rho a_{\pm}^{\dagger} - a_{\pm}^{\dagger}a_{\pm}\rho - \rho a_{\pm}^{\dagger}a_{\pm}\right). \mathbb Notag\\
\widetilde{V}_{\mathbf{C}i,\pm} =& (\widetilde{f}_{\pm}^{*}+ \sqrt{2\gamma\bar{n}}\hbar \mathbf{C}i_\pm^{*}(t))a_{\pm}
+ (\widetilde{f}_{+}+ \sqrt{2\gamma\bar{n}}\hbar \mathbf{C}i_\pm(t))a_{\pm}^{\dagger} \mathbb Notag\\
\rho =& \langle\rho_{\mathbf{C}i}\mathrm{ran}\,gle. \mathbb Notag
\end{align}
In this form the equation is almost identical to Eq.~\eqref{eq:2QubitTimeEvolution}
with the small difference that the thermal noise has been added to the
deterministic driving force.
The solution $\rho_{\xi}$ is similar to the zero temperature case, with the
addition of the random thermal process. Solution can be
expressed in terms of coherent states, whose labels satisfy
\begin{equation}
\label{eq:noisyTrajectory}
\dot{z}_{\pm}^{j}+\gamma z_{\pm}^{j} = \dfrac{1}{i\hbar}\left(\widetilde{f}_{\pm}(j) + \sqrt{2\gamma\bar{n}}\hbar\mathbf{C}i_{\pm}\right).
\end{equation}
However an important difference is that due to the noise the path can no longer be closed reliably by choosing an appropriate force.
As can be seen in Fig.~{\rm Re}f{fig:thermal_traj}, the thermal noise causes
fluctuations around the paths driven by deterministic force ($z_+^1,\, z_-^0$) and
the non-driven paths ($z_+^0,\, z_-^1$) are no longer stationary but fluctuate
around the ground state.
\begin{figure}
\caption{\label{fig:thermal_traj}
\label{fig:thermal_traj}
\end{figure}
\subsection{Fidelity in the finite temperature case}
\label{ssec:fidelityFiniteTemperature}
{One important question is how the fidelity of the phase gate is affected by
the thermal noise and whether the compensation strategy suggested
in~{\rm Re}f{ssec:force} still improves this fidelity.
In this section we seek to answer this question.\\
As in section {\rm Re}f{ssec:fidelity} the fidelity is defined as
\begin{equation*}
\mathcal{F} = \bra{\mathsf{P}si_{\mathrm{d}}}\rho(T)\ket{\mathsf{P}si_{\mathrm{d}}}.
\end{equation*}
Since in the finite temperature case the density operator is given as an average
$ \rho = \langle\rho_{\mathbf{C}i}\mathrm{ran}\,gle$ the fidelity can be obtained by averaging as well,
\begin{equation*}
\mathcal{F} = \langle\bra{\mathsf{P}si_{\mathrm{d}}}\rho_{\mathbf{C}i}(T)\ket{\mathsf{P}si_{\mathrm{d}}}\mathrm{ran}\,gle.
\end{equation*}
We will proceed to first evaluate the fidelity for a general $\rho_{\mathbf{C}i}(T)$ and
then take the average over the stochastic processes $\mathbf{C}i_\pm(t) $.\\
We again consider the overlap with the target state
$\ket{\mathsf{P}si_{\mathrm{d}}} = ae^{i\mathcal{P(H)}i_{{\mathrm{isol}}}}\ket{P} + b\ket{A}$ and
choose $a=b=1/\sqrt{2}$. Other choices for the prefactors will lead to
higher fidelity as discussed in Sec.~{\rm Re}f{ssec:fidelity}. The fidelity of
any $\rho_{\mathbf{C}i}$ after the phase gate operation is given as
\begin{flalign}
\label{eq:fidelityFull}
\mathcal{F} = &\dfrac{1}{4}( \exp(-(\abs{z_{-}^{1}}^2+\abs{z_{+}^{1}}^2)) + \exp(-(\abs{z_{-}^{0}}^2 + \abs{z_{+}^{0}}^2))\mathbb Notag\\
& +2\operatorname{Re}(e^{-i\Delta\mathcal{P(H)}i - \Gamma -\dfrac{1}{2}\left(\abs{z_{-}^{1}}^2 + \abs{z_{+}^{1}}^2 + \abs{z_{-}^{0}}^2 + \abs{z_{+}^{0}}^2 \right)})), \\
\Gamma = &\gamma\int_{0}^{T} \abs{z_{+}^{1}(\tau) - z_{+}^{0}(\tau)}^{2} + \abs{z_{-}^{1}(\tau) - z_{-}^{0}(\tau)}^{2} {\rm d}{\tau}, \mathbb Notag\\
\Delta\mathcal{P(H)}i = &\mathcal{P(H)}i(T) - \mathcal{P(H)}i_{\mathrm{isol}}.\mathbb Notag
\end{flalign}
All quantities in the expression for $\mathcal{F}$ are evaluated at time $T$, that is,
at the end of the phase gate operation.\\
In the expression one can identify the three possible causes for fidelity-loss:
\begin{enumerate}
\item The terms $ \propto \exp(\abs{z_\pm^j(T)}^2) $ arise when the path is not closed, which means that $ z(T) \mathbb Neq 0 $.
\item The term $ \exp(-i\Delta\mathcal{P(H)}i) $ occurs when $ \mathcal{P(H)}i(T) \mathbb Neq \mathcal{P(H)}i_{\mathrm{isol}} $,
\item and the term $ \exp(-\Gamma) $ describes decoherence, which is induced by the damping and can not be prevented.
\end{enumerate}
In the zero temperature case we could close the path and restore the phase
of the isolated case. Therefore only the decoherence was contributing to a
fidelity loss and the formula reduced to Eq.~\eqref{eq:fidelity}. For finite
temperature this is no longer the case and we therefore have to work
with the full expression for $ \mathcal{F} $. We have plotted
the average infidelity as a function of $\gamma\bar{n}T$ over
5000 realizations in Fig.~{\rm Re}f{fig:thermal_fid} (solid line). We checked that our
result coincide with directly solving the master equation ({\rm Re}f{eq:model}) (not shown
in the Figure). The other parameters are as in Fig.~{\rm Re}f{fig:2QubitAlphaPath} c),
except $\gamma/\omega=0.2$. We see that the compensation strategy improves
the fidelity even at finite temperatures. {The reason is that it still increases the average overlap of the final state and the ground state. }{We note, however, that the importance of this effect decreases
as $\bar{n}/(\gamma T)$ increases}.
When $\gamma\bar{n}T\approx 0.1$ (dot) $\mathcal{F}\approx 0.61$ and given the values
$T=0.3\mu$s, and $\gamma/\omega = 0.2$ and $\omega/2\pi=2 $\, MHz, the
average thermal photon number $\bar{n}\approx 2.7$.
This corresponds
to a temperature of about $0.3$ mK (we evaluate $\bar{n}$ at $\omega$).
{These temperatures are within the range of current experimental
capabilities, see for example~\cite{Feldker2020}}.\\
In the appendix {\rm Re}f{sec:AnhangFidelity} we evaluate the $\mathbf{C}i$ dependence of the fidelity.
We use a cumulant expansion to find an approximate analytical expression for the average fidelity.
This approximation is plotted in Fig.~{\rm Re}f{fig:thermal_fid} (dashed line)\footnote{For the
plot we used the results \eqref{eq:avg1stTerm} and \eqref{eq:avg2ndTerm} instead of the final
linearized expression \eqref{eq:avgFidelity} since those provided better agreement in the rather strongly damped regime}. The inset
shows that our approximation gives the first order correction in $\gamma\bar{n} T$
to the zero temperature result. This approximation is motivated by
the fact that for current ion traps the operation time $T$ is much faster then the coherence
time $1/(\bar{n}\gamma)$\cite{Schafer2018, doi:10.1063/1.5088164}.
Our scenario is in the range $\bar{n}\gamma\sim 10^{-1}T$ and
as we can see, {the approximation nicely captures the finite temperature effects
on the fidelity in this regime. The rather complicated approximate expression can
be found in the appendix.}}\\
\begin{figure}
\caption{\label{fig:thermal_fid}
\label{fig:thermal_fid}
\end{figure}
\section{Summary and Outlook}
\label{sec:summary}
We examined how phase gates based on the geometrical phases of driven
trapped ions behave under dissipation. We used forces which depend on an
internal state of the trapped ion in order to construct relative phases
which show up in the density operator. We then showed that in the
special case of a GKSL-type evolution with closed phase space paths
admitting pure state
solutions the total phase will always have an additional third
contribution beyond dynamical- and geometric phases due to dissipation
\begin{flalign*}
\mathcal{P(H)}i &= \mathcal{P(H)}i_{\mathrm{d}} + \mathcal{P(H)}i_{\mathrm{g}} + \xi,\\
\xi &= \mathcal{P(H)}i_{\mathrm{L}} + i\eta.
\end{flalign*}
This third contribution will in general be {complex} and can be directly
related to the Lindblad operators. Applied to the harmonic oscillator
this means that the damping results in a new additional phase that can
however vanish for certain special cases. More severely, dephasing occurs
which cannot be avoided and depends on the amplitude of the
oscillation and the damping strength. We applied our results obtained for a
single trapped ion to a two-qubit phase gate proposed in
\cite{PhysRevA.95.022328} which is based on two trapped ions.
We found that in the presence of damping the phase
produced by the gate depends on the area swept in the interaction picture alone,
if the ion is in the ground state at the
beginning of the operation. However due to the dephasing the fidelity
of the gate is reduced. This loss of fidelity is especially noticeable
for large damping strengths or short operation times. Furthermore, our
calculations show how due to the damping the phase gate no
longer operates independently of the initial motional state. On
the other hand it is possible to maintain the robustness against small
constant offsets of the force $ f \mapsto f + \delta f $ in the damped
case by constructing the force using a Gram-Schmidt procedure that also
ensures that the force produces a closed phase space path evolution. {We considered
also finite temperature effects in order to assess the feasibility of this
scheme. We conclude that this scheme could be soon within reach
of current experimental techniques by reducing the operation time of
the gate or by using colder ion traps, thus preventing the onset of thermal fluctuations.}
{To give fair assessment of the relevance of our results, we point out that
at the moment two qubit gates are typically experimentally implemented
under conditions where the motional state of the ion is rather heated than
damped~\cite{PhysRevLett.117.060504,PhysRevLett.117.060505}.
However, our results predict improvement on the performance of the geometric
phase gate if it would be implemented using buffer gas cooling
scenario~\cite{Feldker2020} where damping can be the main source for gate errors.}
{This work could be extended to more than two ions if the
block structure of the Hamiltonian is kept and only independent
decay processes are considered. In future work we will model
the system from first principles in order to determine the validity of
the phenomenological model presented and analyzed in this article.}
\acknowledgments
The authors would like to thank Valentin Link for fruitful discussions {and
the anonymous Referees for valuable comments.}
\appendix
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\section{Analyzing the phase of a dissipative time evolution}
\label{sec:Anhangphidissipative}
Since we assumed that the evolution can be described by a pure state
we can construct two solutions to Eq.~(\eqref{eq:time evolution}):
\begin{align*}
\rho_{00}(t) &= \kb{0}{0}\otimes\kb{\mathsf{P}si_0(t)}{\mathsf{P}si_0(t)} \\
\rho_{11}(t) &= \kb{1}{1}\otimes\kb{\mathsf{P}si_1(t)}{\mathsf{P}si_1(t)}.
\end{align*}
In order to determine the relative phase between those two states we
examine the evolution of the superposition which means that we have a
density operator of the form
\begin{flalign}
\rho &= \rho_{00} + \rho _{01} + \left(\rho_{01}\right)^{\dagger} + \rho_{11} \mathbb Notag\\
\label{eq:rho01A}
\rho_{01} &= e^{i\mathcal{P(H)}i(t)} \kb{0}{1}\otimes\kb{\mathsf{P}si_0}{\mathsf{P}si_1}.
\end{flalign}
We already know that $ \rho_{00} $ and $ \rho_{11} $ solve the
equation so after inserting $ \rho $ into Eq.~({\rm Re}f{eq:time evolution}) we
are left with
\begin{align}
\label{eq:Anhangrho01dot}
\dot{\rho}_{01} &= \frac{-i}{\hbar}[{H},{\rho_{01}}] + \mathcal[\rho_{01}]\\
\mathcal{L}[\rho_{01}] &= \sum_{l=1}^{N} L_l\rho_{01}L_l^{\dagger} - \dfrac{1}{2}\left(L_l^{\dagger}L_l\rho_{01} + \rho_{01} L_l^{\dagger}L_l\right)\mathbb Notag.\\
\end{align}
In analogy to \cite{PhysRevLett.58.1593} we can use \eqref{eq:rho01A} and calculate
\begin{align}
\dot{{\rho}}_{01} &= -i\dot{\mathcal{P(H)}i}{\rho}_{01} + e^{-i\mathcal{P(H)}i(t)} \frac{{\rm d}}{{\rm d} t} \left(\kb{0}{1}\otimes\kb{\mathsf{P}si_0}{\mathsf{P}si_1}\right)\\
-\dot{\mathcal{P(H)}i} &= -i\bra{\mathsf{P}si_0} \left(\frac{{\rm d}}{{\rm d} t} \kb{\mathsf{P}si_0}{\mathsf{P}si_1}\right) \ket{\mathsf{P}si_1} + ie^{-i\mathcal{P(H)}i}\bra{\mathsf{P}si_0}\dot{\rho}_{01}\ket{\mathsf{P}si_1}.
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\end{align}
Since we want to determine $ \dot{\mathcal{P(H)}i} $ this leaves us with two terms to evaluate:
\begin{equation*}
\bra{{\mathsf{P}si}_0}\left(\frac{{\rm d}}{{\rm d} t} \kb{\mathsf{P}si_0}{\mathsf{P}si_1}\right)\ket{{\mathsf{P}si}_1} = \langle{{\mathsf{P}si}_0}\mathbf{e}rt{\dot{{\mathsf{P}si}}_0}\mathrm{ran}\,gle + \langle{\dot{{\mathsf{P}si}}_1}\mathbf{e}rt{{\mathsf{P}si}_1}\mathrm{ran}\,gle,
\end{equation*}
and
\begin{align*}
e^{-i\mathcal{P(H)}i(t)}\bra{{\mathsf{P}si}_0}\dot{\rho}_{01}\ket{{\mathsf{P}si}_1} &= \frac{-ie^{-i\mathcal{P(H)}i}}{\hbar}\left( \bra{\mathsf{P}si_0}[{H},{\rho_{01}}] + L\left[\rho_{01}\right]\ket{\mathsf{P}si_1} \right)\\
&= \dfrac{-i}{\hbar}\left(\bra{\mathsf{P}si_0}H\ket{\mathsf{P}si_0} - \bra{\mathsf{P}si_1}H\ket{\mathsf{P}si_1}\right) \\
&+ \sum_{l=1}^{N}\bra{\mathsf{P}si_0}L_l\ket{\mathsf{P}si_0}\bra{\mathsf{P}si_1}L_l^{\dagger}\ket{\mathsf{P}si_1} \\
&- \dfrac{1}{2}\left(\bra{\mathsf{P}si_0}L_l^{\dagger}L_l\ket{\mathsf{P}si_0} + \bra{\mathsf{P}si_1}L_l^{\dagger}L_l\ket{\mathsf{P}si_1}\right).
\end{align*}
This leads to the final result
\begin{align*}
-\frac{{\rm d}{\mathcal{P(H)}i}}{{\rm d} t} =& -i\left(\bra{\mathsf{P}si_0}\partial_t\ket{\mathsf{P}si_0} - \bra{\mathsf{P}si_1}\partial_t\ket{\mathsf{P}si_1}\right) +\frac{1}{\hbar}(\langle H_0\mathrm{ran}\,gle - \langle H_1\mathrm{ran}\,gle) \\
& i\sum_{l=1}^{N}\bra{\mathsf{P}si_0}L_l\ket{\mathsf{P}si_0}\bra{\mathsf{P}si_1}L_l^{\dagger}\ket{\mathsf{P}si_1} \\
&- \dfrac{1}{2}\left(\bra{\mathsf{P}si_0}L_l^{\dagger}L_l\ket{\mathsf{P}si_0} + \bra{\mathsf{P}si_1}L_l^{\dagger}L_l\ket{\mathsf{P}si_1}\right).
\end{align*}
Here we also used that $ \langle{\mathsf{P}si}\mathbf{e}rt\dot{\mathsf{P}si}\mathrm{ran}\,gle $ is purely
imaginary and therefore
$ \langle{\mathsf{P}si}\mathbf{e}rt{\dot{\mathsf{P}si}}\mathrm{ran}\,gle = -\langle{\dot{\mathsf{P}si}}\mathbf{e}rt{\mathsf{P}si}\mathrm{ran}\,gle $.
{\section{Fidelity at finite temperature}}
\label{sec:AnhangFidelity}
{In Sec.~{\rm Re}f{sec:finiteTemperatureEffects} we derived an expression
for the fidelity of the gate for finite temperatures. Since this expression depends on
the random noise $ \mathbf{C}i $ we have to take the average of $\mathcal{F}$ over $\mathbf{C}i$ in order to make
meaningful predictions about the fidelity of the phase gate. \\
We will first take the averages of the terms $\propto\exp(-\abs{z_{\pm}^{j}(T)}^2)$ and later
evaluate the average of the term $ \exp(-i\Delta\mathcal{P(H)}i - \Gamma -\dfrac{1}{2}\left(\abs{z_{-}^{1}}^2 + \abs{z_{+}^{1}}^2 + \abs{z_{-}^{0}}^2 + \abs{z_{+}^{0}}^2 \right)) $.\\
In the finite temperature case the presence of noise leads to the following equation for $ z(t) $:
\begin{equation}
\label{eq:z(t)FiniteT}
z_{\pm}^{j}(t) = \dfrac{1}{i\hbar}\int_{0}^{t}(f_{\pm}^{j}+ \sqrt{2\gamma\bar{n}}\hbar\mathbf{C}i_{\pm})e^{i\Omega_{\pm}\tau - \gamma (t-\tau)}{\rm d}{\tau}.
\end{equation}
Note that there are two different uncorrelated noises $ \mathbf{C}i_{\pm} $ for the two
different modes of oscillation (stretch and center of mass mode) that do however
not depend on the internal states. Since the noise is just added to the force
we can split $ z $ in two parts, where one part $ z_{\pm,f}^{j}(t) $ is under full
control of the force (zero temperature path) and the second part $ z_{\pm,\mathbf{C}i}^{j}(t) $ is a fluctuation due to the noise.
\begin{flalign}
\label{eq:splitZ}
z_{\pm}^{j}(t) &= z_{\pm,f}^{j}(t)+z_{\pm,\mathbf{C}i}^{j}(t),\\
z_{\pm,f}^{j}(t) &= \dfrac{1}{i\hbar}\int_{0}^{t}f_{\pm}^{j}e^{i\Omega_{\pm}\tau - \gamma (t-\tau)}{\rm d}{\tau}, \mathbb Notag\\
z_{\pm,\mathbf{C}i}^{j}(t) &= \dfrac{1}{i\hbar}\int_{0}^{t} \sqrt{2\gamma\bar{n}}\hbar\mathbf{C}i_{\pm}e^{i\Omega_{\pm}\tau - \gamma (t-\tau)}{\rm d}{\tau}.\mathbb Notag
\end{flalign}
Since $z_{\pm,f}^{j}$ does not depend on $\mathbf{C}i$ we arrive at
\begin{flalign}
\label{eq:fidelityAbsTerms}
&\langle\exp(-\abs{z_{\pm}^{j}(T)}^2)\mathrm{ran}\,gle = e^{-\abs{z_{\pm,f}^{j}(T)}^2} \mathbb Notag\\
&\times \langle\exp\Big[-z_{\pm,f}^{j*}(T)z_{\pm,\mathbf{C}i}^{j}(T)
- z_{\pm,f}^{j}(T)z_{\pm,\mathbf{C}i}^{j*}(T) \mathbb Notag\\%sry I don't know how to space it better ^^
&\hspace{20pt}- \abs{z_{\pm,\mathbf{C}i}^{j}(T)}^2\Big]\mathrm{ran}\,gle
\end{flalign}
If we define
\begin{flalign}
\mathcal{V}_{\pm}^{j}(\tau) &= -z_{\pm,f}^{j*}(T)\dfrac{1}{i\hbar}\sqrt{2\gamma\bar{n}}\hbar\mathbf{C}i_{\pm}e^{i\Omega_{\pm}\tau - \gamma(T-\tau)} \mathbb Notag\\
&- z_{\pm,f}^{j}(T)\dfrac{1}{-i\hbar}\sqrt{2\gamma\bar{n}}\hbar\mathbf{C}i_{\pm}^{*}e^{-i\Omega_{\pm}\tau - \gamma(T-\tau)}\mathbb Notag\\
&- \int_{0}^{T} 2\gamma\bar{n}\mathbf{C}i_\pm(\tau)\mathbf{C}i^{*}_\pm(s) e^{i\Omega_\pm(\tau-s)-\gamma(T-\tau + T -s)}{\rm d}{s},
\end{flalign}
we can rewrite Eq.~\eqref{eq:fidelityAbsTerms} as
\begin{flalign}
\label{eq:fidelityV}
\langle e^{-\abs{z_{\pm}^{j}(T)}^2}\mathrm{ran}\,gle &= e^{-\abs{z_{\pm,f}^{j}(T)}^2}
\langle e^{\int_{0}^{T} \mathcal{V}_{\pm}^{j}(\tau) {\rm d}{\tau}}\mathrm{ran}\,gle .
\end{flalign}
We now evaluate the average term by taking the cumulant expansion to 2nd order:
\begin{flalign*}
\langle e^{\int_{0}^{T} \mathcal{V}_{\pm}^{j}(t) {\rm d}{t}}\mathrm{ran}\,gle = e^{\int_{0}^{T} \langle\mathcal{V}_{\pm}^{j}(t)\mathrm{ran}\,gle {\rm d}{t}
+ \dfrac{1}{2} \int_{0}^{T}\int_{0}^{T} \langle\mathcal{V}_{\pm}^{j}(t_1)\mathcal{V}_{\pm}^{j}(t_2)\mathrm{ran}\,gle{\rm d}{t_1} {\rm d}{t_2} + \dots}
\end{flalign*}
Using the Gaussian nature of the processes $\mathbf{C}i_\pm$ we get
\begin{flalign}
\langle e^{-\abs{z_{\pm}^{j}(T)}^2}\mathrm{ran}\,gle\approx& \exp\Big[-\bar{n}\left(1-e^{-2\gamma T}\right) \mathbb Notag\\
&-\abs{z_{\pm,f}^{j}}^{2}(1-\bar{n}\left(1-e^{-2\gamma T}\right)) \mathbb Notag\\
\label{eq:avg1stTerm}
&+ \dfrac{3}{2} \left(\bar{n}\left(1-e^{-2\gamma T}\right)\right)^{2}\Big] .
\end{flalign}
From the result we can see that the finite temperature leads to additional decoherence
which increases even further for $ z_{\pm,f}^{j}(T) \mathbb Neq 0 $. Note that $ z_{\pm,f}^{j}(T) = 0 $
when the path is closed in the zero temperature case. The
third term in the expansion above is of the order $ (\gamma\bar{n}T)^{2} = (T/\tau_{\mathrm{d}})^{2} $.
Since the coherence time $ \tau_{\mathrm{d}} $ is much larger than the
operation time for current ion-traps \cite{Schafer2018, lucas2007longlived}
we will only keep terms in the first order of $ \gamma\bar{n}T $ from here on.\\
We are now going to take a look at the average of the term
\begin{equation}
\label{eq:2ndTerm}
\langle e^{-i\Delta\mathcal{P(H)}i - \Gamma -\dfrac{1}{2}\left(\abs{z_{-}^{1}}^2 + \abs{z_{+}^{1}}^2 + \abs{z_{-}^{0}}^2 + \abs{z_{+}^{0}}^2 \right)}\mathrm{ran}\,gle.
\end{equation}
At first one finds that $ \Gamma $ is actually independent of $ \mathbf{C}i $ since it is
defined as the difference between two paths of different internal states
(see Eq.~\eqref{eq:fidelityFull}) and since the noise is independent
of the internal state it cancels out.
We than follow the same strategy as before and introduce a $\mathcal{V}'(t)$
and approximate the exponential with a cumulant expansion
\begin{equation*}
\mathcal{V}'(\tau) = -i\dot{\mathcal{P(H)}i_{\mathbf{C}i}}(\tau) +\dfrac{1}{2}\left(\mathcal{V}_{-}^{1}(\tau) + \mathcal{V}_{+}^{1}(\tau)+ \mathcal{V}_{-}^{0}(\tau)+\mathcal{V}_{+}^{0}(\tau)\right) .
\end{equation*}
Like the trajectory before we also split the phase into a part that is determined by the force
alone $ \mathcal{P(H)}i_{f} $ and a part that depends on the noise $ \mathcal{P(H)}i_{\mathbf{C}i} $: $ \mathcal{P(H)}i = \mathcal{P(H)}i_f + \mathcal{P(H)}i_{\mathbf{C}i} $
With this definition we can write Eq.~\eqref{eq:2ndTerm} as
\begin{flalign*}
e^{-i(\mathcal{P(H)}i_f - \mathcal{P(H)}i_{\mathrm{isol}}) - \Gamma -\dfrac{1}{2}\left(\abs{z_{f,-}^{1}}^{2}
+\abs{z_{f,+}^{1}}^{2}+\abs{z_{f,-}^{0}}^{2}+\abs{z_{f,+}^{0}}^{2} \right)}
\langle e^{\int_{0}^{T} \mathcal{V}'(\tau) {\rm d}{\tau}}\mathrm{ran}\,gle .
\end{flalign*}
We again take the cumulant expansion up to 2nd order and only keep terms up to first order of $ \gamma\bar{n}T $ to arrive at:
\begin{widetext}
\begin{flalign}
\label{eq:avg2ndTerm}
&\langle \exp({-i\Delta\mathcal{P(H)}i - \Gamma -\frac{\abs{z_{-}^{1}}^2 + \abs{z_{+}^{1}}^2 + \abs{z_{-}^{0}}^2 + \abs{z_{+}^{0}}^2}{2}})\mathrm{ran}\,gle
\approxeq \exp\Bigg[ -i(\mathcal{P(H)}i_f(T) - \mathcal{P(H)}i_{\mathrm{isol}}) - \Gamma -2\bar{n}\left({1-e^{-2\gamma T}}\right) \mathbb Notag\\
&\hspace{10pt}+ \dfrac{\gamma\bar{n}}{\hbar}\operatorname{Im}\left(\int_{0}^{T}{\rm d}{t_1}\int_{0}^{t_1}{\rm d}{t_2} \left(z_{-,f}^{1*}(t_2)\widetilde{f}_{-}^{1}(t_1) + z_{+,f}^{0*}(t_2)\widetilde{f}_{+}^{0}(t_1) \right)e^{-\gamma(T+t_1 - 2t_2)} \right) \mathbb Notag\\
&\hspace{10pt}-\dfrac{\bar{n}}{\hbar^2}\operatorname{Re}\left(\int_{0}^{T}{\rm d}{t_1}\int_{0}^{t_1}{\rm d}{t_2} \left(\widetilde{f}_{-}^{1}(t_1)\widetilde{f}_{-}^{1*}(t_2) +
\widetilde{f}_{+}^{0}(t_1)\widetilde{f}_{+}^{0*}(t_2)\right)e^{-\gamma t_1}\sinh(\gamma t_2)\right)\mathbb Notag\\
&\hspace{10pt}+ \dfrac{\gamma\bar{n}}{2}\int_{0}^{T}{\rm d}{t_1}\abs{z_{-,f}^{1}(t_1)}^{2} + \abs{z_{+,f}^{0}(t_1)}^{2}
+ \dfrac{i\bar{n}}{\hbar}\operatorname{Re}\left(\int_{0}^{T}{\rm d}{t_1}\left(z_{-,f}^{1}(T)\widetilde{f}_{-}^{1}(t_1) - z_{+,f}^{0}(T)\widetilde{f}_{+}^{0}(t_1)\right)e^{-\gamma T}\sinh(\gamma t_1)\right)\mathbb Notag\\
&\hspace{10pt}-\dfrac{\abs{z_{-,f}^{1}(T)}^{2}+ \abs{z_{+,f}^{0}(T)}^{2}}{2}{\left(1-\bar{n}\left(1-e^{-2\gamma T}\right)\right)}\Bigg] .
\end{flalign}
With Eq.~\eqref{eq:avg1stTerm}, \eqref{eq:avg2ndTerm} we can now express the average fidelity in first order of $ \bar{n}\gamma T $, where we also expanded the exponentials of $\gamma T$:
\begin{flalign}\label{eq:avgFidelity}
\langle\mathcal{F}\mathrm{ran}\,gle \approxeq& \dfrac{\exp(-4\bar{n}\gamma T - \abs{z_{-,f}^{1}(T)}^{2}(1-2\bar{n}\gamma T))+\exp(-4\bar{n}\gamma T - \abs{z_{+,f}^{0}(T)}^{2}(1-2\bar{n}\gamma T))}{4} \mathbb Notag\\
&+ \frac{1}{2}\operatorname{Re}\Bigg[ \exp\Bigg\{ -i(\mathcal{P(H)}i_f(T) - \mathcal{P(H)}i_{\mathrm{isol}}) - \Gamma -4\bar{n}\gamma T - \dfrac{\gamma\bar{n}}{2}\int_{0}^{T}{\rm d}{t_1}\abs{z_{-,f}^{1}(t_1)}^{2} + \abs{z_{+,f}^{0}(t_1)}^{2} \mathbb Notag\\
&+ \dfrac{\gamma\bar{n}}{\hbar}\operatorname{Im}\left(\int_{0}^{T}{\rm d}{t_1}\int_{0}^{t_1}{\rm d}{t_2} z_{+,f}^{0*}(t_2)\widetilde{f}_{+}^{0}(t_1)+z_{-,f}^{1*}(t_2)\widetilde{f}_{-}^{1}(t_1)\right) \mathbb Notag\\
&+ \dfrac{i{\gamma}\bar{n}}{\hbar}\operatorname{Re}\left(\int_{0}^{T}{\rm d}{t_1}\left(z_{-,f}^{1}(T)\widetilde{f}_{-}^{1}(t_1) - z_{+,f}^{0}(T)\widetilde{f}_{+}^{0}(t_1)\right){t_1}\right)\mathbb Notag\\
&{-}\dfrac{{\gamma}\bar{n}}{\hbar^2}{\operatorname{Re}}\left(\int_{0}^{T}{\rm d}{t_1}\int_{0}^{t_1}{\rm d}{t_2} \left(\widetilde{f}_{-}^{1}(t_1)\widetilde{f}_{-}^{1*}(t_2) +
\widetilde{f}_{+}^{0}(t_1)\widetilde{f}_{+}^{0*}(t_2)\right){t_2}\right)\mathbb Notag\\
&-\dfrac{\abs{z_{-,f}^{1}(T)}^{2}+ \abs{z_{+,f}^{0}(T)}^{2}}{2}\left(1-{2}\bar{n}\gamma T\right)\Bigg\}\Bigg] .
\end{flalign}
{Note that this expression still contains the endpositions of the zero temperature paths $\abs{z_{\pm,f}^{j}(T)}^{2}$. Since these vanish if the compensation strategy is employed, this suggests that the strategy improves the fidelity even at finite temperatures. Furthermore, although all of the terms in the exponential are of first order of $\gamma\bar{n}T$ we found that for our set of forces the dominating temperature dependent contribution would be the $-4\bar{n}\gamma T$ terms. They arise because the finite temperature paths can no longer be closed reliably.}
\end{widetext}}
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\begin{document}
\pagestyle{myheadings}
\markboth{
Twisted Alexander polynomials
associated to metacyclic representations
}
{
\ \ M. Hirasawa \& K. Murasugi}
\title{Twisted Alexander polynomials of $2$-bridge knots
associated to metacyclic representations}
\author{Mikami Hirasawa}
\address{Department of Mathematics,
Nagoya Institute of Technology\\
Nagoya Aichi 466-8555 Japan\\
{\it E-mail: [email protected]}
}
\author{Kunio Murasugi}
\address{Department of Mathematics,
University of Toronto\\
Toronto, ON M5S2E4 Canada\\
{\it E-mail: [email protected]}
}
\maketitle
\begin{abstract}
Let $p=2n+1$ be a prime and $D_p$ a dihedral group of order $2p$.
Let $\widehat{\rho} : G(K) \rightarrow D_p \rightarrow
GL(p,\ZZ)$ be
a non-abelian representation of the knot group $G(K)$ of a knot $K$ in
3-sphere.
Let $\widetilde{\Delta}_{\widehat{\rho},K} (t)$ be the twisted
Alexander polynomial of $K$ associated to $\widehat{\rho}$.
Then we prove that for any 2-bridge knot $K(r)$ in $H(p)$,
$\widetilde{\Delta}_{\widehat{\rho},K}(t)$ is of the form
$\left\{\dfrac{{\Delta}_{K(r)} (t)}{1-t}\right\}
f(t) f(-t)$ for
some integer polynomial $f(t)$, where $H(p)$ is the set of $2$-bridge knots
$K(r), 0<r<1$, such that $G(K(r))$ is mapped onto a non-trivial free
product $\ZZ/2 * \ZZ/p$. Further, it is proved that
$f(t) \equiv \left\{\dfrac{{\Delta}_{K} (t)} {1+t}\right\}^n$ (mod $p$),
where ${\Delta}_{K} (t)$ is the Alexander polynomial of $K$.
Later we discuss the twisted Alexander polynomial
associated to the general metacyclic representation.
\end{abstract}
\keywords{$2$-bridge knot, twisted Alexander polynomial, dihedral representation,
metacyclic representation.}
\ccode{Mathematics Subject Classification 2000: 57M25, 57M27}
\section{Introduction}
In the previous paper \cite{HM},
we studied the parabolic representation of the
group of a $2$-bridge knot and showed some properties of its twisted
Alexander polynomial.
In this paper, we consider a metacyclic
representations of the knot group.
Let $G(m,p|k)$ be a (non-abelian) semi-direct product of two cyclic groups
$\ZZ/m$ and $\ZZ/p$, $p$ an odd prime,
with the following presentation:
\begin{equation}
G(m,p|k)=\langle s,a|s^m=a^p=1,sas^{-1}=a^k\rangle,
\end{equation}
where $k$ is a primitive $m$-th root of 1 (mod $p$), i.e.
$k^m \equiv 1$ (mod $p$), but $k^q \not\equiv 1$ (mod $p$) for any
$q, 0<q<m$ and $k \ne 0, 1$.
If $k=-1$, then $m=2$ and hence $G(2,p|-1)$ is a dihedral group $D_p$.
Since $k$ is a primitive $m$-th root of 1 (mod $p$),
$G(m,p|k)$ is imbedded in the symmetric
group $S_p$ and hence in $GL(p,\ZZ)$ via permutation matrices.
Now suppose that the knot group $G(K)$ of a knot $K$ is mapped onto
$G(m,p|k)$ for some $m,p$ and $k$.
Then, we have a representation $f:\ G(K)
\rightarrow G(m,p|k) \rightarrow GL(p,\ZZ)$
and the twisted Alexander polynomial
$\widetilde{\Delta}_{f,K}(t)$
associated to $f$ is defined
\cite{L} \cite{W} \cite{KL}.
One of our objectives is to
characterize these twisted Alexander polynomials.
In fact, we propose the following conjecture.
\noindent{\bf Conjecture A}.
{\it
$\widetilde{\Delta}_{f,K}(t)=
\left\{\dfrac{\Delta_{K} (t)}{1-t}\right\} F(t)$,
where $\Delta_{K}(t)$ is the Alexander polynomial of $K$ and $F(t)$ is an
integer polynomial in $t^m$.
}
First we study the case $k=-1$, dihedral representations of the knot group.
Let $D_p$ be a dihedral group of order $2p$,
where $p=2n+1$ and $p$ is a prime.
Then the knot group $G(K)$ of a knot $K$ is mapped onto $D_p$
if and only if ${\Delta}_{K} (-1) \equiv 0$ (mod $p$)
\cite{Fox62}, \cite{Ha}.
Therefore, if ${\Delta}_{K} (-1) \ne \pm 1$,
$G(K)$ has at least one representation on a certain dihedral group $D_p$.
For these cases, we can make Conjecture A slightly sharper:
\noindent{\bf Conjecture B}.
{\it
Let $\widehat{\rho}:\
G(K) \rightarrow D_p \rightarrow
GL(p, \ZZ)$ be a non-abelian
representation of the knot group $G(K)$ of a knot $K$ and let
$\widetilde{\Delta}_{\widehat{\rho}, K} (t)$ be the
twisted Alexander polynomial of $K$ associated to $\widehat{\rho}$.
Then
\begin{equation}
\widetilde{\Delta}_{\widehat{\rho}, K} (t) =
\left\{\frac{{\Delta}_{K}(t)}{1-t}\right\}f(t) f(-t),
\end{equation}
where $f(t)$ is an integer polynomial and further,
\begin{equation}
f(t) \equiv \left\{\frac{{\Delta}_{K} (t)}{1+t}\right\}^{n}
({\rm mod\ } p)
\end{equation}
}
We should note that $(1+t)^2$ divides $\Delta_K(t)$ (mod $p$)
if and only if $\Delta_K(-1)\equiv 0$ (mod $p$).
The main purpose of this paper is to prove (1.2) for a
$2$-bridge knot $K(r)$
in $H(p)$, $p$ a prime, and (1.3) for a $2$-bridge knot with
${\Delta}_{K}(-1) \equiv 0$ (mod $p$).
(See Theorem 2.2.)
Here $H(p)$ is the set of $2$-bridge knots $K(r), 0<r<1,$
such that $G(K(r))$ is mapped onto a free product $\ZZ/2 * \ZZ/p$.
We note that knots in $H(p)$ have been studied extensively in \cite{GR} and \cite{ORS}.
A proof of the main theorem (Theorem 2.2) is given in
Section 2 through
Section 7.
Since this paper is a sequel of \cite{HM},
we occasionally skip some details if
the argument used in \cite{HM} also works in this paper.
In Section 8, we consider another type of metacyclic groups, denoted by
$N(q,p)$.
$N(q,p)$ is a semi-direct product of two cyclic groups,
$\ZZ/2q$ and
$\ZZ/p$
defined by
\begin{equation}
N(q,p)=\langle s, a| s^{2q}=a^p=1, sas^{-1}=a^{-1}\rangle,
\end{equation}
where $q \geq 1$ and $p$ is an odd prime and $\gcd(q,p)=1$. We note that
$N(1,p) =D_p$ and $N(2,p)$ is called a binary dihedral group.
Let $\widetilde{\nu}:\ G(K) \longrightarrow N(q,p) \longrightarrow
GL(2pq,\ZZ)$ be a representation of $G(K)$.
(For details, see Section 8.)
Then we show that for a $2$-bridge knot $K(r)$, the twisted Alexander
polynomial $\widetilde{\Delta}_{\widetilde{\nu},K(r)}(t)$
associated to $\widetilde{\nu}$ is completely determined by
the Alexander polynomial
$\Delta_{K(r)}(t)$ and the twisted Alexander polynomial
$\widetilde{\Delta}_{\widehat{\rho},K(r)}(t)$ associated to
$\widehat{\rho}$. (Proposition \ref{prop:8.5})
In Section 9, we give examples that illustrate our main
theorem and Proposition \ref{prop:8.5}.
It is interesting to observe that
$\widetilde{\Delta}_{\widetilde{\nu},K(r)}(t)$
is an integer polynomial in $t^{2q}$.
In Section 10, we briefly discuss general
$G(m,p|k)$-representations of
the knot group and give several examples,
one of which is not a $2$-bridge knot,
that support Conjecture A.
In Section 11, we prove Proposition 2.1 and Lemma 5.2
that plays a key role
in our proof of the main theorem.
Finally, for convenience, we draw a diagram below consisting of homomorphisms that
connect various groups and rings.
\begin{center}
$
\begin{array}{
cccc
ccc}
& & GL(p,\ZZ) & & GL(2n,\ZZ)& & \\
& & \mbox{\large $\pi$}\uparrow& \nearrow& \hspace*{-11mm}
\mbox{\large $\pi_0$}
\ \ \ \ \ \uparrow \mbox{\large $\gamma$}& & \\
G(K)&\underset{\mbox{\large $\rho$}}{\longrightarrow}&D_p&
\underset{\mbox{\large $\xi$}}{\longrightarrow}&GL(2,\CC)& & \\
\downarrow& & \downarrow& & & & \\
\ZZ G(K)&\longrightarrow&\ZZ D_p&
\underset{\mbox{\large $\zeta$}}{\longrightarrow}&\widetilde{A}(\omega)& &
M_{2n,2n}(\ZZ[t^{\pm 1}])\\
& \mbox{\large $\rho^{*}$}\hspace{-1mm}\searrow&\downarrow& &\downarrow&&\uparrow
\mbox{\large $\gamma^{*}$}\\
& & \ZZ D_p[t^{\pm1}]&\underset{\mbox{\large $\zeta^{*}$}}{\longrightarrow}&
\widetilde{A}(\omega)[t^{\pm 1}]&
\underset{\mbox{\large $\xi^{*}$}}{\longrightarrow}&
M_{2,2}\bigl((\ZZ [\omega])[t^{\pm 1}]\bigr)
\end{array}
$
\end{center}
Here, $\tau=\rho\circ\xi, \widehat\rho=\rho\circ\pi, \rho_0=\rho\circ\pi_0,
\eta=\xi\circ\gamma,
\Phi^*=\rho^*\circ\zeta^*\circ\xi^*$ and $\nu=\rho\circ\xi\circ\gamma$.
Unmarked arrows indicate natural extensions of homomorphisms.
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\section{
Dihedral representations and statement of the main theorem}
We begin with a precise formulation of representations.
Let $p=2n+1$ and $D_p$ be a dihedral group of order $2p$
with a presentation:
$D_p = \langle x,y| x^2 = y^2 = (xy)^p = 1\rangle$.
As is well known, $D_p$ can be faithfully represented
in $GL(p,\ZZ)$ by the map $\pi$ defined by:
\begin{center}$
x\mapsto
\left[
\begin{array}{ccccccc}
1&0&0&\cdots&0&0&0\\
0&0&0&\cdots&0&0&1\\
0&0&0&\cdots&0&1&0\\
\vdots & \vdots& \vdots& &\nana & &\vdots \\
\vdots &\vdots & &\nana& & \vdots& \vdots\\
\vdots & & \nana & & \vdots&\vdots & \vdots\\
0&1&0&\cdots&0&0&0
\end{array}
\right]
\
y\mapsto
\left[
\begin{array}{ccccccc}
0&1&0&\cdots&0&0&0\\
1&0&0&\cdots&0&0&0\\
0&0&0&\cdots&0&0&1\\
0&0&0&\cdots&0&1&0\\
\vdots & \vdots& \vdots& & \nana& & \vdots\\
\vdots & \vdots& &\nana & \vdots& \vdots&\vdots \\
0&0&1&\cdots&0&0&0
\end{array}
\right]
$
\end{center}
However, $\pi$ is reducible.
In fact, $\pi$ is equivalent to $id \ast{\pi}_0$, where
\begin{equation}
\pi_0: x\mapsto
\left[
\begin{array}{cccccc}
0&0&\cdots&0&0&1\\
0&0&\cdots&0&1&0\\
\vdots & \vdots& &\nana & & \vdots\\
\vdots& &\nana & & \vdots& \vdots\\
0&1&\cdots&0&0&0\\
1&0&\cdots&0&0&0
\end{array}
\right]
y\mapsto
\left[
\begin{array}{ccccccc}
-1&0&0&\cdots&0&0&0\\
-1&0&0&\cdots&0&0&1\\
-1&0&0&\cdots&0&1&0\\
\vdots & \vdots& \vdots& & \nana&& \vdots\\
\vdots & \vdots& & \nana & & \vdots& \vdots\\
-1&0&1&\cdots&0&0&0\\
-1&1&0&\cdots&0&0&0
\end{array}
\right]
\end{equation}
For convenience, ${\pi}_0$ is called the
{\it irreducible representation} of $D_p$ (of degree $p-1=2n$).
Now let $K(r), 0<r<1, r= \frac{\beta}{\alpha}$ and $\gcd(\alpha,\beta)=1$,
be a $2$-bridge knot and consider a Wirtinger presentation of
the group $G(K(r))$:
\begin{align}
&G(K(r)) = \langle x,y| R \rangle,\
{\rm where}\nonumber\\
&R = WxW ^{-1} y^{-1},
W = x^{\epsilon_1}
y^{\epsilon_2} \cdots x^{\epsilon_{\alpha-2}} y^{\epsilon_{\alpha-1}}\
{\rm and}\nonumber\\
&\epsilon_j = \pm 1\ {\rm for}\
1\leq j \leq \alpha -1.
\end{align}
Suppose $p$ be a prime. If $\alpha \equiv 0$ (mod $p$), then a mapping
\begin{equation}
\rho: x \mapsto x\ {\rm and}\ y \mapsto y
\end{equation}
defines a surjection from $G(K(r))$ to $D_p$.
Therefore $\rho_0 = \rho \circ \pi_0$ defines a representation of
$G(K(r))$ into $GL(2n,\ZZ)$ and we can define the twisted Alexander
polynomial $\widetilde{\Delta}_{\rho_0, K(r)} (t)$ associated
to $\rho_0$.
Since $\pi = id \ast \pi_0$, the twisted Alexander polynomial
associated to $\widehat{\rho} = \rho \circ \pi $ is given by
$\left[\dfrac{\Delta_{K(r)}(t)}{1-t}\right] \widetilde{\Delta}_{\rho_0, K(r)}(t)$ and
hence (1.2) becomes
\begin{equation}
\widetilde{\Delta}_{\rho_0, K(r)} (t) = f(t) f(-t).
\end{equation}
Now there is another representation of $D_p$ in $GL(2,\CC)$. To be more
precise, consider $\xi: D_p \rightarrow GL(2,\CC)$ given by
\begin{equation}
\xi (x) =\mtx{-1}{1}{0}{1}\ {\rm and}\ \xi (y)
=\mtx{-1}{0}{\omega}{1},
\end{equation}
where $\omega \in \CC$ is determined as follows.
First we set $\xi (x) = \mtx{-1}{1}{0}{1}$
and $\xi (y) =\mtx{-1}{0}{z}{1}$, and write
$\xi((xy)^k) = \mtx{a_k(z)}{b_k(z)}{c_k(z)}{d_k(z)}$.
Since $\xi(xy) = \mtx{1+z}{1}{z}{1}$,
we see that $a_k ,b_k ,c_k$
and $d_k$ are exactly the same polynomials found in \cite[(4.1)]{HM}.
Further, as is
mentioned in \cite{HM}, $a_n(z)$ and $b_n(z)$ are given as follows:
\cite[Propositions 10.2 and 2.4]{HM}:
\begin{equation}
a_n(z) = \sum_{k=0}^{n} \binom{n+k}{2k}z^k\
{\rm and}\
b_n(z)=\sum_{k=0}^{n-1}
\binom{n+k}{2k+1} z^k.
\end{equation}
Since $(xy)^{2n+1} =1$, we have $(xy)^n x= y (xy)^n$ and
hence, a simple calculation shows that
$\xi ((xy)^n x)=\xi (y (xy)^n)$ yields $a_n(z) + 2 b_n(z)= 0$.
Therefore, the number $\omega$ we are looking for is a root
of $\theta_n (z) = a_n (z) + 2 b_n (z)$. Write
$\theta_n (z)={c_0}^{(n)} + {c_1}^{(n)} z + \cdots +
{c_{n-1}}^{(n)} z^{n-1} + {c_n}^{(n)} z^n$.
Then we see
\begin{equation}
{\displaystyle
{c_k}^{(n)}=\binom{n+k}{2k}+ 2\binom{n+k}{2k+1}
=\frac{2n+1}{2k+1}\binom{n+k}{n-k}}.
\end{equation}
If $p = 2n+1$ is prime, then, for $0 \leq k \leq n-1$,
${c_k}^{(n)} \equiv 0$
(mod $p$), but ${c_0}^{(n)} = p$ and ${c_n}^{(n)} = 1$.
Therefore, by Eisenstein's criterion,
$\theta_n (z)$ is irreducible and it is the minimal
polynomial of $\omega$.
Let $C_n$ be the companion matrix of $\theta_n (z)$. By substituting
$C_n$ for $\omega$, we have a homomorphism $\gamma : GL(2,\CC)
\rightarrow GL(2n,\ZZ)$, namely, $\gamma (1) = E_n$ and
$\gamma (\omega)= C_n$, where $E_n$ is the identity matrix, and hence we obtain another
representation $\eta=\xi \circ \gamma : D_p \rightarrow GL(2n,\ZZ)$.
The following proposition is likely known, but since we are unable to find a
reference, we prove it in Section 11.
\begin{prop}\label{prop:2.1}
Two representations $\pi_0$ and $\eta$ are equivalent.
In other words, there is a matrix $U_n \in GL(2n,\ZZ)$ such that
\begin{equation}
U_n \pi_0 (x) {U_n}^{-1}= \eta (x)\ {\rm and}\
U_n \pi_0 (y)
{U_n}^{-1} = \eta (y).
\end{equation}
\end{prop}
Let $K(r)$ be a $2$-bridge knot in $H(p)$.
Then $\tau = \rho \circ \xi :
G(K(r)) \rightarrow D_p \rightarrow GL(2,\CC)$ defines a
representation of $G(K(r))$ and let
$\widetilde{\Delta}_{\tau, K(r)} (t |\omega)$ be
the twisted Alexander polynomial associated to $\tau$.
Sometimes, we use the notation
$\widetilde{\Delta}_{\tau, K(r)} (t|\omega)$ to emphasize that the
polynomial involves $\omega$.
Let $\omega_1, \omega_2, \cdots, \omega_n$
be all the roots of $\theta_n (t)$.
Since $\theta_n (t)$ is
irreducible, the total $\tau$-twisted Alexander polynomial
$D_{\tau, K(r)} (t)$ defined in \cite{SW} is given by
\begin{equation}
D_{\tau, K(r)} (t)=\prod_{j=1}^n
\widetilde{\Delta}_{\tau, K(r)} (t|\omega_j).
\end{equation}
It is known that the polynomial $D_{\tau, K(r)} (t)$ is rewritten as
\begin{equation}
D_{\tau, K(r)} (t)=\det[ \widetilde{\Delta}_{\tau,
K(r)} (t|\omega)]^{\gamma}.
\end{equation}
By (2.5), we see that $D_{\tau, K(r)} (t)$ is exactly the twisted
Alexander polynomial of $K(r)$ associated to $\nu = \rho \circ \eta
:G(K) \rightarrow GL(2n,\ZZ)$.
Since, by Proposition \ref{prop:2.1},
$\pi_0$ and $\eta$ are equivalent,
$\rho_0$ and $\nu$ are equivalent, and
hence $\widetilde{\Delta}_{\rho_0, K(r)} (t) =
D_{\tau, K(r)}(t)$.
Conjecture A now becomes the following theorem under our
assumptions that will be proven in Sections 5-7.
\begin{thm}\label{thm:2.2}
If a 2-bridge knot $K(r)$ is in $H(p)$, then
\begin{equation}
D_{\tau, K(r)} (t) = f(t) f(-t)
\end{equation}
for some integer polynomial $f(t)$,
and further, for any $2$-bridge knot
$K(r)$ with $\Delta_{K(r)}(-1) \equiv 0$ (mod $p$),
\begin{align}
&(1)\ D_{\tau, K(r)} (t) \equiv f(t) f(-t)\ {\rm (mod}\ p)\
{\rm and}\nonumber\\
&(2)\ f(t) \equiv \Bigl\{\dfrac{\Delta_K(t)}{1+t}\Bigr\}^n\
{\rm (mod}\ p),
\end{align}
where $\Delta_{K(r)}(t)$ is the Alexander polynomial of $K(r)$.
\end{thm}
We note that $\Delta_{K(r)}(t)$ is divisible by $1+t$ in $(\ZZ/p)[t^{\pm 1}]$.
\begin{rem}
If $n=1$, i.e., $p=3$, $\theta_1(z)=z+3$, and hence $\omega=-3$.
Therefore, $\gamma$ is an identity homomorphism and
$\widetilde\Delta_{\rho_0,K(r)}(t)=D_{\tau,K(r)}(t)$.
\end{rem}
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\section{Basic formulas}
In this section, we list various formulas involving
$a_k, b_k, c_k$ and $d_k $
which will be used throughout this paper.
Most of these materials are
collected from Section 4 in \cite{HM}.
For simplicity, let
$\xi (x) =X= \mtx{-1}{1}{0}{1}$ and
$\xi (y)=Y = \mtx{-1}{0}{\omega}{1}$,
where $\omega$ is a root of $\theta_n (z)$.
First we list several formulas which are similar to \cite[Proposition 4.2]{HM}
\begin{prop}\label{prop:3.1}
As before, write $(XY)^k$ = $
\mtx{a_k}{b_k}{c_k}{d_k}$.
\begin{align}
&(I)\ a_0 = d_0 = 1\ {\it and}\ b_0 = c_0 = 0.\nonumber\\
&(II)\ a_1 = 1+ \omega, b_1 =1, c_1 = \omega\ {\it and}\ d_1 = 1.\nonumber\\
&(III)\ (i)\ {\it For}\ k \geq 2,\nonumber\\
&\ \ (1)\ a_k = (2+ \omega) a_{k-1} - a_{k-2},\nonumber\\
&\ \ (2)\ \omega b_k = (1+ \omega) a_{k-1} - a_{k-2},\nonumber\\
&\ \ \ \ \ \ \ (ii)\ {\it For}\ k \geq 1,\nonumber\\
&\ \ (3)\ \omega b_k = a_k - a_{k-1},\nonumber\\
&\ \ (4)\ \omega b_k = c_k,\nonumber\\
&\ \ (5)\ a_k = \omega b_k + d_k,\nonumber\\
&\ \ (6)\ d_k = a_{k-1},\nonumber\\
&\ \ (7)\ b_k = b_{k-1} + a_{k-1},\nonumber\\
&\ \ (8)\ c_k + d_k = a_k,\nonumber\\
&\ \ (9)\ a_0 + a_1 + \cdots + a_{k-1} = b_k.
\end{align}
\end{prop}
Since a proof of Proposition \ref{prop:3.1}
is exactly the same as that of Proposition 4.2 in \cite{HM}, we omit the details.
Next three propositions are different from the corresponding proposition
\cite[Proposition 4.4]{HM}, since they depend on defining relations of $D_p$.
\begin{prop}\label{prop:3.2}
Let $p=2n+1$.
\begin{align}
&(1)\ {\it For}\
0 \leq k \leq 2n, a_k = a_{2n-k}\ {\it and}\
a_{2n+1} = a_0.\nonumber\\
&(2)\ {\it For}\ 0 \leq k \leq 2n,
b_k = -b_{p-k}\ {\it and}\ b_p =0.
\end{align}
\end{prop}
{\it Proof.}
Since $(XY)^k = (YX)^{p-k} = Y(XY)^{p-k}Y$, we have\\
$\mtx{a_k}{b_k}{c_k}{d_k}=\mtx{a_{p-k}-\omega b_{p-k}}{-b_{p-k}}
{-\omega a_{p-k}-c_{p-k}+\omega^2 b_{p-k}+\omega d_{p-k}}
{\omega b_{p-k}+d_{p-k}}$
and hence $a_k = a_{p-k} - \omega b_{p-k}$ and $b_k = - b_{p-k}$ which
proves (2). Further, $a_k = a_{p-k} - \omega b_{p-k} = a_{p-k} + \omega
b_k$ and thus, $a_{p-k} = a_{k-1}$ by (3.1)(III)(3). This proves (1).
Finally, it is obvious that $a_p = a_0$.
\fbox{}
\begin{prop}\label{prop:3.3}
Let $p=2n+1$. Then we have the following
\begin{align}
&(1)\ a_0 + a_1 + \cdots + a_{2n} = 0,\nonumber\\
&(2)\ b_1 + b_2 + \cdots + b_{2n} = 0,\nonumber\\
&(3)\ d_0 + d_1 + \cdots + d_{2n} = 0,\nonumber\\
&(4)\ a_n + 2b_n =0.\nonumber\\
&(5)\ {\it If}\
k \equiv \ell\ {\rm (mod}\ p), \
{\it then}\ a_k = a_{\ell}, b_k =
b_{\ell}, c_k = c_{\ell}\ {\it and}\ d_k = d_{\ell}.
\end{align}
\end{prop}
{\it Proof.} First, we see that $(XY)^n X= Y(XY)^n$ implies\\
$\mtx{-a_n}{a_n+b_n}{-c_n}{c_n+d_n}=
\mtx{-a_n}{-b_n}{\omega a_n+c_n}{\omega b_n+d_n}$,
and hence
$a_n + b_n = - b_n$ that proves (4). (5) is immediate, since
$(XY)^p$ = 1. (1) follows from (3.1)(III)(9), since $a_0 + a_1 +
\cdots + a_{2n}=b_{2n+1}=0$. To show (2), use (3.1)(III)(3). Since
$b_0$ = 0, we see $\omega (b_1 + b_2 + \cdots + b_{2n})
=(a_1 - a_0) + (a_2 - a_1) + \cdots + (a_{2n-1} - a_{2n-2}) + (a_{2n} -
a_{2n-1})
=a_{2n} - a_0$ = 0, by (3.2)(1). (3) follows from
(3.1)(III)(6), since $d_0 = 1 = a_0 = a_{2n}$ and $d_0 + d_1 + \cdots
+ d_{2n}
=1 + a_0 + a_1 + \cdots + a_{2n-1}$ = $a_0 + a_1 + \cdots +
a_{2n-1} + a_{2n} = 0$.
\fbox{}
Now we define an algebra $\widetilde {A}(\omega)$ using the group ring
$\ZZ D_p$.
Consider the linear extension $\widehat {\xi}$ of $\xi: \ZZ D_p
\rightarrow M_{2,2}(\ZZ[\omega])$ given by $\widehat {\xi} (x)=X$ and
$\widehat {\xi} (y)=Y$, where $M_{k,k} (R)$ denotes the ring of
$k \times k$ matrices over a commutative ring $R$.
Let ${\widehat {\xi}}^{-1} (0)$
be the kernel of $\widehat {\xi}$.
Then $\widetilde {A}(\omega)=
\ZZ D_p / {\widehat {\xi}}^{-1} (0)$ is a non-commutative
$\ZZ[\omega]$-algebra.
Some elements of ${\widehat {\xi}}^{-1} (0)$ can be
found in Proposition \ref{prop:3.4}
below.
We define $\zeta: \ZZ D_p \rightarrow \widetilde {A}(\omega)$
to be the natural projection.
\begin{prop}\label{prop:3.4}
In $\widetilde{A}(\omega)$, the following formulas hold,
where $1$ denotes the identity of $\widetilde{A}(\omega)$.
\begin{equation}
{\it For}\ 1 \leq k \leq n,
(xy)^k + (yx)^k = ( a_{k-1} + a_k) 1.
\end{equation}
\begin{align}
&(1)\ {\it For}\
1 \leq k \leq n-1, (xy)^k x + y(xy)^k = a_k (x+y),\nonumber\\
&(2)\ (xy)^n x = y(xy)^n=\frac{a_n}{2} (x+y) = -b_n (x+y).
\end{align}
\end{prop}
{\it Proof.}
To prove (3.4), it suffices to show that
$(XY)^k + (YX)^k =(a_{k-1}+ a_k) E_n$.
In fact, for $1 \leq k \leq n$,
\begin{equation*}
(XY)^k + (YX)^k =(XY)^k + (XY)^{p-k}
=
\mtx{a_k+a_{p-k}}{b_k+b_{p-k}}{c_k+c_{p-k}}{d_k+d_{p-k}}.
\end{equation*}
Since $a_k + a_{p-k} = a_k + a_{k-1}$ by (3.2)(1), $b_k + b_{p-k}=0$ by
(3.2)(2), $c_k + c_{p-k} = \omega (b_k + b_{p-k}) =0$ and
$d_k +d_{p-k} = a_{k-1} + a_{2n-k} = a_{k-1} + a_k$ by (3.1)(6) and (3.2)(1),
(3.4) follows immediately.
Next, for $1 \leq k \leq n-1$,
$(XY)^kX + Y(XY)^k= \mtx{-2a_k}{a_k}{\omega a_k}{2(c_k+d_k)}=a_k(X+Y)$,
which proves (3.5)(1). Finally, (3.5)(2) follows, since $(xy)^n x
=y(xy)^n$ and $a_n = -2b_n$.
\fbox{}
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\section{Polynomials over $\widetilde{A}(\omega)$}
In this section, as the first step toward a proof of
Theorem \ref{thm:2.2},
we introduce one of our key concepts in this paper.
\begin{dfn}
Let $\varphi (t)$ be a polynomial on $t^{\pm 1}$ with
coefficients in the non-commutative algebra $\widetilde{A}(\omega)$. We
say $\varphi (t)$ is {\it split} if $\varphi (t)$ is of the form:\\
$\varphi (t)=\sum_j \alpha_j t^{2j} + \sum_k \beta_k (x+y)
t^{2k+1}$, where $\alpha_j, \beta_k \in \ZZ[\omega]$.
The set of split polynomials is denoted by $S(t)$.
For example, $\varphi(t)$ = $1+t^2, (x+y)t$ are split.
\end{dfn}
First we show that $S(t)$ is a commutative ring.
\begin{prop}\label{prop:4.2}
If $\varphi (t)$ and ${\varphi}^{\prime}(t)$ are split,
so are $\varphi (t) + {\varphi}^{\prime}(t)$ and $\varphi (t)
{\varphi}^{\prime}(t)$.
\end{prop}
{\it Proof.} Let
$\varphi (t)=\sum_j \alpha_j t^{2j} + \sum_k \beta_k (x+y)
t^{2k+1}$ and ${\varphi}^{\prime}(t)=\sum_{\ell}
{\alpha_{\ell}}^{\prime}t^{2\ell} + \sum_m {\beta_m}^{\prime}(x+y)
t^{2m+1}$.
Then obviously $\varphi (t)+ {\varphi}^{\prime}(t)$ is split.
Further,
\begin{align*}
\varphi (t) {\varphi}^{\prime}(t)& = \sum_{j, \ell} \alpha_j
{\alpha_{\ell}}^{\prime}t^{2j+2\ell} + \sum_{j,m} \alpha_j
{\beta_m}^{\prime}(x+y) t^{2j+2m+1}\\
&\ \ + \sum_{k,\ell} \beta_k {\alpha_\ell}^{\prime} (x+y) t^{2k+2\ell+1} +
\sum_{k,m} \beta_k {\beta_m}^{\prime}(x+y) (x+y) t^{2k+2m+2}.
\end{align*}
Since $(x+y)(x+y)=2+xy+yx=(2+b_2)1$ by (3.4) and (3.1)(III)(9), it follows
that $\varphi (t) {\varphi}^{\prime}(t)$ is split.
\fbox{}
Next, to obtain the proposition corresponding to Lemma 4.5 in \cite{HM}, we
define the polynomials over $\widetilde{A}(\omega)$.
Let
$Q_k (t)=1 + (yx)t^2 + (yx)^2 t^4 + \cdots + (yx)^k t^{2k}$ and\\
$P_k (t)=1 + (xy)t^2 + (xy)^2 t^4 + \cdots + (xy)^k t^{2k}$. Note
$Q_k (t) = y P_k (t) y$.
The following proposition is a slight modification of Lemma 4.5 in \cite{HM}.
\begin{prop}\label{prop:4.3}
Let $p=2n+1$.\\
(1) $(y^{-1} t^{-1}) (1-yt) Q_{2n} (t) yt(1-xt) \in S(t)$.\\
(2) $(y^{-1} t^{-1}) \{(1-yt) Q_n (t) yt +
(yx)^{n+1}t^{2n+2}\} (1-xt)
\in S(t)$.\\
(3) $(y^{-1} t^{-1}) \{(1-yt) Q_{3n+1}(t) yt +
(yx)^{3n+2}t^{6n+4}\}
(1-xt) \in S(t)$.\\
(4) $(y^{-1}t^{-1}) (1-yt) Q_{4n}(t) yt (1-xt) \in S(t)$.
\end{prop}
{\it Proof.}
First we prove (2). Since
\begin{align*}
(1-yt) Q_n(t) yt + (yx)^{n+1}t^{2n+2}
&=(1-yt) yP_n(t) t + (yx)^{n+1} t^{2n+2}\\
&=yt (1-yt) P_n (t) + yt (xy)^n x t^{2n+1}\\
&=yt\{(1-yt)P_n (t) + (xy)^n x t^{2n+1}\},
\end{align*}
{\rm it\ suffices\ to\ show}
\begin{equation}
\{(1-yt) P_n (t) + (xy)^n x t^{2n+1}\}(1-xt) \in S(t).
\end{equation}
Now a simple computation shows that
\begin{align*}
&\{(1-yt) P_n (t) + (xy)^n x t^{2n+1}\}(1-xt)\\
&=\left\{\susum{k=0}{n} (xy)^k t^{2k}
- \susum{k=0}{n-1} y(xy)^k t^{2k+1}\right\} (1-xt)\\
&=1 + \susum{k=1}{n} \left\{(xy)^k
+ (yx)^k\right\} t^{2k} - \susum{k=0}{n-1}
\left\{y(xy)^k + (xy)^k x\right\} t^{2k+1}\\
&=1+
\susum{k=1}{n} (a_{k-1} + a_k ) t^{2k} -
\susum{k=0}{n-1} (x+y) a_k t^{2k+1}
\in S(t),
\end{align*}
by (3.4) and (3.5).
This proves (4.1).
{\it Proof of (1).}
Since
\begin{align*}(1-yt) Q_{2n}(t) yt (1-xt)
&=(1-yt) yP_{2n}(t) t
(1-xt)\\
&=yt (1-yt) P_{2n} (t) (1-xt),
\end{align*} it suffices to show
\begin{equation}
(1-yt) P_{2n} (t) (1-xt) \in S(t).
\end{equation}
However, the following straightforward
calculation proves (4.2):
\begin{align*}
&(1-yt) P_{2n}(t) (1-xt)\\
&= {\textstyle\sum\limits_{k=0}^{2n}}(xy)^k t^{2k} -
{\textstyle\sum\limits_{k=0}^{2n}} y(xy)^k t^{2k+1}
- {\textstyle\sum\limits_{k=0}^{2n}} (xy)^k x t^{2k+1}
+ {\textstyle\sum\limits_{k=0}^{2n}} (yx)^{k+1} t^{2k+2}\\
&=1 + {\textstyle\sum\limits_{k=1}^{2n}} \left\{
(xy)^k + (yx)^k\right\} t^{2k} + (yx)^p t^{2p}
- {\textstyle\sum\limits_{k=0}^{2n}}\left\{ y(xy)^k +
(xy)^k x\right\} t^{2k+1}\\
&=1+ {\textstyle\sum\limits_{k=1}^{2n}} (a_{k-1} + a_k) t^{2k} +
t^{2p} - {\textstyle\sum\limits_{k=0}^{2n}}
a_k (x+y) t^{2k+1} \in S(t).
\end{align*}
{\it Proof of (3).}
Since
\begin{align*}
&\left\{(1-yt) Q_{3n+1}(t) yt + (yx)^{3n+2} t^{6n+4}
\right\}(1-xt)\\
&=yt \{(1-yt)
P_{3n+1}(t) + (xy)^{3n+1} x t^{6n+3}\}(1-xt),
\end{align*}
it suffices to show\\
\begin{equation}
\{(1-yt) P_{3n+1}(t) + (xy)^{3n+1}x t^{6n+3}\}
(1-xt) \in S(t).
\end{equation}
Since $P_{3n+1}(t)=P_{2n}(t) + t^{4n+2} P_n (t)$ and $(xy)^{3n+1} x =
(xy)^n x$, we must show\\
$\Bigl\{(1-yt) \{P_{2n}(t) + P_n (t) t^{4n+2}\} +
(xy)^n x t^{6n+3}\Bigr\}(1-xt) \in
S(t)$.
However, since $(1-yt) P_{2n}(t) (1-xt) \in S(t)$ by (4.2), it suffices
to show that
\begin{equation}
\{(1-yt) P_n (t) t^{4n+2} + (xy)^n x t^{6n+3}\}(1-xt) \in S(t).
\end{equation}
Now, (4.4) follows from (4.1), since $t^{4n+2}$ is split.
{\it Proof of (4).}
Since $(yx)^{2n+1} = 1$, we have
\begin{equation*}
Q_{4n}(t)=\susum{k=0}{2n}
(yx)^k t^{2k} + \susum{k=2n+1}{4n} (yx)^k t^{2k}
=(1+t^{2p})Q_{2n}(t).
\end{equation*}
Since $(1+t^{2p})$ is split, it follows that
\begin{equation*}
(y^{-1}t^{-1}) (1-yt) Q_{4n}(t) yt (1-xt)
=(1+t^{2p})(y^{-1}t^{-1})
(1-yt) Q_{2n}(t) yt (1-xt)
\end{equation*} is split by (1).
\fbox{}
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\section{Proof of Theorem 2.2.(I)}
In this section we prove Theorem \ref{thm:2.2}
(2.11) for a torus knot $K(1/p)$, $p =2n+1$ a prime.
First we define various homomorphisms among group rings.\\
Let $g=x^{m_1} y^{m_2} x^{m_3}y^{m_4} \cdots x^{m_{k-1}}
y^{m_k}$, where $m_j$ are integers and let $m = \susum{j=1}{k}
m_j$ and $\ell$ is arbitrary. Then we have:
\begin{align}
&(1)\ {\rho}^\ast : \ZZ G(K) \rightarrow \ZZ D_p [t^{\pm 1}]\
{\rm is\ defined\ by}\ {\rho}^\ast (g)=\rho (g) t^m,\nonumber\\
&(2)\ {\zeta}^\ast:\ZZ D_p [t^{\pm 1}] \rightarrow
\widetilde{A}(\omega)[t^{\pm 1}]\
{\rm is\ defined\ by}\ {\zeta}^\ast
(gt^{\ell})=\zeta (g) t^{\ell},\nonumber\\
&(3)\
{\xi}^\ast: \widetilde{A}(\omega) [t^{\pm 1}] \rightarrow M_{2,2}
(\ZZ[\omega][t^{\pm 1}])\ {\rm is\ defined\ by}\
{\xi}^\ast (gt^{\ell})=\xi (g) t^{\ell},\nonumber\\
&(4)\
{\gamma}^\ast: M_{2,2}(\ZZ[\omega] [t^{\pm 1}]) \rightarrow M_{2n,2n}
(\ZZ[t^{\pm 1}])\ {\rm is\ defined\ by} \nonumber\\
&\ \ \ {\gamma}^\ast
\mtx{\sum_j p_j t^j}{\sum_j q_j t^j}
{\sum_j r_j t^j}{\sum_j s_j t^j}
=\mtx{\sum_j \gamma(p_j) t^j}{\sum_j \gamma(q_j) t^j}
{\sum_j \gamma(r_j) t^j}{\sum_j \gamma(s_j) t^j}.
\end{align}
Now we show the following proposition.
\begin{prop}\label{prop:5.1}
Let $p=2n+1$, a prime. Then $D_{\tau,K(1/p)}$(t) is of
the form $q(t) q(-t)$ for some integer polynomial $q(t)$.
\end{prop}
{\it Proof.}
We write $G(K(1/p))= \langle x,y|
R_0 = W_0 x {W_0}^{-1} y^{-1} =1\rangle$, where
$W_0 = (xy)^n$.
Consider the free derivative of $R_0$ with respect to $x$;
\begin{equation*}
\dfrac{\partial R_0}{\partial x}
=(1-y)
\dfrac{\partial W_0}{\partial x} + W_0
=(1-y) \susum{k=0}{n-1}
(xy)^k + (xy)^n,
\end{equation*}
and we write
\begin{equation*}
{\Phi}^\ast\left( \dfrac{\partial R_0}{\partial x}\right)
=\mtx{h_{11}(t)}{h_{12}(t)}{h_{21}(t)}{h_{22}(t)},
\end{equation*}
where ${\Phi}^\ast=
{\rho}^\ast \circ {\zeta}^\ast \circ {\xi}^\ast$. \\
Then we see;
\begin{align}
(1)\ h_{11}(t)&=\susum{k=0}{n} a_k t^{2k} +
\susum{k=0}{n-1}
a_k t^{2k+1}
=(1+t) \susum{k=0}{n-1} a_k t^{2k} + a_n t^{2n},\nonumber\\
(2)\ h_{12}(t)&=\susum{k=0}{n} b_k t^{2k} +
\susum{k=0}{n-1}
b_k t^{2k+1}
=(1+t) \susum{k=0}{n-1} b_k t^{2k} + b_n t^{2n},\nonumber\\
(3)\
h_{21}(t)&=\susum{k=0}{n} c_k t^{2k} - \omega
\susum{k=0}{n-1} a_k t^{2k+1} - \susum{k=0}{n-1}
c_k t^{2k+1}\nonumber\\
& =- \omega
t \susum{k=0}{n-1} a_k t^{2k} + (1-t)
\susum{k=0}{n-1} c_k t^{2k} + c_n
t^{2n},\nonumber\\
(4)\ h_{22}(t)&=\susum{k=0}{n} d_k t^{2k} - \omega
\susum{k=0}{n-1} b_k t^{2k+1} - \susum{k=0}{n-1} d_k t^{2k+1}
\nonumber\\
&=- \omega t \susum{k=0}{n-1} b_k t^{2k} + (1-t)
\susum{k=0}{n-1} d_k t^{2k} + d_nt^{2n}.
\end{align}
Since $h_{11}(1) = 0$ and $h_{21}(1) = 0$, both $h_{11} (t)$ and
$h_{21}(t)$ are divisible by $1-t$.
In fact, we have:
\begin{align*}
h_{11}(t)&=(1-t) \Bigl\{\ \ \susum{k=0}{n-1}
(2a_0 + 2a_1 + \cdots +2a_{k-1} + a_k)t^{2k}\\
&\hspace*{18mm}
+ \susum{k=0}{n} (2a_0 + 2a_1
+ \cdots + 2a_k)t^{2k+1}\Bigr\}\\
&
=(1-t) \left\{\susum{k=0}{n-1} (b_k + b_{k+1}) t^{2k} +
\susum{k=0}{n-1}
2b_{k+1} t^{2k+1}\right\},\ {\rm and}\\
h_{21}(t)&=- \omega t(1-t^2)
\susum{k=0}{n-2}( a_0 + a_1 + \cdots
+a_k) t^{2k} \\
&\hspace*{15mm}- \omega t(1-t)
(a_0 + a_1 + \cdots +a_{n-1}) t^{2n-2}
+(1-t) \susum{k=1}{n-1} c_k t^{2k}\\
&
=(1-t)\left\{- \omega t(1+t)
\susum{k=0}{n-2} b_{k+1} t^{2k} - \omega tb_n t^{2n-2} + \susum{k=1}{n-1}
c_k t^{2k}\right\}.
\end{align*}
Since $c_k = \omega b_k$, we see
$h_{21}(t) = (1-t) \left\{ - \omega t
\susum{k=0}{n-1}b_{k+1}t^{2k}\right\}$, and hence,\\
$\dfrac{1}{1-t} \det
\left(\dfrac{\partial R_0}{\partial x}\right)^{{\Phi}^\ast}
=\dfrac{1}{1-t} \det \mtx{h_{11}(t)}{h_{12}(t)}
{h_{21}(t)}{h_{22}(t)}
=\det \mtx{h_{11}^{\prime}(t)}{ h_{12}(t)}
{h_{21}^{\prime}(t)}{h_{22}(t)}$,
where
\begin{align*}
{h_{11}}^{\prime}(t)&=\susum{k=0}{n-1}
(b_k + b_{k+1}) t^{2k} +
\susum{k=0}{n-1} 2b_{k+1} t^{2k+1}\\
&=\susum{k=1}{n-1} b_k (1+t^2)t^{2k-2} +
b_nt^{2n-2}+\susum{k=1}{n} 2b_k t^{2k-1},\ {\rm and}\\
{h_{21}}^{\prime}(t)&=- \omega t \susum{k=0}{n-1}
b_{k+1} t^{2k}.
\end{align*}
Let $g(t)= \susum{k=1}{n-1} b_k (1+t^2) t^{2k+2}
+ b_n t^{2n-2}$ and
$h(t) = \susum{k=1}{n} b_k t^{2k-1}$.
Then
\begin{equation*}
{h_{11}}^{\prime}(t) = g(t) +2h(t)\ {\rm and}\
{h_{21}}^{\prime}(t) = - \omega h(t).
\end{equation*}
Further a straightforward computation shows that
\begin{equation*}
{h_{11}}^{\prime}(t) + h_{12}(t)=(1+t) (g(t) + h(t)).
\end{equation*}
And,
\begin{align*}
{h_{21}}^{\prime}(t) + h_{22}(t)
&=- \omega t \susum{k=1}{n} b_k
t^{2k-2} - \omega t \susum{k=1}{n-1}b_k t^{2k} +(1-t)
\susum{k=1}{n-1}
d_k t^{2k} + d_n t^{2n}\\
&=- \susum{k=1}{n} c_k t^{2k-1} -
\susum{k=1}{n-1} c_k t^{2k+1} + (1-t)
\susum{k=0}{n-1} d_k t^{2k} + d_n
t^{2n}.
\end{align*}
Since $c_k + d_k = a_k$ and $d_0 = a_0$, we see
\begin{equation*}
- \susum{k=1}{n-1} c_k t^{2k+1} -
\susum{k=0}{n-1}d_k t^{2k+1} =
- \susum{k=0}{n-1} a_k t^{2k+1},
\end{equation*}
and hence
\begin{equation*}
{h_{21}}^{\prime}(t) + h_{22}(t)=\susum{k=0}{n}
d_k t^{2k} -
\susum{k=0}{n-1}(a_k + c_{k+1}) t^{2k+1}.
\end{equation*}
Now, ${h_{21}}^{\prime}(t) + h_{22}(t)$ is divisible by
$1+t$, and in
fact, we have
\begin{equation*}
{h_{21}}^{\prime}(t) + h_{22}(t)=(1+t)\{g(t) - 2h(t)
- \omega h(t)\}.
\end{equation*}
Therefore,
\begin{align*}
\dfrac{1}{(1-t)(1+t)} \det \left({\Phi}^\ast \dfrac{\partial
R_0}{\partial x}\right)
&=\det
\mtx{g(t) + 2h(t)}{g(t) + h(t)}
{-\omega h(t)}{g(t) - 2h(t) - \omega h(t)}\\
&=\det
\mtx{g(t) + 2h(t)}{- h(t)}
{- \omega h(t)}{ g(t) - 2h(t)},
\end{align*}
and hence
\begin{equation}
\widetilde{\Delta}_{\tau,K(1/p)}(t) = g(t)^2 - (4+w) h(t)^2.
\end{equation}
Now we apply the following key lemma.
\begin{lemm}\label{lem:5.2}
Let $C_n$ be the companion matrix of $\theta_n (z)$, the
minimal polynomial of $\omega$. Then there exists a matrix $V_n \in
GL(n,\ZZ)$ such that
${V_n}^2 = 4 E_n + C_n$.
\end{lemm}
Since our proof involves a lot of computations,
the proof is postponed to Section 11.
Since the total twisted Alexander polynomial of $K(1/p)$ at $\tau$ is
$D_{\tau,K(1/p)}(t) = \det
[\widetilde{\Delta}_{\tau,K(1/p)}(t)]^{{\gamma}^\ast}$, we obtain,
noting that $V_n$ commutes with $C_n$,
\begin{align*}
D_{\tau, K(1/p)}(t)&=\det[g(t|C_n)^2 - {V_n}^2 h(t|C_n)^2]\\
&=\det [g(t|C_n) - V_n h(t|C_n)] \det [g(t|C_n) +V_n h(t|C_n)].
\end{align*}
Let $q(t) = \det [g(t|C_n) - V_n h(t|C_n)]$.
Then since $g(-t) = g(t)$
and $h(-t)= - h(t)$, it follows that
\begin{equation*}
D_{\tau,K(1/p)}(t) = q(t) q(-t).
\end{equation*}
This proves
Theorem \ref{thm:2.2}
(2.11) for $K(1/p)$.
\begin{rem}
It is quite likely that
\begin{equation}
q(t) = (1+t)^n \left\{\Delta_{K(1/p)}(t)
\right\}^{n-1},
\end{equation}
where $\Delta_{K(1/p)}(t)$ is the Alexander polynomial of $K(1/p)$.
\end{rem}
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\section{Proof of Theorem 2.2 (II)}
Now we return to a proof of Theorem \ref{thm:2.2}
(2.11) for a $2$-bridge knot $K(r)$ in $H(p)$.
Let $G(K(r)) = \langle x,y|R\rangle, R=WxW^{-1} y^{-1}$,
be a Wirtinger presentation of $G(K(r))$.
Then as is shown in \cite{HM},
$R$ is written freely as a product of conjugates
of $R_0$:
$R=\prod_{j=1}^s u_j {R_0}^{\epsilon_j}{u_j}^{-1}$,
where for $1 \leq j \leq s$, $\epsilon_j = \pm 1$ and $u_j \in F(x,y)$,
the free group generated by $x$ and $y$, and
$\frac{\partial R}{\partial x}=
\sum_{j} \epsilon_j u_j (\frac{\partial R_0}{\partial x})$, and
hence
\begin{align*}
\widetilde{\Delta}_{\tau, K(r)}(t)
&=\det\left(\frac{\partial R}{\partial
x}\right)^{{\Phi}^\ast}/ \det (y^{{\Phi}^{\ast}}- E_2)\\
&=
\widetilde{\Delta}_{\tau, K(1/p)}(t) \det \bigl(\sum_{j} \epsilon_j u_j\bigr)
^{{\Phi}^{\ast}}.
\end{align*}
As we did in \cite{HM}, we study $\lambda (r)=(\sum_{j} \epsilon_j u_j)
^{{\tau}^\ast}
\in \widetilde{A}(\omega)[t^{\pm 1}]$,
where ${\tau}^\ast =
{\rho}^{\ast} \circ {\zeta}^{\ast}$.
For simplicity, we denote
${\tau}^{\ast}(\lambda (r))$ by
${\lambda_r}^{\ast}(t)$. In fact, it is
a polynomial in $t^{\pm 1}$.\\
Since $K(r) \in H(p)$, the continued fraction of $r$ is of the form:\\
$r=[pk_1, 2m_1, pk_2, \cdots, 2m_{\ell}, pk_{\ell +1}]$, where $k_j$ and
$m_j$ are non-zero integers.\\
First we state the following proposition.
\begin{prop}\label{prop:6.1}
Suppose $K(r)$ and $K(r^{\prime})$ belong to $H(p)$ and
let\\
$r=[pk_1, 2m_1, pk_2, \cdots, 2m_{\ell}, pk_{\ell +1}]$,
$r^{\prime}=[{pk_1}^{\prime}, 2m_1, {pk_2}^{\prime}, \cdots, 2m_{\ell},
pk^{\prime}_{\ell +1}]$ be continued fractions of $r$ and $r^{\prime}$.
Suppose that $k_j \equiv {k_j}^{\prime}$ (mod $4$)
for each $j, 1 \leq j \leq \ell +1$. Then if $y^{-1}t^{-1}
\lambda_r^{\ast}(t)$ is split, so is $y^{-1} t^{-1}
\lambda_{r^{\prime}}^{\ast}(t)$.
\end{prop}
Since a proof is analogous to that of Proposition 6.3 in \cite{HM},
we omit the
details.
Now we study the polynomial ${\lambda_r}^{\ast}(t) \in
\widetilde{A}(\omega) [t^{\pm 1}]$ and we prove that
$y^{-1}t^{-1}
{\lambda_r}^{\ast}(t)$ is split.
As is seen in Section 7 in \cite{HM},
${\lambda_r}^{\ast}(t)$ is written as
${w^\ast}_{2\ell +1}(t)$ and we will prove the following proposition. The
same
notation employed in Section 7 in \cite{HM}
will be used in this section.
\begin{prop}\label{prop:6.2}
$ y^{-1} t^{-1}w^\ast_{2\ell +1}(t) \in S(t)$.
\end{prop}
{\it Proof.} Use induction on $j$.
First we prove $y^{-1}t^{-1}{w^\ast}_1 (t) \in
S(t)$.\\
(1) If $w_1(t) = yt$, then $y^{-1}t^{-1}{w^\ast}_1 (t) = 1$ and hence
$y^{-1}t^{-1}{w^\ast}_1 (t) \in S(t)$.\\
(2) If $w_1 = y - (yx)^{n+1}$, then ${w^\ast}_1 (t) = yt - (yx)^{n+1}
t^{2n+2}$ and\\
$y^{-1}t^{-1}{w^\ast}_1 (t)=1-(xy)^n x t^{2n+1}$ = $1
+ b_n (x+y) t^{2n+1}$ and hence\\
$y^{-1}t^{-1}{w^\ast}_1 (t) \in S(t)$.\\
(3) If $w_1 = - (yx)^{n+1}$, then ${w^\ast}_1 (t) = - (yx)^{n+1}
t^{2n+2}$ and\\ $y^{-1}t^{-1}{w^\ast}_1 (t)
=-(xy)^n x t^{2n+1}=b_n
(x+y) t^{2n+1}$ and hence\\
$y^{-1}t^{-1}{w^\ast}_1 (t) \in S(t)$.
Now suppose $y^{-1}t^{-1}{w^\ast}_{2j - 1} (t) \in S(t)$ for $j \leq
\ell$, and we claim \\
$y^{-1}t^{-1}{w^\ast}_{2\ell + 1}(t) \in S(t)$.
There are three cases to be considered. (See \cite[Proposition 7.1.]{HM}
Case 1. $k_{\ell+1}=1$.
$w_{2\ell + 1}
=
\{(1-y) Q_n y + (yx)^{n+1}\}
\sum_{j} m_j (x-1)
y^{-1} w_{2j-1} - (yx)^{n+1} y^{-1} w_{2\ell-1} + y$.\\
Then
\begin{align*}
y^{-1}t^{-1}{w^\ast}_{2\ell + 1}(t)
&=
y^{-1}t^{-1}\{(1-yt) Q_n
(t) yt \\
&\ \ + (yx)^{n+1}t^{2n+2}\}
\sum_{j} m_j (xt-1)
y^{-1}t^{-1}{w^\ast}_{2j-1}(t)\\
&\ \ - (xy)^n x
t^{2n+1}(y^{-1}t^{-1}{w^\ast}_{2\ell - 1}(t)) + 1.
\end{align*}
By Proposition \ref{prop:4.3}(2),
each summand is split.
Further, $-(xy)^n x t^{2n+1}= b_n (x+y) t^{2n+1} \in S(t)$ and
$1 \in S(t)$. Therefore, the sum of them is split.
Proofs of the other cases are essentially the same.
Case 2. $k_{\ell+1}=2$.\\
$w_{2\ell+1}
=(1-y) Q_{2n} y\{\sum_{j}m_j (x-1)
y^{-1}w_{2j-1}\} + (yx)^{2n+1}w_{2\ell-1} - (yx)^{n+1} + y$.\\
Then
\begin{align*}
y^{-1}t^{-1}{w^\ast}_{2\ell + 1}(t)
&=y^{-1} t^{-1}(1-yt)
Q_{2n}(t) yt \{ \sum_{j} m_j (xt-1)y^{-1}t^{-1}{w^\ast}_{2j - 1}(t)\}\\
&\ \ +y^{-1}t^{-1} t^{4n+2} {w^\ast}_{2\ell - 1}(t) - x(yx)^n t^{2n+1}+1.
\end{align*}
Again, $y^{-1}t^{-1}(1-yt) Q_{2n}(t) yt (xt-1) \in S(t)$ by Proposition
\ref{prop:4.3}(1)
and\\ $y^{-1}t^{-1}{w^\ast}_{2j- 1}(t) \in S(t)$ by induction
hypothesis and $t^{4n+2}$,\\
$- x(yx)^n t^{2n+1} = b_n (x+y) t^{2n+1}$ and
1 are split. Thus, $y^{-1}t^{-1}{w^\ast}_{2\ell + 1}(t) \in S(t)$.
Case 3. $k_{\ell+1}=3$.
\begin{align*}
w_{2\ell+1}
& = \{(1-y) Q_{3n+1}y + (yx)^{3n+2}\}
\sum_{j} m_j
(x-1) y^{-1} w_{2j-1} \\
&\ \ - (yx)^{3n+2} y^{-1}w_{2\ell-1}+(yx)^{p}y - (yx)^{n+1}+y.
\end{align*}
Then
\begin{align*}
y^{-1}t^{-1}{w^\ast}_{2\ell + 1}(t)
&=y^{-1} t^{-1}\{(1-yt)Q_{3n+1}(t) yt\\
&\ \ + (yx)^{3n+2}t^{6n+4}\} \sum_{j} m_j (xt-1)
y^{-1}t^{-1}{w^\ast}_{2j-1}(t)\\
&\ \ - (xy)^n x
t^{6n+3}(y^{-1}t^{-1}{w^\ast}_{2\ell - 1}(t))+t^{2p} - (xy)^n x t^{2n+1}+1.
\end{align*}
We see that $y^{-1}t^{-1}{w^\ast}_{2\ell + 1}(t)$ is split, since
each of
$y^{-1}t^{-1}\{(1-yt) Q_{3n+1}(t) yt
+ (yx)^{3n+2} t^{6n+4}\} (xt-1)$,
$y^{-1}t^{-1}{w^\ast}_{2j - 1}(t)$ and
$- (xy)^n x t^{6n+3}=b_n (x+y) t^{6n+3}$
and $- (xy)^n x t^{2n+1} = b_n (x+y)
t^{2n+1}$ is split.
This proves Proposition \ref{prop:6.1}
\fbox{}
Now a proof of (2.11) for our knots is exactly the same as we did in Section 5.
Since $y^{-1}t^{-1}{w^\ast}_{2\ell + 1}(t) \in S(t)$, we can write
\begin{equation*}
y^{-1}t^{-1}{w^\ast}_{2\ell + 1}(t)
=\sum_{j} \alpha_j t^{2j} +
\sum_{k} \beta_k (x+y) t^{2k+1},
\end{equation*}
where $\alpha_j, \beta_k \in \ZZ[\omega]$.
Define $g(t) = \sum_{j} \alpha_j t^{2j}$ and $h(t) = \sum_{k} \beta_k
t^{2k+1}$.
Since $X+Y =
\mtx{-2}{1}{\omega}{2}$,
\begin{equation*}
{\xi}^\ast [y^{-1}t^{-1}{w^\ast}_{2\ell + 1}(t)]
=\mtx{g(t) - 2h(t)}{h(t)}
{\omega h(t)}{g(t)+2h(t)}\
{\rm and}
\end{equation*}
\begin{equation*}
\det (y^{-1}t^{-1}{w^\ast}_{2\ell + 1}
(t))^{{\xi}^\ast}
=
g(t)^2 - (\omega+4)
h(t)^2.
\end{equation*}
Thus $\widetilde{\Delta}_{\tau,K(r)}(t|\omega)
=\widetilde{\Delta}_{\tau, K(1/p)}(t|\omega)
\bigl\{g(t)^2 - (w+4) h(t)^2\bigr\}$,
and hence, we have
\begin{equation*}
D_{\tau, K(r)}(t)
= D_{\tau, K(1/p)}(t) \det[g(t|C_n)^2 - (C_n +4E_n)
h(t|C_n)^2].
\end{equation*}
Now by Lemma \ref{lem:5.2},
there exists a matrix $V_n \in GL(n, \ZZ)$ such
that ${V_n}^2 = C_n + 4E_n$. Since $V_n$ commutes with $C_n$, we see
\begin{align*}
g(t|C_n)^2 - (C_n +4E_n) h(t|C_n)^2
&=g(t|C_n)^2 - {V_n}^2 h(t|C_n)^2\\
&=\{g(t|C_n) - V_n h(t|C_n)\} \{g(t|C_n) + V_n h(t|C_n)\}.
\end{align*}
Let $f(t) = \det [g(t|C_n) - V_n h(t|C_n)]$.
Since $h(-t|C_n) = -
h(t|C_n)$ and $g(- t|C_n)= g(t|C_n )$, $f(-t) = \det [g(t|C_n) + V_n
h(t|C_n)]$, and thus,
\begin{equation*}
\det [g(t|C_n)^2 - (C_n +4E_n ) h(t|C_n)^2] = f(t) f(-t).
\end{equation*}
Therefore, $D_{\tau, K(r)}(t) = D_{\tau, K(1/p)}(t) f(t) f(-t)$. Since
$D_{\tau, K(1/p)}(t)$ is of the form $q(t) q(-t)$, it follows that
$D_{\tau, K(r)}(t) = F(t) F(-t)$, where $F(t)=q(t) f(t)$.
This proves (2.11) for $K(r)$ in $H(p)$.
\fbox{}
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\section{Proof of Theorem 2.2 (III)}
In this section, we prove (2.12) for a $2$-bridge knot $K(r)$ with
$\Delta_{K(r)} (-1) \equiv 0$ (mod $p$).
First we state the following easy lemma without proof.
\begin{lemm}\label{lem:7.1}
Let $M$ be a $2n \times 2n$ matrix over a commutative ring
which is
decomposed into four $n \times n$ matrices, $A, B, C$ and $D$:
$M =\mtx{A}{B}{C}{D}$.
Suppose that each matrix is lower triangular and in particular, $C$ is
strictly
lower triangular, namely, all diagonal entries are 0. Then
$\det M = (\det A) (\det D)$, and hence, $\det M$ is the product of all
diagonal entries of $M$.
\end{lemm}
Lemma \ref{lem:7.1}
can be proven easily by induction on $n$.
Now let $K(r), 0<r<1$, be a $2$-bridge knot and
consider a Wirtinger
presentation $G(K(r)) = \langle x,y| R\rangle$,
where $R=x^{\epsilon_1}y^{\eta_1} x^{\epsilon_2} y^{\eta_2} \cdots
x^{\epsilon_{\alpha}} y^{\eta_{\alpha}}$ and $\epsilon_j, \eta_j = \pm 1$
for $1\leq j \leq \alpha$.
Applying the free differentiation, we have
$\dfrac{\partial R}{\partial x}
=\susum{i=1}{\alpha} g_i, g_i \in
\ZZ G(K)$,
where
\begin{equation}
g_i=
\begin{cases}\begin{array}{ll}
x^{\epsilon_1}y^{\eta_1} x^{\epsilon_2} y^{\eta_2}
\cdots x^{\epsilon_{i-1}} y^{\eta_{i-1}}& {\rm if\ } \epsilon_i =1\\
-x^{\epsilon_1}y^{\eta_1} x^{\epsilon_2} y^{\eta_2} \cdots
x^{\epsilon_{i-1}} y^{\eta_{i-1}} x^{-1}& {\rm if\ } \epsilon_i = -1.
\end{array}
\end{cases}
\end{equation}
Let $\Psi: \ZZ G(K) \rightarrow \ZZ[t^{\pm 1}]$ be the homomorphism defined by
$ \Psi( g_i)=\epsilon_i t^{m_i}$, where $m_i = \susum{j=1}{i-1}
(\epsilon_j + \eta_j ) + \dfrac{\epsilon_i - 1}{2}$.
Then $\left(\dfrac{\partial R}{\partial x}\right)^\Psi$ gives the Alexander
polynomial $\Delta_{K(r)}(t)$ of $K(r)$.
On the other hand,
$\dfrac{1}{(1-t)(1+t)} \det\left(
\dfrac{\partial R}{\partial x}\right)^{{\Phi}^\ast}$ gives
the twisted Alexander polynomial
$\widetilde{\Delta}_{\rho_0,K(r)}(t|w)$ associated to the irreducible
dihedral representation $\rho_0$, and
further, we see $D_{\tau,K(r)}(t)=\det\left[
\dfrac{1}{(1-t)(1+t)}
(\dfrac{\partial R}{\partial x})^{{\Phi}^\ast}\right]^{
{\gamma}^\ast}$.
Now using (7.1), we compute $\left(\dfrac{\partial R}{\partial
x}\right)^{
{\Phi}^\ast}=\sum_{i} {{\Phi}^\ast}(g_i)$.\\
If $\epsilon_i = 1$, then $m_i$ is even and
\begin{align*}
{{\Phi}^\ast}(g_i) &=
[(xy)^{i-1}]^{\xi} t^{m_i}\\
&=\mtx{a_{i-1}}{b_{i-1}}{c_{i-1}}{d_{i-1}}t^{m_i}.
\end{align*}
If $\epsilon_i = - 1$, then
$m_i$ is odd and
\begin{align*}
{{\Phi}^\ast}(g_i) &= - [(xy)^{i-1} x]^{\xi} t^{m_i}\\
& =-\mtx{-a_{i-1}}{ a_{i-1} + b_{i-1}}
{-c_{i-1}}{ c_{i-1} + d_{i-1}}t^{m_i}.
\end{align*}
Therefore we have
\begin{align*}
(\dfrac{\partial R}{\partial x})^{{\Phi}^\ast}&=\susum{i}{}
{{\Phi}^\ast}(g_i)\\
&={\displaystyle \sum_{m_i = even}}
\mtx{a_{i-1}}{b_{i-1}}{c_{i-1}}{d_{i-1}} t^{m_i}
- {\displaystyle \sum_{m_j = odd}}
\mtx{-a_{j-1}}{a_{j-1} + b_{j-1}}{-c_{j-1}}{c_{j-1} +
d_{j-1}}t^{m_j}.
\end{align*}
We note that as polynomials on $\omega$, the constant terms of $a_{i-1}$
and $d_{i-1}$ both are $1$.
Further, since $c_{i-1}= \omega b_{i-1}$, the
constant term of $c_{i-1} + d_{i-1}$ is also 1, and hence
\begin{equation*}
\sum_{i} [{g_i}^{{\Phi}^\ast}]^{{\gamma}^\ast}
=\mtx{\Delta_{K(r)}(-t)
+ \omega \mu_{11}}{\mu_{12}}
{\omega \mu_{21}}
{\Delta_{K(r)}(t) +
\omega \mu_{11}},\
{\rm where}\ \mu_{ij} \in (\ZZ[\omega])[t^{\pm 1}].
\end{equation*}
If we replace $\ZZ$ by $\ZZ/p$, then $C_n$ is reduced to
$\left[\begin{array}{ccc|c}
0&\cdots&0&0\\ \hline
&&&0\\
&\Hsymb{E} & &\vdots\\
&&&0
\end{array}
\right]
$
and hence
$\sum_{i} [{g_i}^{\Phi^\ast}]^{\gamma^\ast} \equiv \mtx{A}{B}{C}{D}$
(mod $p$),
where $A,B,C$ and $D$ are lower triangular and in particular, $C$ is
strictly lower
triangular, and each diagonal entry of $A$ and $D$ is $\Delta_{K(r)}(t)$
(mod $p$) and $\Delta_{K(r)}(-t)$ (mod $p$), respectively.
Therefore, by Lemma \ref{lem:7.1}, we have
\begin{align*}
D_{\tau,K(r)}(t) &\equiv
\det (\susum{i}{} [{g_i}^{\Phi^\ast}]^{{\gamma}^\ast}) /
\det [(1-t)(1+t)]^{\gamma^\ast}\\
& \equiv
\left\{\dfrac{\Delta_{K(r)}(t)}{1+t}\right\}^n
\left\{
\dfrac{\Delta_{K(r)}(-t)}{1-t}\right\}^n\
{\rm (mod}\ p).
\end{align*}
This proves (2.12) for any $2$-bridge knot $K(r)$ with $\alpha \equiv 0$
(mod $p$).
We note that $\Delta_{K(r)}(t)$ is divisible by $1+t$ over
$(\ZZ/p)[t^{\pm1}]$.
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\section{$N(q,p)$-representations}
In this section, we discuss another type of
metacyclic representations
and the twisted Alexander polynomial associated to
these representations.
Let $q \geq 1$ and $p = 2n+1$ be an odd prime.
Consider a metacyclic group,
$N(q,p)= \ZZ/2q {\small \marusen} \ZZ/p$
that is a semi-direct product of $\ZZ/2q$ and
$\ZZ/p$ defined by
\begin{equation}
N(q,p) = \langle
s,a| s^{2q}= a^p = 1 , sas^{-1}=a^{-1}
\rangle.
\end{equation}
Note that $N(1,p)=D_p$ and
$N(2,p)$ is a binary dihedral group, denoted by $N_p$.
Since $s^2$
generates the center of $N(q,p)$, we see that
$N(q,p)/\langle s^2\rangle =D_p$
and hence
$|N(q,p)| = 2pq$.
For simplicity, we assume hereafter that $\gcd(q,p)= 1$.
Now it is known \cite{Ha-M}, \cite{Ha}
that the knot group $G(K)$ of a knot $K$ is mapped
onto $N(q,p)$ if and only if $G(K)$ is mapped onto $D_p$,
namely, $\Delta_K (-1) \equiv 0$ (mod $p$).
For a 2-bridge knot $K(r)$, if
$\Delta_{K(r)} (-1) \equiv 0$ (mod $p$), then we may
assume without loss of generality that there is an epimorphism
$\widetilde{\rho}: G(K(r))
\longrightarrow N(q,p)$ for any $q \geq 1$ such that
\begin{equation}
\widetilde{\rho}(x)= s\ {\rm and}\
\widetilde{\rho}(y) = sa.
\end{equation}
As before, we draw a diagram below consisting of various groups and
connecting homomorphisms.
$
\begin{array}{
cc
ccc
cc}
& & & &GL(2qp,\ZZ)& &\\
& & &\hspace*{5mm} \mbox{\Large $\nearrow$}\mbox{{\large $\tilde{\xi}$}}& & &\\
& &\hspace*{5mm} N(q,p) &
\hspace*{5mm} \overset{\mbox{\large $\tilde{\pi}$}}{\longrightarrow}&
GL(2n,\CC)&
\overset{\mbox{\large $\tilde{\gamma}$}}{\longrightarrow}&GL(2nm,\ZZ)\\
& \mbox{\large $\tilde{\rho}$}\mbox{\Large $\nearrow$}& & &&& \\
G(K)& \overset{\mbox{{\large $\rho_p$}}}{\longrightarrow}&N_p&
\overset{\mbox{\large $\xi_p$}}{\longrightarrow}&SU(2,\CC)&
\overset{\mbox{\large $\gamma_p$}}{\longrightarrow}&
GL(4n,\ZZ)\\
& \mbox{\large $\rho$}\mbox{\Large $\searrow$}&&
\hspace*{5mm} &&&\\
&&D_p&\hspace*{5mm}\overset{\mbox{{\large $\pi_0$}}}\longrightarrow&GL(2n,\ZZ)&&\\
&&&\hspace*{5mm} \mbox{\Large $\searrow$}\mbox{\large$\pi$}&&&\\
&&&&GL(p,\ZZ)&&
\end{array}
$\\
Here, $p=2n+1, \hat{\rho}=\rho\circ\pi,\rho_0=\rho\circ\pi_0,
\tilde\nu=\tilde\rho\circ\tilde\xi,\tilde\tau=\tilde\rho\circ\tilde\pi$,$\tau_p=\rho_p\circ\xi_p$
and $m$ is the degree of the minimal polynomial of
$\zeta$ over $\QQ$.
\\
Using the irreducible representation $\pi_0$ of $D_p$ on
$GL(2n,\ZZ)$, we can
define a representation of $N(q,p)$ on $GL(2n,\CC)$.
In fact, we have
\begin{lemm}\label{lem:newSec8.1}
Let $\zeta$ be a primitive $2q$-th root of $1$, $q \geq 1$.
Then the mapping
$\widetilde{\pi}: N(q,p)
\longrightarrow GL(2n,\CC)$ defined by
\begin{align}
\widetilde{\pi} (s) &= \zeta \pi_0 (x)\ {\it and}\nonumber\\
\widetilde{\pi} (sa) &= \zeta \pi_0 (y)
\end{align}
gives a representation of $N(q,p)$ on $GL(2n,\CC)$.
\end{lemm}
Since a proof is straightforward, we omit details.
Now $\widetilde{\tau}= \widetilde{\rho} \circ \widetilde{\pi}:
G(K(r)) \longrightarrow GL(2n,\CC)$ defines a metacyclic
representation of $G(K(r))$. Then the twisted Alexander polynomial
$\widetilde{\Delta}_{\tilde{\tau}, K(r)}(t|\zeta)$
of $K(r)$ associated to $\widetilde{\tau}$ is given by
\begin{equation}
\widetilde{\Delta}_{\tilde\tau, K(r)}(t|\zeta) =
\widetilde{\Delta}_{\rho_0, K(r)}(\zeta t),
\end{equation}
where ${\rho}_0 =\rho \circ {\pi}_0$.
Therefore, the total twisted Alexander polynomial is
\begin{equation}
D_{\tilde{\tau}, K(r)}(t)= \prod_{(2q,k)=1}
\widetilde{\Delta}_{\rho_0, K(r)}({\zeta}^k t).
\end{equation}
This proves the following theorem.
\begin{thm}
Let $p=2n+1$ be an odd prime and $q \geq 1$.
Let $K(r)$ be a 2-bridge knot.
Suppose $\Delta_{K(r)}(-1)\equiv 0$ (mod $p$).
Then $G(K(r))$ has a metacyclic representation
\begin{equation*}
\widetilde{\tau}=\widetilde{\rho} \circ \widetilde{\pi}:
G(K(r)) \longrightarrow N(q,p) \longrightarrow GL(2n,\CC).
\end{equation*}
Let $\zeta$ be a primitive $2q$-th root of $1$.
Then the twisted Alexander
polynomial $\widetilde{\Delta}_{\tilde{\tau}, K(r)}(t)$ and the total
twisted Alexander polynomial $D_{\tilde{\tau}, K(r)}(t)$
associated to $\widetilde{\tau}$ are given by
\begin{align}
&(1)\ \widetilde{\Delta}_{\tilde{\tau}, K(r)}(t)
=\widetilde{\Delta}_{\rho_0, K(r)}(\zeta t).\nonumber\\
&(2)\ D_{\tilde{\tau}, K(r)}(t)=
\prod_{(2q,k)=1}\widetilde{\Delta}_{\rho_0,K(r)}({\zeta}^k t).
\end{align}
\end{thm}
We conclude this section with a few remarks.
First, as we mentioned earlier, if $q=2$, $N(2,p)$ is a binary dihedral
group, denoted by $N_p$.
It is known \cite{Kl} \cite{L2}
that generators $s$ and $sa$ of $N_p$ are represented
in $SU(2, \CC)$ by trace free matrices.
In fact, the mapping $\xi_p$:
\begin{equation}
\xi_p (s) =\mtx{0}{1}{-1}{0}\
{\rm and}\
\xi_p (sa) =\mtx{0}{v_p}{-v_p^{-1}}{0}
\end{equation}
gives a representation of $N_p$ into $SU(2, \CC)$,
where $v_p=e^{\frac{2\pi i}{p}}$.
Then we will show that the total twisted Alexander polynomial
$D_{{\tau}_p, K(r)}(t)$
associated to ${\tau}_p = {\rho}_p \circ {\xi}_p$ is given by
\begin{equation}
D_{{\tau}_p, K(r)}(t)= \widetilde{\Delta}_{\rho_0,K(r)}(it)
\widetilde{\Delta}_{\rho_0, K(r)}(-it),
\end{equation}
where $i = \sqrt{-1}$.
Therefore we have the following corollary.
\begin{cor}
If $q=2$, then $D_{\tilde\tau, K(r)}(t) =
D_{\tau_p, K(r)}(t)$.
\end{cor}
{\it Proof of (8.8).}
Let $C_p$ be the companion matrix of the minimal
polynomial of $v_p$, namely,
$C_p=
\left[\begin{array}{ccc|c}
0&\cdots&0&-1\\\hline
& & &-1\\
& \hsymb{E}& &\vdots\\
& & &-1
\end{array}
\right].
$
Then, by definition, we have
\begin{equation}
D_{\tau_p, K(r)} (t) = \det[\widetilde{\Delta}_{\tau_p,K(r)}(t|C_p)].
\end{equation}
And (8.8) follows from the following lemma.
\begin{lemm}\label{lem:newSec8.4}
Let $E^*_{2n} = \left[a_{j,k}\right]$ be a
$2n \times 2n$ matrix such that
$a_{j,k}= 1$, if $k+j =2n+1$ and $0$, otherwise
($E_{2n}^{*}$ is the \lq mirror image\rq\ of $E_{2n}$.)
Denote\\
\begin{align*}
&A = \mtxc{0}{E_{2n}}{-E_{2n}}{0},
B = \mtxc{0}{C_p}{-C_p^{-1}}{ 0},\ {\it and}\\
&\widehat A = \mtxc{i E_{2n}^{*}}{0}{0}{ -iE_{2n}^{*}},
\widehat B =\mtxc{i\pi_0(y)}{0}{0}{-i\pi_0(y)}.
\end{align*}
Then there exists a matrix $M_{4n} \in GL(4n,\CC)$ such that
$M_{4n}A {M_{4n}}^{-1}=\widehat A$ and $M_{4n} B {M_{4n}}^{-1} =
\widehat B$.
\end{lemm}
{\it Proof.}
A simple computation shows that
$M_{4n}=\frac{1}{\sqrt{2}}\mtx{E_{2n}}{-iE_{2n}^{*}}{E_{2n}}{iE_{2n}^{*}}$
is what we sought.
\fbox{}
Secondly, the metacyclic group $N(q,p)$ is also represented by $\widetilde{\xi}$
in
$GL(2qp,\ZZ)$ via
{\it maximum} permutation representation
on the symmetric group $S_{2qp}$.
To be more precise, let
\begin{align*}
S=\{1,s,s^2,\cdots,s^{2q-1},\
& a,sa,s^2a,\cdots,s^{2q-1}a,\
a^2,sa^2,s^2a^2,\cdots,s^{2q-1}a^2,\ \cdots,\\
& a^{p-1},sa^{p-1},s^2a^{p-1},\cdots,s^{2q-1}a^{p-1}\}
\end{align*}
be the ordered set of the elements of $N(q,p)$.
Then the right
multiplication by an element $g$ of $N(q,p)$
on $S$ induces a permutation associated to $g$, and by taking the
permutation matrix corresponding to this permutation,
we obtain the representation $\widetilde\xi$
of $N(q,p)$ on $GL(2qp,\ZZ)$.
Then we have the following:
\begin{prop}\label{prop:8.5}
For any $q\ge 1$, the twisted Alexander polynomial
$\widetilde{\Delta}_{\tilde\nu,K(r)}(t)$ of $K(r)$
associated to $\widetilde{\nu}=\widetilde{\rho} \circ \widetilde{\xi}$
is given by
\begin{equation}
\widetilde{\Delta}_{\tilde{\nu}, K(r)}(t)
=
\frac{\displaystyle{\prod_{k=0}^{2q-1}} \Delta_{K(r)}(\zeta^k t)}{1-t^{2q}}
\prod_{k=0}^{2q-1} \widetilde{\Delta}_{\rho_0, K(r)}(\zeta^k t),
\end{equation}
where $\zeta$ is a primitive $2q$-th root of $1$.
Therefore,
$\widetilde{\Delta}_{\tilde{\nu},K(r)}(t)$ is an integer polynomial
in $t^{2q}$ and $D_{\tilde{\tau},K(r)}(t)$ divides
$\widetilde{\Delta}_{\tilde{\nu},K(r)}(t)$.
\end{prop}
{\it Proof.}
By construction, $\widetilde\xi(s)=\rho(x) \otimes C$ and
$\widetilde\xi(sa)=\rho(y) \otimes C$, where $C$ is the transpose of the
companion matrix of $t^{2q}-1$ and $[a_{i,j}] \otimes C=[a_{i,j}C]$,
the tensor product of $[a_{i,j}]$ and $C$.
Therefore (8.10) follows immediately.
\fbox{}
If Conjecture A holds for $K(r)$, $\widetilde{\Delta}_{\tilde{\nu},
K(r)}(t)$ is of the form:
\begin{equation*}
\widetilde{\Delta}_{\tilde{\nu}, K(r)}(t) =
\frac{\prod_{k=0}^{2q-1} \Delta_{K(r)}({\zeta}^k t)}{1-t^{2q}}
f(t^{2q})^2,
\end{equation*}
for some integer polynomial $f(t^{2q})$ in $t^{2q}$.
If coefficients are taken from a finite field, then (8.10)
becomes much simpler. The following proposition is a metacyclic version of
(2.12). Since a proof is easy, we omit details.
\begin{prop}
Let $p$ be an odd prime. Suppose $\Delta_{K(r)}(-1)\equiv 0$ (mod $p$). Then we have
\begin{equation}
\widetilde\Delta_{\tilde\nu,K(r)}(t)\equiv
\left\{\prod_{k=0}^{2q-1} \Delta_{K(r)}(\zeta^k t)\right\}^p / (1-t^{2q})^p\ {\rm (mod}\
p\ {\rm )}.
\end{equation}
\end{prop}
\begin{rem}
In \cite{cha},
Cha defined the twisted Alexander invariant of a fibred
knot $K$ using its Seifert fibred surface. Evidently, this invariant is
closely related to our twisted Alexander polynomial.
For example, as is described in \cite[Example]{cha}, if we
consider a regular dihedral representation $\rho$, then the invariant he
defined is essentially
the same as the twisted Alexander polynomial associated to a regular
dihedral representation $\widetilde{\nu}=\widetilde{\rho} \circ
\widetilde{\xi}: G(K) \rightarrow D_p = N(1,p) \rightarrow
GL(2p,\ZZ)$ we discussed
in this section. More precisely, let $A_{\rho,K}(t)$ be Cha's twisted
Alexander
invariant associated to $\rho$ and
$\widetilde{\Delta}_{\widetilde{\nu},K}(t)$ the twisted Alexander
polynomial of
a knot $K$ associated to $\widetilde{\nu}$. Then we have
\begin{equation}
(1-t^2) \widetilde{\Delta}_{\widetilde{\nu},K}(t)=
A_{\rho,K}(t^2).
\end{equation}
We should note that $\widetilde{\Delta}_{\widetilde{\nu},K}(t)$ is an
integer polynomial in $t^2$.
(See Proposition \ref{prop:8.5}) Details will appear elsewhere.
\end{rem}
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\section{Example}
The following examples illustrate our main theorem.
\begin{ex}
Dihedral representations
$\tau: G(K(r))\longrightarrow D_p\longrightarrow GL(2,\CC)$.
(I) Let $p=3$ and $n=1$. Then $\theta_1(z) = z+3$ and
$\omega = -3$.\\
(a) $r=1/3$. $D_{\tau,K(1/3)}(t)=\widetilde{\Delta}_{\rho_0,K(1/3)}(t) = 1-t^2$.\\
(b) $r= 1/9$. $D_{\tau,K(1/9)}(t)=
\widetilde{\Delta}_{\rho_0,K(1/9)}(t)=(1-t^2)(1-t^3+t^6)(1 +t^3 +t^6)$.\\
(c) $r=5/27$. $D_{\tau,K(5/27)}(t)=\widetilde{\Delta}_{\rho_0,K(r)}(t)
=(1-t^2) (1+t -t^2
+t^3 +t^4) (1 -t -t^2 -t^3 +t^4)$.
Note $\Delta_{K(5/27)}(t)=(1 -t +t^2)
(2 - 2t +t^2 -2t^3 +2t^4)$ and\\
$2 - 2t +t^2 -2t^3 +2t^4 \equiv - (1 -t -t^2 -t^3 +t^4)$ (mod $3$), and
\begin{align*}
D_{\tau,K(5/27)}(t)
&\equiv \dfrac{\Delta_{K(r)}(t)}{1+t}
\dfrac{\Delta_{K(r)}(-t)}{1-t}\\
&\equiv
\dfrac{(1+t)^2 (1 -t -t^2-t^3 +t^4)}{1+t}
\dfrac{(1-t )^2 (1+t -t^2+t^3 +t^4)}{1-t} \\
&\equiv (1-t^2) (1-t -t^2 -t^3 +t^4)
(1+t -t^2 +t^3 +t^4) \
{\rm (mod}\ 3).
\end{align*}
(II) Let $p=5$ and $n=2$. Then $\theta_2 (z) = z^2 + 5z +5$.\\
(a) $r=1/5$. $D_{\tau,K(r)}(t)=(1-t^2)^2
\Delta_{K(1/5)}(t) \Delta_{K(1/5)}(-t)$.\\
(b) $r= 19/85$ $D_{\tau,K(r)}(t)
=D_{\tau,K(1/5)}(t)f(t) f(-t)$,
where $f(t) = 1- 3t - 2t^2 + 4t^3 - t^4 - 4t^6 -3t^7 + 7t^8 - 3t^9
-4t^{10} -t^{12} +4t^{13} -2t^{14} - 3t^{15}+ t^{16}$, and
$\Delta_{K(r)}(t) = \Delta_{K(1/5)}(t) g(t)$,
where $g(t) = 2 -2t +2t^2 - 2t^3 + t^4 - 2t^5
+ 2t^6 - 2t^7 + 2t^8$, and
$f(t) \equiv g(t)^2$ (mod $5$).
Since $\Delta_{K(1/5)}(t) \equiv (1+t)^4$ (mod $5$),
we see
\begin{align*}
D_{\tau,K(r)}(t)
&= D_{\tau,K(1/5)}(t) f(t) f(-t)\\
&=\left\{(1+t)^2 \Delta_{K(1/5)}(t) f(t) \right\}
\left\{ (1-t)^2
\Delta_{K(1/5)}(-t) f(-t) \right\}\\
&\equiv \{(1+t)^6 g(t)^2\}\{(1-t)^6 g(-t)^2\}\\
&\equiv \{(1+t)^3 g(t)\}^2 \{(1-t)^3 g(-t)\}^2\\
&\equiv\left\{\dfrac{\Delta_{K(1/5)}(t) g(t)}{1+t}\right\}^2
\left\{\dfrac{\Delta_{K(1/5)}(-t) g(-t)}{1-t}\right\}^2\\
&\equiv \left\{\dfrac{\Delta_{K(r)}(t)}{1+t}\right\}^2
\left\{\dfrac{\Delta_{K(r)}(-t) }{1-t}\right\}^2\
{\rm (mod}\ 5).
\end{align*}
(c) $r=21/115$. $D_{\tau,K(r)}(t)= D_{\tau,K(1/5)}(t)f(t) f(-t)$,
where $f(t) = 4 +2t - 3t^2 - t^3 - 8t^5 - 3t^6+4t^7 + t^9 + 9t^{10}
+t^{11} +4t^{13} -3t^{14} -8t^{15} - t^{17} -3 t^{18} +2t^{19}
+4t^{20}$, and
$ \Delta_{K(r)}(t) = \Delta_{K(1/5)}(t) g(t)$,
where $g(t) = 2 -2t +2t^2 - 2t^3 +2t^4 -3t^5 +2 t^6 - 2t^7 + 2t^8 - 2t^9 +
2t^{10}$, and
$f(t) \equiv g(t)^2$ (mod $5$). Therefore, we see
\begin{equation*}
D_{\tau, K(r)}(t) \equiv \left\{
\dfrac{\Delta_{K(r)}(t)}{1+t}
\right\}^2
\left\{\dfrac{\Delta_{K(r)}(-t)}{1-t}\right\}^2\
{\rm (mod}\ 5).
\end{equation*}
\end{ex}
\begin{ex}\label{ex:9.2n}
Binary dihedral representations.
$\tau_p:G(K(r))\longrightarrow N_p
\longrightarrow GL(2n,\CC)$
(I) Let $p=3$ and $n=1$.\\
(a)When $r=1/9$, $D_{\tau_p, K(r)}(t)$ = $(1+t^2)^2(1-t^6+t^{12})^2$.\\
(b)When $r=5/27$, $D_{\tau_p, K(r)}(t) = (1+t^2)^2(1+3t^2+t^4+3t^6+t^8)^2$.
(II) Let $p=5$ and $n=2$.\\
(a) When $r=1/5$, $D_{\tau_p, K(r)}(t) = (1+t^2)^4(1-t^2+t^4-t^6+t^8)^2$.\\
(b) When $r =19/85$, $D_{\tau_p, K(r)}(t) = (1+t^2)^4(1-t^2+t^4-t^6+t^8)^2
f(t)^2$,
where $f(t)=
1+13t^2+26t^4+20t^6+13t^8+22t^{10}+40t^{12}
+33t^{14}+25t^{16}+33t^{18}+40t^{20}+22t^{22}
+13t^{24}+20t^{26}+26t^{28}+13t^{30}+t^{32}$.
\end{ex}
\begin{ex}\label{ex:9.3}
$N(q,p)$-representations.
$\widetilde{\nu}:G(K(r))
\longrightarrow N(q,p)
\longrightarrow GL(2pq,\ZZ)$.
(I) Let $q=4,p=3,N(4,3)=\ZZ/8{\small \marusen} \ZZ/3$.\\
(a) $r=1/3$. $\widetilde\Delta_{\tilde{\nu},K(1/3)}(t)=
(1-t^8)(1+t^8+t^{16})$.\\
(b) $r=1/9$.
$\widetilde\Delta_{\tilde{\nu},K(1/9)}=
(1-t^8)(1+t^8+t^{16})(1+t^{24}+t^{48})^3$.\\
(c) $r=5/27$.
\begin{align*}
&\ \ \ \ \widetilde\Delta_{\tilde{\nu},K(5/27)}\\
&\ \ \ \ \ \ \ \ =
(1-t^8)(1+t^8+t^{16})(16+31t^{8}+16t^{16})^2(1-79t^8+129t^{16}
-79t^{24}+t^{32})^2.
\end{align*}
(II) Let $q=5,p=3, N(5,3)=\ZZ/10{\small \marusen} \ZZ/3$.\\
(a) $r=1/3$. $\widetilde\Delta_{\tilde{\nu},K(1/3)}(t)=
(1-t^{10})(1+t^{10}+t^{20})$.\\
(b) $r=1/9$. $\widetilde\Delta_{\tilde{\nu},K(1/9)}=
(1-t^{10})(1+t^{10}+t^{20})(1+t^{30}+t^{60})^3$.\\
(c) $r=5/27$.
\begin{align*}
&\ \ \ \ \widetilde\Delta_{\tilde{\nu},K(5/27)}\\
&\ \ \ \ \ \ \ \ =
(1-t^{10})(1+t^{10}+t^{20})(1-228t^{10}-314t^{20}-228t^{30}+t^{40})^2\\
&\ \ \ \ \ \ \ \ \ \ \times
(1024+1201t^{20}+1024t^{40}).
\end{align*}
(III) Let $q=3,p=5$, $N(3,5)=\ZZ/6{\small \marusen} \ZZ/5$\\
(a) $r=1/5$. $\widetilde\Delta_{\tilde\nu,K(1/5)}(t)=
(1-t^6)^3(1+t^6+t^{12}+t^{18}+t^{24})^3$.\\
(b) $r=19/85$.
\begin{align*}
&\ \ \widetilde\Delta_{\tilde{\nu},K(19/85)}\\
&\ \ \ \ =
(1-t^6)^3(1+t^6+t^{12}+t^{18}+t^{24})^3\\
&\ \ \ \ \ \ \times
(64+64t^6+48t^{12}+12t^{18}+49t^{24}+12t^{30}+48t^{36}+64t^{42}+64t^{48})\\
&\ \ \ \ \ \ \times
(1-1243t^6+3335t^{12}+1570t^{18}-2423t^{24}+6320t^{30}-992t^{36}\\
&\ \ \ \ \ \ \ \ -2181t^{42} +9451t^{48}-2181t^{54}-992t^{60}+6320t^{66}-2423t^{72}\\
&\ \ \ \ \ \ \ \ +1570t^{78}+3335t^{84}-1243t^{90}+t^{96})^2
\end{align*}
\end{ex}
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\section{$K$-metacyclic representations}
In this section, we briefly discuss $K$-metacyclic
representations of the
knot group.
Let $p$ be an odd prime. Consider a group $G(p-1,p|k)$ that has the
following presentation:
\begin{equation}
G(p-1,p|k) = \langle s,a | s^{p-1}= a^p = 1, sas^{-1} =a^k
\rangle,
\end{equation}
where $k$ is a primitive $(p-1)$-st root of $1$ (mod $p$).
We call $G(p-1,p| k)$ a $K$-metacyclic group according to
\cite{Fox}.
\begin{prop}
Two $K$-metacyclic groups of the same order, $p(p-1)$ say,
are isomorphic.
\end{prop}
{\it Proof.} Let $G(p-1,p| \ell)=\langle
u,b | u^{p-1}= b^p = 1, u b u^{-1} =
b^{\ell}\rangle$ be
another $K$-metacyclic group.
Since $\ell$ is also a primitive $(p-1)$-st
root (mod $p$),
we see that $\ell \equiv k^m$ (mod $p$), $1 \leq m \leq p-2$ for some $m$,
where $m$ and $p-1$ are coprime.
Take two integers $\lambda$ and $\mu$ such that
$m \lambda + (p-1) \mu= 1$.
Then it is easy to show that a homomorphism
$h:\ G(p-1,p|k) \rightarrow G(p-1,p|\ell)$ defined
by $h(s) = u^{\lambda}$ and $h(a) = b$ is in fact an isomorphism.
\fbox{}
The following proposition is also well-known.
\begin{prop} \cite{Fox}\cite{Ha}
Let $p$ be an odd prime.
Suppose that $k$ is a primitive $(p-1)$-st root of 1
(mod $p$). Then the knot group $G(K)$ is
mapped onto $G(p-1,p|k)$
if and only if $\Delta_K(k) \equiv 0$ (mod $p$).
\end{prop}
As is shown in \cite{Fox},
$G(p-1,p|k)$ is faithfully represented in $S_p$ by
\begin{equation}
\sigma (a) = (1 2 3 \cdots p)\ {\rm and}\
\sigma (s) = (k^{p-1} k^{p-2} \cdots k^2 k).
\end{equation}
Let $\pi_*:\ G(p-1,p |k) \rightarrow GL(p, \ZZ)$
be a matrix representation of
$G(p-1,p|k)$ via $\sigma$.
Now, let $K(r)$ be a $2$-bridge knot.
Suppose that $\Delta_K(k) \equiv 0$
(mod $p$) for some
primitive $(p-1)$-st root of $1$ (mod $p$).
Then a homomorphism $\delta:\ G(K(r)) \rightarrow
G(p-1,p | k)$ given by
\begin{equation}
\delta (x) = s\ {\rm and}\ \delta (y)=sa,
\end{equation}
induces a
$K$-metacyclic representation
$\Theta = \delta \circ \pi_* :\
G(K(r)) \rightarrow GL(p,\ZZ)$.
Then Conjecture A states that
\begin{equation}
\widetilde{\Delta}_{\Theta,K(r)} (t) =
\left[\frac{\Delta_{K(r)}(t)}{1-t}\right] F(t^{p-1}).
\end{equation}
We will see that (10,4) holds for the following knots
including a non-$2$-bridge knot.
\begin{ex}
(1) Consider a trefoil knot $K$.
Since $\Delta_K(-2) \equiv 0$ (mod 7) and
$-2$ is a primitive $6$th root of 1 (mod 7),
$G(K)$ is mapped onto $G(6,7|-2)$.
Then $(\delta \circ \sigma) (x)=
\sigma(s) = (132645)$ and
$(\delta \circ \sigma) (y) = \sigma (sa) = (146527)$ and we see
$\widetilde{\Delta}_{\Theta,K} (t)
= \left[\frac{\Delta_K(t)}{1-t}\right] (1-t^6)$.
(2) Let $K=K(1/9)$. Since $K(1/9) \in H(3)$, $G(K(1/9))$ is mapped onto
$G(6,7|-2)$, and
$\widetilde{\Delta}_{\Theta,K} (t) =\left[\frac{\Delta_K(t)}{1-t}\right]
(1-t^6)(1-t^6+t^{12})$.
(3) Let $K=K(5/27) \in H(3)$. Then
$\widetilde{\Delta}_{\Theta,K} (t) =\left[\frac{\Delta_K(t)}{1-t}
\right] (1-t^6)
(1-7t^6+9t^{12}-7t^{18}+t^{24})$.
\end{ex}
\begin{ex}
Consider a knot $K=K(5/9)$.
Since $\Delta_K (t)=2 - 5t + 2t^2$,
$\Delta_K (2)$ = 0 and hence $G(K(5/9))$ is mapped onto $G(m,p|2)$ for
any odd prime $p$, where $m$ is a divisor of $p-1$.
If $p=5$ or $11$,
then $2$ is a primitive $(p-1)$-st root of 1 (mod $p$). We
see then:
(i) For $p=5, \widetilde{\Delta}_{\Theta,K} (t) =
\left[\frac{\Delta_K(t)}{1-t}\right] (1-t^4)$.
(ii) For $p=11, \widetilde{\Delta}_{\Theta,K} (t) =
\left[\frac{\Delta_K(t)}{1-t}\right] (1-t^{10})$.
It is quite likely that we have $\widetilde{\Delta}_{\Theta,K} (t)
=\left[\frac{\Delta_K(t)}{1-t}\right]
(1-t^{p-1})$, for any odd prime $p$ such that $2$
is a primitive $(p-1)$-st root of 1
(mod $p$).
(iii) If $p=7$,then $2$ is a primitive third root of $1$
(mod $7$) and hence
$G(K)$ has a representation $\Theta:\
G(K) \rightarrow G(3,7|2)
\rightarrow GL(7,\ZZ)$ and we obtain
$\widetilde{\Delta}_{\Theta,K} (t) =
\left[\frac{\Delta_K(t)}{1-t}\right] (1-t^3)^2$.
\end{ex}
\begin{ex}
Consider a non-2-bridge knot $K=8_5$ in
Reidemeister-Rolfsen table.
We have a Wirtinger presentation
$G(K)=\langle x,y,z|R_1,R_2\rangle$, where\\
\hspace*{2cm}
$R_1=
(x^{-1}y^{-1}zyxy^{-1}x^{-1}y^{-1})
x(yxyx^{-1}y^{-1}z^{-1}yx)y^{-1}$ and\\
\hspace*{2cm}
$R_2 = (yx^{-1}y^{-1}z^{-1}x^{-1})y (xzyxy^{-1})z^{-1}$.
(10.5)
\setcounter{equation}{5}
Since $\Delta_{K} (t) =
(1-t+t^2 )(1-2t+t^2 -2t^3 +t^4)$, it follows that
$\Delta_{K}(-1) \equiv 0$ (mod $3$) and
$\Delta_{K} (-1) \equiv 0$ (mod $7$), and
further $\Delta_{K} (-2) \equiv 0$ (mod $7$).
Therefore, $G(K)$ is mapped
onto each of the following groups:
$D_3, D_7, N(2,3), N(2, 7)$ and $G(6,7|-2)$,
since $-2$ is a primitive $6$-th root of $1$ (mod 7).
Now we have five representations and computed their twisted Alexander
polynomials.
(1) For $\rho_1:\ G(K) \rightarrow D_3 \rightarrow GL(3,\ZZ)$,
defined
by $\rho_1(x)= \rho_1(z)= \pi \rho (x)$ and
$\rho_1(y) = \pi \rho(y)$,we have
$\widetilde{\Delta}_{\rho_1, K}(t) =
\left[ \frac{\Delta_K(t)}{1-t}\right] f_1(t)
f_1(-t)$, where $f_1 (t)= (1+t)(1+t - 2t^2 +t^3 +t^4 )$.
(2) For $\rho_2:\
G(K) \rightarrow D_7 \rightarrow GL(7,\ZZ)$, defined by
$\rho_2(x)=\rho_2 (y)=\pi \rho (x)$ and
$\rho_2(z) =\pi \rho(y)$, we have
$\widetilde{\Delta}_{\rho_2, K}(t) =
\left[ \frac{\Delta_K(t)}{1-t}\right] f_2 (t)
f_2 (-t)$, where $f_2 (t)=(1+t)^3 (1+2t - 7t^3 -13t^4
-13t^5 -11t^6 -13t^7 -13t^8 -7t^9 +2t^{11} +t^{12})$.
(3) For $\rho_3:\
G(K) \rightarrow N(2,3) \rightarrow GL(12,\ZZ)$, defined
by $\rho_3(x)= \rho_3(z)= \widetilde{\nu} (x)$ and $\rho_3 (y)=
\widetilde{\nu} (y)$, we have
$\widetilde{\Delta}_{\rho_3, K}(t) =
(1+t^2 )^2 (1+5t^2 +4t^4 +5t^6+t^8)^2$.
(4) For $\rho_4:\ G(K) \rightarrow N(2,7) \rightarrow GL(28,\ZZ)$,
defined by $\rho_4(x)=\rho_4 (y)=\widetilde{\nu} (x)$ and $\rho_4
(z)=\widetilde{\nu} (y)$, we have
$\widetilde{\Delta}_{\rho_4, K}(t)=(1+t^2)^6 (1+4t^2 + 2t^4 +19t^6
+13t^8 +37t^{10} +17t^{12} +37t^{14} +13t^{16} +19t^{18} +2t^{20}
+4t^{22} +t^{24})^2$.
(5) For $\rho_5:\
G(K) \rightarrow G(6,7|-2) \rightarrow GL(7,\ZZ)$,
defined by $\rho_5(x)= \rho_5 (z)=\Theta (x)$ and
$\rho_5 (y)=\Theta (y)$, we have
$\widetilde{\Delta}_{\rho_5, K}(t) =
\left[ \frac{\Delta_K(t)}{1-t}\right]F(t)$,
where $F(t)=(1-t^6)(1-72t^6 -82t^{12} -72t^{18} +t^{24})$.
\end{ex}
We note that this example also supports Conjecture A.
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\section{Appendix}
\noindent
{\bf 11.1. Proof of Proposition 2.1.}\\
Let $\theta_n (z) = c_0^{(n)} + c_1^{(n)} z + \cdots + c_n^{(n)}z^n$ be
the polynomial
defined in Section 2.
Here $c_k^{(n)} =\binom{n+k}{2k} +2\binom{n+k}{2k+1}$.
Now we define four $n \times n$
integer matrices $A, A^*, B, B^*$ as follows:
$A=[A_{i,j}]$, where $A_{i,j} = a_{i,n-j+1}$,
$A^* = [A^{*}_{i,j}]$, where $A_{i,j}^{*} = - a_{i,j-1}$,
$B = [B_{i,j}]$, where $B_{i,j} = b_{i,n-j+1}$, and
$B^* = [B_{i,j}^*]$, where $B_{i,j}^* = b_{i,j}$.
Here $a_{j,k}$ and $b_{j,k}$ are given as follows.
\begin{align}
&(1)\ a_{j,j} = b_{j,j} = 1\ {\rm for}\ 1 \leq j \leq n.\nonumber\\
&(2)\ {\rm For}\ 1 \leq j \leq k,
a_{j,k}=\binom{j+k-1}{2j-1}\ {\rm and}\
b_{j,k} = \binom{j+k-2}{2j-2}.\nonumber\\
&(3)\ {\rm If}\ 0 \leq k < j, a_{j,k} = b_{j,k} = 0.
\end{align}
\begin{lemm}\label{lemm:a1}
{\it The following formulas hold.}
\begin{align}
{\it For}\ &0 \leq k \leq n, \nonumber\\
&(1)\ c_k^{(n)} = a_{k+1,n+1} +
a_{k+1,n}.\nonumber\\
{\it For}\ &1 \leq j \leq k,\nonumber\\
&(2)\ b_{j,k} = a_{j,k} - a_{j,k-1},\nonumber\\
&(3)\ b_{j,k} = a_{j-1,k-1} + b_{j,k-1}\ {\it and}\nonumber\\
&(4)\ -2\sum_{k=j}^n b_{j,k} = a_{j-1,n} - c_{j-1}^{(n)} + b_{j,n}.
\end{align}
\end{lemm}
{\it Proof.} Only (4) needs a proof.
Since $\sum_{k=j}^n b_{j,k} =
\sum_{k=j}^n (a_{j,k} - a_{j,k-1}) = a_{j,n}$,
we need to show that $-2a_{j,n} = a_{j-1,n} - c_{j-1}^{(n)} + b_{j,n}$.
However, it follows easily from (11.2) (1)-(3).
\fbox{}
Now these formulas are sufficient to show that the
$2n \times 2n$ matrix $U_n =\mtx{A}{A^*}{B}{B^*}$
is what we sought. Since a proof is straightforward,
we omit the details.
\arraycolsep=1.7pt
\begin{ex}\label{ex:a2}
For $n=4, 5$, $U_n$ are given by
\begin{center}
$U_4=\left[\begin{array}{rrrrrrrr}
4& 3& 2& 1& 0& -1& -2& -3\\
10& 4& 1& 0& 0& 0& -1& -4\\
6& 1& 0& 0& 0& 0& 0& -1\\
1& 0& 0& 0& 0& 0& 0& 0\\
1& 1& 1& 1& 1& 1& 1& 1\\
6& 3& 1& 0& 0& 1& 3& 6\\
5& 1& 0& 0& 0& 0& 1& 5\\
1& 0& 0& 0& 0& 0& 0& 1
\end{array}\right]$, and
$U_5=\left[\begin{array}{rrrrrrrrrr}
5& 4& 3& 2& 1& 0& -1& -2& -3& -4\\
20& 10& 4& 1& 0& 0& 0& -1& -4& -10\\
21& 6& 1& 0& 0& 0& 0& 0& -1& -6\\
8& 1& 0& 0& 0& 0& 0& 0& 0& -1\\
1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\
10& 6& 3& 1& 0& 0& 1& 3& 6& 10\\
15& 5& 1& 0& 0& 0& 0& 1& 5& 15\\
7& 1& 0& 0& 0& 0& 0& 0& 1& 7\\
1& 0& 0& 0& 0& 0& 0& 0& 0& 1
\end{array}\right].
$
\end{center}
\end{ex}
\arraycolsep\labelsep
\noindent
{\bf
11.2. Proof of Lemma 5.2.
}
First we write down a solution $X=V_n$ of the equation
$X^2 = 4E_n+C_n$.
Let us begin with the alternating Catalan series
\begin{equation}
\mu (y) = \sum_{k=0}^\infty b_k y^k,\ {\rm where}\
b_k=\frac{(-1)^k}{k+2} \binom{2k+2}{k+1}.
\end{equation}
Therefore,
$\mu (y) = 1- 2y +5y^2 - 14y^3 + 132y^4 - 429y^5+ 1430y^6
+ \cdots$.
Let $\theta_n (z)=c_0^{(n)}+ c_1^{(n)} z+ \cdots + c_n^{(n)}z^n$
be the polynomial defined in
Section 2.
Using $\theta_n (z)$, we define a new polynomial
$f_n (x)=x^n \theta (x^{-1})=
a_0^{(n)} + a_1^{(n)} x + a_2^{(n)}x^2 + \cdots + a_n^{(n)} x^n$.
For example, $f_1(x) = x \theta_1(x^{-1})=
x(3+x^{-1}) = 3x + 1$, and $f_2(x)= 5x^2+ 5x +1$.
Since $a_k^{(n)} = c_{n-k}^{(n)}$, we see that
\begin{equation}
a_k^{(n)}=
\frac{2n+1}{2n-2k+1} \binom{2n-k}{2n-2k}
=\binom{2n-k+1}{2n-2k+1} + \binom{2n-k}{2n-2k+1}.
\end{equation}
Next, we compute $f_n(x) \mu (y)$ = $\sum_{r,s \geq 0} c_{r,s}^{(n)} x^r
y^s$, where $c_{r,s}^{(n)} = a_r^{(n)}b_s$,
and define integers $d_{k,\ell}^{(n)}$, $0 \leq k, \ell$, as follows:
\begin{equation}
d_{k,\ell}^{(n)}=c_{k,\ell}^{(n)} + c_{k-1,\ell+1}^{(n)}
+ c_{k-2,\ell+2}^{(n)} + \cdots+ c_{0,k+ \ell}^{(n)}
={\displaystyle \sum_{i=0,i+j=k+\ell}^k} a_i^{(n)}b_j.
\end{equation}
Then we claim:
\begin{prop}\label{prop:a3}
$V_n$ = $[ v_{j,k}^{(n)}]_{1 \leq j,k \leq n}$, where
$v_{j,k}^{(n)} = d_{n-j,k-1}^{(n)}$, is a solution.
\end{prop}
\begin{ex}\label{ex:a4}
The following is the list of solutions $V_n, n=1,\dots,5$. \\
\arraycolsep=1.7pt
\begin{center}
$[1]$,
$\mtx{3}{-5}{1}{-2}$,
$
\left[\begin{array}{rrr}
5&-7&14\\
5&-9&21\\
1&-2&5
\end{array}\right]$,
$
\left[\begin{array}{rrrr}
7& -9&18& -45\\
14&-23& 51&-132\\
7&-13& 31& -84\\
1& -2& 5& -14
\end{array}\right]$,
$
\left[\begin{array}{rrrrr}
9& -11& 22& -55& 154\\
30& -46& 99& -253& 715\\
27&-47&108&-286&825\\
9&-17& 41& -112& 330\\
1& -2& 5& -14& 42
\end{array}\right]$.
\end{center}
\end{ex}
\arraycolsep\labelsep
Now, to prove Proposition
\ref{prop:a3}, we need several technical lemmas.
\begin{lemm}\label{lem:a5}
For $n \geq 2$ and $0 \leq k \leq n$, the following recursion
formula holds.
\begin{equation}
a_k^{(n)}=a_k^{(n-1)} + 2a_{k-1}^{(n-1)} -a_{k-2}^{(n-2)}.
\end{equation}
\end{lemm}
For convenience, we define $a_0^{(0)}$ = 1.
Since a direct computation using (11.4) verifies (11.6) easily, we omit
details.
Next, for $n, m \geq 0$, we define a number $F(n, m)$ as follows.
\begin{equation}
F(n, m) = \sum_{j=0}^{n}a_{n-j}^{(n)} b_{m+j}.
\end{equation}
\begin{ex}\label{ex:a6}
We have the following values for $F(n,m)$;
\begin{align*}
(1)\ &(i)\ F(0, 0) = a_0^{(0)} b_0= 1.\\
&(ii)\ F(0, m) = a_0^{(n)}b_m = b_m.\\
(2)\ &(i)\ F(1,0) = a_1^{(1)}b_0 + a_0^{(1)} b_1 = 3 - 2 = 1.\\
&(ii)\ F(1,1) = a_1^{(1)}b_1 +a_0^{(1)}b_2 = -6 + 5 = -1.\\
&(iii)\ F(1, m) = a_1^{(1)} b_m + a_0^{(1)}b_{m+1} = 3 b_m +
b_{m+1}.\\
(3)\ &(i)\
F(2, 0) = a_2^{(2)}b_0 +a_1^{(2)}b_1 + a_0^{(2)}b_2 = 0.\\
&(ii)\ F(2, 1) = 1.\\
&(iii)\ F(2, 2) = - 3.
\end{align*}
\end{ex}
\begin{lemm}\label{lem:a7}
For $n \geq 2$ and $m \geq 0$, the following recursion formula
holds.
\begin{equation}
F(n, m) = F(n-1, m+1) + 2F(n-1, m) - F(n-2, m).
\end{equation}
\end{lemm}
\noindent
{\it Proof.} Use (11.6) to show (11.8) as follows:
\begin{align*}
F(n, m)&=\sum_{j=0}^{n}a_{n-j}^{(n)}b_{m+j}
=\sum_{j=0}^{n}[a_{n-j}^{(n-1)}
+ 2a_{n-1-j}^{(n-1)} -a_{n-2-j}^{(n-2)}]b_{m+j}\\
&=\sum_{j=0}^{n-1}a_{n-1-j}^{(n-1)}b_{m+1+j} +
2\sum_{j=0}^{n-1}a_{n-1-j}^{(n-1)} b_{m+j} -
\sum_{j=0}^{n-2}a_{n-2-j}^{(n-2)} b_{m+j}\\
&=F(n-1, m+1) + 2F(n-1, m) - F(n-2, m).\ \ \ \
\fbox{}
\end{align*}
\begin{lemm}\label{lem:a8}
The following formulas hold.
\begin{align}
&(1)\ {\it For}\ n \geq 1\ {\it and}\
0 \leq k \leq n,
\sum_{j=0}^{n}a_{k-j}^{(k)}b_{j} = a_k^{(n-1)}.\nonumber\\
&(2)\ {\it For}\ n \geq 2\ {\it and}\
0 \leq m \leq n-2, F(n, m) =0.\nonumber\\
&(3)\ {\it For}\ n \geq 1, F(n, n- 1) = 1.\nonumber\\
&(4)\ {\it For}\ n \geq 1, F(n, n) = - (2n-1).
\end{align}
\end{lemm}
\noindent
{\it Proof.}
(1) Use induction on $n$. Since (1) holds for $n = 1$, we may
assume
that it holds for $n$. Further, if $k = 0$, (1) holds trivially, and
hence it suffices
to show (1) for $n = n+1$ and $k = k+1$. Then, by (11.6),
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\arraycolsep\labelsep
Now, to prove Proposition
\ref{prop:a3}, we need several technical lemmas.
\begin{lemm}\label{lem:a5}
For $n \geq 2$ and $0 \leq k \leq n$, the following recursion
formula holds.
\begin{equation}
a_k^{(n)}=a_k^{(n-1)} + 2a_{k-1}^{(n-1)} -a_{k-2}^{(n-2)}.
\end{equation}
\end{lemm}
For convenience, we define $a_0^{(0)}$ = 1.
Since a direct computation using (11.4) verifies (11.6) easily, we omit
details.
Next, for $n, m \geq 0$, we define a number $F(n, m)$ as follows.
\begin{equation}
F(n, m) = \sum_{j=0}^{n}a_{n-j}^{(n)} b_{m+j}.
\end{equation}
\begin{ex}\label{ex:a6}
We have the following values for $F(n,m)$;
\begin{align*}
(1)\ &(i)\ F(0, 0) = a_0^{(0)} b_0= 1.\\
&(ii)\ F(0, m) = a_0^{(n)}b_m = b_m.\\
(2)\ &(i)\ F(1,0) = a_1^{(1)}b_0 + a_0^{(1)} b_1 = 3 - 2 = 1.\\
&(ii)\ F(1,1) = a_1^{(1)}b_1 +a_0^{(1)}b_2 = -6 + 5 = -1.\\
&(iii)\ F(1, m) = a_1^{(1)} b_m + a_0^{(1)}b_{m+1} = 3 b_m +
b_{m+1}.\\
(3)\ &(i)\
F(2, 0) = a_2^{(2)}b_0 +a_1^{(2)}b_1 + a_0^{(2)}b_2 = 0.\\
&(ii)\ F(2, 1) = 1.\\
&(iii)\ F(2, 2) = - 3.
\end{align*}
\end{ex}
\begin{lemm}\label{lem:a7}
For $n \geq 2$ and $m \geq 0$, the following recursion formula
holds.
\begin{equation}
F(n, m) = F(n-1, m+1) + 2F(n-1, m) - F(n-2, m).
\end{equation}
\end{lemm}
\noindent
{\it Proof.} Use (11.6) to show (11.8) as follows:
\begin{align*}
F(n, m)&=\sum_{j=0}^{n}a_{n-j}^{(n)}b_{m+j}
=\sum_{j=0}^{n}[a_{n-j}^{(n-1)}
+ 2a_{n-1-j}^{(n-1)} -a_{n-2-j}^{(n-2)}]b_{m+j}\\
&=\sum_{j=0}^{n-1}a_{n-1-j}^{(n-1)}b_{m+1+j} +
2\sum_{j=0}^{n-1}a_{n-1-j}^{(n-1)} b_{m+j} -
\sum_{j=0}^{n-2}a_{n-2-j}^{(n-2)} b_{m+j}\\
&=F(n-1, m+1) + 2F(n-1, m) - F(n-2, m).\ \ \ \
\fbox{}
\end{align*}
\begin{lemm}\label{lem:a8}
The following formulas hold.
\begin{align}
&(1)\ {\it For}\ n \geq 1\ {\it and}\
0 \leq k \leq n,
\sum_{j=0}^{n}a_{k-j}^{(k)}b_{j} = a_k^{(n-1)}.\nonumber\\
&(2)\ {\it For}\ n \geq 2\ {\it and}\
0 \leq m \leq n-2, F(n, m) =0.\nonumber\\
&(3)\ {\it For}\ n \geq 1, F(n, n- 1) = 1.\nonumber\\
&(4)\ {\it For}\ n \geq 1, F(n, n) = - (2n-1).
\end{align}
\end{lemm}
\noindent
{\it Proof.}
(1) Use induction on $n$. Since (1) holds for $n = 1$, we may
assume
that it holds for $n$. Further, if $k = 0$, (1) holds trivially, and
hence it suffices
to show (1) for $n = n+1$ and $k = k+1$. Then, by (11.6),
\begin{align*}
\sum_{j=0}^{k+1}a_{k+1-j}^{(n+1)}b_j
&=\sum_{j=0}^{k+1}\{a_{k+1-j}^{(n)}
+ 2a_{k-j}^{(n)} -a_{k-1-j}^{(n-1)}\} b_j\\
&=\sum_{j=0}^{k+1}a_{k+1-j}^{(n)}b_j
+ 2\sum_{j=0}^{n}a_{k-j}^{(n)} b_j
- \sum_{j=0}^{k-1}a_{k-1-j}^{(n-1)} b_j\\
&=a_{k+1}^{(n-1)} + 2a_k^{(n-1)} - a_{k-1}^{(n-2)}\\
& = a_{k+1}^{(n)}.
\end{align*}
\noindent
{\it Proof of (2).}
Since
$F(n,m+1)=F(n+1,m)-2F(n,m)+F(n-1,m)$,
it suffices to show that $F(n,0)=0$ if $n\ge 2$.
Now
\begin{align*}
F(n, 0)&=
\sum_{j=0}^{n}a_{n-j}^{(n)} b_{j}
=
\sum_{j=0}^{n}\frac{(2n+1)}{(2j+1)}\binom{n+j}{2j}
\frac{(-1)^{j}}{(j+2)} \binom{2j+2}{j+1}\\
&= (2n+1) \sum_{j=0}^{n} (-1)^j \frac{(n+j)!}{(2j+1)!(n-j)!}
\frac{(2j+2)!}{(j+2)!(j+1)!}\\
&=(2n+1) \sum_{j=0}^{n} (-1)^j \frac{(n+j)! (2j+2)}{(n-j)!(j+2)!(j+1)!}\\
&= (2n+1) \sum_{j=0}^{n} (-1)^j\frac{2 (n+j)!}{(n-j)!(j+2)! j!}.
\end{align*}
Therefore, to prove (2),
it suffices to show
\begin{equation}
\sum_{j=0}^{n} (-1)^j\frac{(n+j)!}{(n-j)!(j+2)! j!} =0
\end{equation}
or equivalently, by multiplying both sides through $n!/(n-2)!$, to show
\begin{equation}
\sum_{j=0}^{n} (-1)^j \binom{n}{j}
\binom{n+j}{j+2} = 0.
\end{equation}
To show (11.11), we apply the following lemma
\cite[Lemma 5.3]{HM2}.
\begin{lemm}\label{lem:a9}
For $N \geq M \geq 0$ and $N \geq K \geq 0$,
\begin{align}
&\binom{N}{K}
\binom{M}{M} - \binom{N-1}{K-1} \binom{M}{M-1}
+\binom{N-2}{K-2} \binom{M}{M-2} - \cdots \nonumber\\
&+(-1)^M \binom{N-M}{K-M} \binom{M}{0} \nonumber\\
&=
\binom{N-M}{K}.
\end{align}
\end{lemm}
Put $N = 2n$, $K = n+2$ and $M = n$ in (11.12).
Since $N - M = n < K$, we see
\begin{equation*}
\binom{2n}{n+2} \binom{n}{n}-\binom{2n-1}{n+1}
\binom{n}{n-1}+
\cdots + (-1)^{n} \binom{n}{2}
\binom{n}{0}
=\binom{n}{n+2}=0,
\end{equation*}
and hence $\sum_{j=0}^{n} (-1)^j
\binom{n+j}{j+2} \binom{n}{j}=0$.
This proves (2).
{\it Proof of (3)}. By (11.8), we see that for $n \geq 2$,
\begin{equation*}
F(n+1, n-2) = F(n, n-1) + 2F(n, n-2) - F(n-1, n-2).
\end{equation*}
Since $F(n+1, n-2)
=F(n, n-2) =0$ by (11.9) (2), it
follows that $F(n, n-1) = F(n-1, n-2)$ , and hence $F(n, n-1)=F(1, 0)= 1$
by Example \ref{ex:a6} (2)(i).
{\it Proof of (4)}. Use (11.8) for $n \geq 1$ to see
\begin{equation*}
F(n+1, n-1) = F(n, n) +
2F(n, n-1) - F(n-1, n-1).
\end{equation*}
Since $F(n+1, n-1)=0$ and $F(n, n-1)= 1$, it follows that
\begin{equation*}
F(n, n)=F(n-1, n-1) - 2
\end{equation*}
and hence,
\begin{equation*}
F(n, n)=F(1,1)-2(n-1)= -1-2n+2=-(2n-1).
\end{equation*}
\fbox{}
We define another number $H_k^{(n)}$ as follows.
For any $n \geq 1$ and $k \geq 2$, we define
\begin{equation}
H_k^{(n)}=\sum_{j=0}^{k} a_j^{(n)} F(n-1,n+k-2-j) -
\sum_{j=0}^{k-2} a_j^{(n-1)} F(n,n+k-3-j).
\end{equation}
For example, $H_2^{(5)}=a_0^{(5)} F(4,5) + a_1^{(5)} F(4,4) +
a_2^{(5)} F(4,3) - a_0^{(4)} F(5,4)= 0$.\\
In particular, we should note;
\begin{equation}
{\rm For}\ k \geq 2, H_k^{(1)} = 0.
\end{equation}
\begin{align*}
{\rm In\ fact},\ H_k^{(1)}&=a_0^{(1)} F(0, k-1) + a_1^{(1)} F(0, k-2)
-a_0^{(0)} F(1, k-2)\\
&=b_{k-1} + a_1^{(1)} b_{k-2} - 3b_{k-2} -b_{k-1}\\
&= 0.
\end{align*}
The last formula we need is the following lemma.
\begin{lemm}\label{lem:a10}
For any $n \geq 1$ and $k \geq 2$, we have
$H_k^{(n)} = 0.$
\mbox{{\rm (11.15)}}
\end{lemm}
\setcounter{equation}{15}
\noindent
{\it Proof.}
We compute $H=H_k^{(n)} - H_k^{(n-1)}$.
By definition, for $n \geq 2$,
\begin{align*}
H &=- \sum_{j=0}^{k-2} a_j^{(n-1)} F(n, n+k-3-j) + a_0^{(n)}F(n-1,
n+k-2)\\
&\ \ \ +a_1^{(n)} F(n-1, n+k-3)
+ \sum_{j=0}^{k-2}(a_{j+2}^{(n)} +a_j^{(n-2)})F(n-1, n+k-4-j) \\
&\ \ \ -\sum_{j=0}^{k} a_j^{(n-1)} F(n-2, n+k-3-j).
\end{align*}
Since $a_{j+2}^{(n)} + a_j^{(n-2)}=a_{j+2}^{(n-1)} + 2a_{j+1}^{(n-1)}$
and $a_1^{(n)}=a_1^{(n-1)}+2a_0^{(n-1)}$ by (11.6),
we see
\begin{align*}
H &=- \sum_{j=0}^{k-2} a_j^{(n-1)} F(n, n+k-3-j) + a_0^{(n)}F(n-1,
n+k-2) \\
&\ \ \ \ +(a_1^{(n-1)}+2a_0^{(n-1)})F(n-1, n+k-3)\\
&\ \ \ \ + \sum_{j=0}^{k-2}(a_{j+2}^{(n-1)} + 2a_{j+1}^{(n-1)} )F(n-1, n+k-4-j)\\
&\ \ \ \ -\sum_{j=0}^{k} a_j^{(n-1)} F(n-2, n+k-3-j).
\end{align*}
Note $a_0^{(n)} =a_0^{(n-1)}$ = 1 to see
\begin{align*}
H & =a_0^{(n-1)}\{-F(n, n+k-3) + F(n-1, n+k-2)\\
&\ \ \ \ \ \ \ \ +2F(n-1, n+k-3) - F(n-2,
n+k-3)\}\\
&\ \ \ \ + a_1^{(n-1)}\{-F(n, n+k-4) + F(n-1, n+k-3) \\
&\ \ \ \ \ \ \ \ +2F(n-1, n+k-4) - F(n-2,
n+k-4)\} +
\cdots \\
&\ \ \ \
+ a_{k-2}^{(n-1)}\{-F(n, n-1) + F(n-1, n) \\
&\ \ \ \ \ \ \ \ +2F(n-1, n-1) - F(n-2,
n-1)\}\\
&\ \ \ \ + a_{k-1}^{(n-1)}\{F(n-1, n-1) +2F(n-1, n-2) - F(n-2, n-2)\}\\
&\ \ \ \
+a_k^{(n-1)} \{ F(n-1, n-2)-F(n-2, n-3)\}.
\end{align*}
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{\it Proof of (4)}. Use (11.8) for $n \geq 1$ to see
\begin{equation*}
F(n+1, n-1) = F(n, n) +
2F(n, n-1) - F(n-1, n-1).
\end{equation*}
Since $F(n+1, n-1)=0$ and $F(n, n-1)= 1$, it follows that
\begin{equation*}
F(n, n)=F(n-1, n-1) - 2
\end{equation*}
and hence,
\begin{equation*}
F(n, n)=F(1,1)-2(n-1)= -1-2n+2=-(2n-1).
\end{equation*}
\fbox{}
We define another number $H_k^{(n)}$ as follows.
For any $n \geq 1$ and $k \geq 2$, we define
\begin{equation}
H_k^{(n)}=\sum_{j=0}^{k} a_j^{(n)} F(n-1,n+k-2-j) -
\sum_{j=0}^{k-2} a_j^{(n-1)} F(n,n+k-3-j).
\end{equation}
For example, $H_2^{(5)}=a_0^{(5)} F(4,5) + a_1^{(5)} F(4,4) +
a_2^{(5)} F(4,3) - a_0^{(4)} F(5,4)= 0$.\\
In particular, we should note;
\begin{equation}
{\rm For}\ k \geq 2, H_k^{(1)} = 0.
\end{equation}
\begin{align*}
{\rm In\ fact},\ H_k^{(1)}&=a_0^{(1)} F(0, k-1) + a_1^{(1)} F(0, k-2)
-a_0^{(0)} F(1, k-2)\\
&=b_{k-1} + a_1^{(1)} b_{k-2} - 3b_{k-2} -b_{k-1}\\
&= 0.
\end{align*}
The last formula we need is the following lemma.
\begin{lemm}\label{lem:a10}
For any $n \geq 1$ and $k \geq 2$, we have
$H_k^{(n)} = 0.$
\mbox{{\rm (11.15)}}
\end{lemm}
\setcounter{equation}{15}
\noindent
{\it Proof.}
We compute $H=H_k^{(n)} - H_k^{(n-1)}$.
By definition, for $n \geq 2$,
\begin{align*}
H &=- \sum_{j=0}^{k-2} a_j^{(n-1)} F(n, n+k-3-j) + a_0^{(n)}F(n-1,
n+k-2)\\
&\ \ \ +a_1^{(n)} F(n-1, n+k-3)
+ \sum_{j=0}^{k-2}(a_{j+2}^{(n)} +a_j^{(n-2)})F(n-1, n+k-4-j) \\
&\ \ \ -\sum_{j=0}^{k} a_j^{(n-1)} F(n-2, n+k-3-j).
\end{align*}
Since $a_{j+2}^{(n)} + a_j^{(n-2)}=a_{j+2}^{(n-1)} + 2a_{j+1}^{(n-1)}$
and $a_1^{(n)}=a_1^{(n-1)}+2a_0^{(n-1)}$ by (11.6),
we see
\begin{align*}
H &=- \sum_{j=0}^{k-2} a_j^{(n-1)} F(n, n+k-3-j) + a_0^{(n)}F(n-1,
n+k-2) \\
&\ \ \ \ +(a_1^{(n-1)}+2a_0^{(n-1)})F(n-1, n+k-3)\\
&\ \ \ \ + \sum_{j=0}^{k-2}(a_{j+2}^{(n-1)} + 2a_{j+1}^{(n-1)} )F(n-1, n+k-4-j)\\
&\ \ \ \ -\sum_{j=0}^{k} a_j^{(n-1)} F(n-2, n+k-3-j).
\end{align*}
Note $a_0^{(n)} =a_0^{(n-1)}$ = 1 to see
\begin{align*}
H & =a_0^{(n-1)}\{-F(n, n+k-3) + F(n-1, n+k-2)\\
&\ \ \ \ \ \ \ \ +2F(n-1, n+k-3) - F(n-2,
n+k-3)\}\\
&\ \ \ \ + a_1^{(n-1)}\{-F(n, n+k-4) + F(n-1, n+k-3) \\
&\ \ \ \ \ \ \ \ +2F(n-1, n+k-4) - F(n-2,
n+k-4)\} +
\cdots \\
&\ \ \ \
+ a_{k-2}^{(n-1)}\{-F(n, n-1) + F(n-1, n) \\
&\ \ \ \ \ \ \ \ +2F(n-1, n-1) - F(n-2,
n-1)\}\\
&\ \ \ \ + a_{k-1}^{(n-1)}\{F(n-1, n-1) +2F(n-1, n-2) - F(n-2, n-2)\}\\
&\ \ \ \
+a_k^{(n-1)} \{ F(n-1, n-2)-F(n-2, n-3)\}.
\end{align*}
By (11.8) and (11.9)(3),(4), we see easily that each term of the summation
is
equal to $0$. This proves $H = 0$.
\fbox{}
Now we are in position to prove Proposition \ref{prop:a3}.
Let $\mathbf{u}_j=(v_{j,1}^{(n)}, v_{j,2}^{(n)}, \cdots, v_{j,n}^{(n)})$
and $\mathbf{w}_k=(v_{1,k}^{(n)}, v_{2,k}^{(n)}, \cdots,
v_{n,k}^{(n)})^{T}$
be, respectively, the $j$-th row vector and the $k$-th column vector of $V_n$.
Then
we must show
\begin{align}
&(1)\ \mathbf{u}_{n-j} \cdot
\mathbf{w}_k = 0\ {\it for}\ (i)\ 0 \leq j \leq
n-3, 1 \leq k \leq n-j-2\ {\it and}\ \nonumber\\
&\ \ \ \
(ii)\ 2\leq j \leq n-1, n-j+1 \leq k \leq n-1.\nonumber\\
&(2)\
\mathbf{u}_{n-j}\cdot \mathbf{w}_{n-j-1}= 1\ {\it for}\
0 \leq j \leq n-2.\nonumber\\
&(3)\
\mathbf{u}_{n-j}\cdot \mathbf{w}_{n-j}= 4,\
{\it for}\ 1 \leq j \leq n-1.\nonumber\\
&(4)\
\mathbf{u}_n\cdot \mathbf{w}_n =
4 - a_1^{(n)} = 4 - c_{n-1}^{(n)},\nonumber\\
&(5)\
\mathbf{u}_{n-j} \cdot
\mathbf{w}_n = - a_{j+1}^{(n)} = - c_{n-j-1}^{(n)}\
{\it for}\ 1 \leq j \leq n-1.
\end{align}
Since (11.16) is obviously true for $n=1$,
we assume hereafter that $n \geq2$.
We introduce new vectors,
$\mathbf{b}_j=(b_j, b_{j+1}, \cdots,
b_{j+n-1})$ for $j \geq 0$ and $\mathbf{a}_k^{(n)}=
(a_k^{(n)}, a_{k-1}^{(n)}, \cdots, a_0^{(n)}, 0 ,
\cdots,0)^T$ for $0 \leq
k \leq n$. Then, from the definition
of $v_{j,k}^{(n)}$, it is easy to see the following:
\begin{align}
&(1)\ {\rm For}\
0 \leq j \leq n-1,
\mathbf{u}_{n-j} = a_j^{(n)} \mathbf{b}_0
+ a_{j-1}^{(n)} \mathbf{b}_1 + \cdots + a_0^{(n)}
\mathbf{b}_j.\nonumber\\
&(2)\ {\rm For}\
1 \leq k \leq n, \mathbf{w}_k = b_{k-1}
\mathbf{a}_{n-1}^{(n)}
+ b_k \mathbf{a}_{n-2}^{(n)} + \cdots + b_{n+k-2}
\mathbf{a}_0^{(n)}.
\end{align}
Since $\mathbf{u}_{n-j}\cdot \mathbf{w}_k=
\sum_{i=0}^{j} a_{j-i}^{(n)}
(\mathbf{b}_i \cdot \mathbf{w}_k)$, we first compute
$\mathbf{b}_i \cdot
\mathbf{w}_k$.
In fact, a straightforward computation shows\\
\hspace*{1.5cm}$\mathbf{b}_i\cdot \mathbf{w}_k=
b_{k-1}( a_{n-1}^{(n)} b_i +
a_{n-2}^{(n)} b_{i +1} + \cdots + a_0^{(n)} b_{n+i-1})\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_k( a_{n-2}^{(n)} b_i + a_{n-3}^{(n)} b_{i +1} + \cdots + a_0^{(n)}b_{n+i-2})
+ \cdots + b_{n+k-2}(a_0^{(n)} b_i)$\\
\hspace*{1.5cm}\hspace*{1.1cm}
$= b_{k-1}(F(n, i-1) - a_n^{(n)} b_{i-1})\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_k ( F(n, i-2) - a_{n-1}^{(n)} b_{i-1} - a_n^{(n)}
b_{i-2})\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ \cdots\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_{k+i-2} ( F(n, 0) - a_{n-i+1}^{(n)} b_{i-1}- \cdots -
a_n^{(n)} b_0)\\
\hspace*{1.5cm}\hspace*{1.5cm}
+b_{k+i-1}(a_{n-1}^{(n-1)} - a_{n-i}^{(n)} b_{i-1}- \cdots
- a_{n-1}^{(n)} b_0)\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ \cdots\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_{n+k-2}(a_i^{(n-1)} - a_1^{(n)} b_{i-1} -
a_2^{(n)} b_{i-2} - \cdots - a_i^{(n)} b_0)\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_{n+k-1}(a_{i-1}^{(n-1)} - a_0^{(n)} b_{i-1}-
a_1^{(n)} b_{i-2} - \cdots - a_{i-1}^{(n)} b_0)\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ \cdots \\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_{n+k+i-2}(a_0^{(n-1)} - a_0^{(n)} b_0)$.
Note that in the above summation, each of the last $i$ terms is
0 by (11.9)(1).
By rearranging this summation, we obtain\\
\hspace*{1.5cm}$\mathbf{b}_i \cdot \mathbf{w}_k=b_{k-1}F(n, i-1) + b_k F(n, i-2) + \cdots\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_{k+i-2}F(n, 0) + F(n-1, k+i-1)\\
\hspace*{1.5cm}\hspace*{1.5cm} - b_{i-1}F(n, k-1) - b_{i-2}F(n, k) -
\cdots - b_0 F(n,k+i-2)$.
Since $0 \leq i \leq j \leq n-1$, we have for $\ell \geq 0$, $i-1- \ell
\leq n-2$ and hence $F(n, i-1- \ell)= 0$. Therefore
\begin{equation}
\mathbf{b}_i \cdot \mathbf{w}_k=F(n-1, k+i-1) -
\sum_{\ell=0}^{i-1} b_{i-1-\ell} F(n, k-1+\ell).
\end{equation}
Case 1. $i= 0$. Then $\mathbf{b}_0 \cdot \mathbf{w}_k=F(n-1, k-1)$. If
$1\leq k \leq n-2$, then
$F(n-1, k-1)= 0$, and hence $\mathbf{u}_n \cdot \mathbf{w}_k=
a_0^{(n)}(\mathbf{b}_0 \cdot \mathbf{w}_k)= 0$.
Further,
\begin{align*}
&\mathbf{u}_n \cdot \mathbf{w}_{n-1}=a_0^{(n)}F(n-1, n-2)=
a_0^{(n)} = 1,\ {\rm and}\\
&\mathbf{u}_n \cdot \mathbf{w}_n=a_0^{(n)}F(n-1, n-1)
= -(2n-3) =4-(2n+1) = 4 - a_1^{(n)}.
\end{align*}
This proves (11.16) for $j= 0$.
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Since $\mathbf{u}_{n-j}\cdot \mathbf{w}_k=
\sum_{i=0}^{j} a_{j-i}^{(n)}
(\mathbf{b}_i \cdot \mathbf{w}_k)$, we first compute
$\mathbf{b}_i \cdot
\mathbf{w}_k$.
In fact, a straightforward computation shows\\
\hspace*{1.5cm}$\mathbf{b}_i\cdot \mathbf{w}_k=
b_{k-1}( a_{n-1}^{(n)} b_i +
a_{n-2}^{(n)} b_{i +1} + \cdots + a_0^{(n)} b_{n+i-1})\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_k( a_{n-2}^{(n)} b_i + a_{n-3}^{(n)} b_{i +1} + \cdots + a_0^{(n)}b_{n+i-2})
+ \cdots + b_{n+k-2}(a_0^{(n)} b_i)$\\
\hspace*{1.5cm}\hspace*{1.1cm}
$= b_{k-1}(F(n, i-1) - a_n^{(n)} b_{i-1})\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_k ( F(n, i-2) - a_{n-1}^{(n)} b_{i-1} - a_n^{(n)}
b_{i-2})\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ \cdots\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_{k+i-2} ( F(n, 0) - a_{n-i+1}^{(n)} b_{i-1}- \cdots -
a_n^{(n)} b_0)\\
\hspace*{1.5cm}\hspace*{1.5cm}
+b_{k+i-1}(a_{n-1}^{(n-1)} - a_{n-i}^{(n)} b_{i-1}- \cdots
- a_{n-1}^{(n)} b_0)\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ \cdots\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_{n+k-2}(a_i^{(n-1)} - a_1^{(n)} b_{i-1} -
a_2^{(n)} b_{i-2} - \cdots - a_i^{(n)} b_0)\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_{n+k-1}(a_{i-1}^{(n-1)} - a_0^{(n)} b_{i-1}-
a_1^{(n)} b_{i-2} - \cdots - a_{i-1}^{(n)} b_0)\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ \cdots \\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_{n+k+i-2}(a_0^{(n-1)} - a_0^{(n)} b_0)$.
Note that in the above summation, each of the last $i$ terms is
0 by (11.9)(1).
By rearranging this summation, we obtain\\
\hspace*{1.5cm}$\mathbf{b}_i \cdot \mathbf{w}_k=b_{k-1}F(n, i-1) + b_k F(n, i-2) + \cdots\\
\hspace*{1.5cm}\hspace*{1.5cm}
+ b_{k+i-2}F(n, 0) + F(n-1, k+i-1)\\
\hspace*{1.5cm}\hspace*{1.5cm} - b_{i-1}F(n, k-1) - b_{i-2}F(n, k) -
\cdots - b_0 F(n,k+i-2)$.
Since $0 \leq i \leq j \leq n-1$, we have for $\ell \geq 0$, $i-1- \ell
\leq n-2$ and hence $F(n, i-1- \ell)= 0$. Therefore
\begin{equation}
\mathbf{b}_i \cdot \mathbf{w}_k=F(n-1, k+i-1) -
\sum_{\ell=0}^{i-1} b_{i-1-\ell} F(n, k-1+\ell).
\end{equation}
Case 1. $i= 0$. Then $\mathbf{b}_0 \cdot \mathbf{w}_k=F(n-1, k-1)$. If
$1\leq k \leq n-2$, then
$F(n-1, k-1)= 0$, and hence $\mathbf{u}_n \cdot \mathbf{w}_k=
a_0^{(n)}(\mathbf{b}_0 \cdot \mathbf{w}_k)= 0$.
Further,
\begin{align*}
&\mathbf{u}_n \cdot \mathbf{w}_{n-1}=a_0^{(n)}F(n-1, n-2)=
a_0^{(n)} = 1,\ {\rm and}\\
&\mathbf{u}_n \cdot \mathbf{w}_n=a_0^{(n)}F(n-1, n-1)
= -(2n-3) =4-(2n+1) = 4 - a_1^{(n)}.
\end{align*}
This proves (11.16) for $j= 0$.
Case 2. $i=1$. Then $\mathbf{b}_1 \cdot \mathbf{w}_k=F(n-1, k) - b_0
F(n,k-1)$.
If 1 $\leq k \leq n-3$, then $F(n-1, k) = F(n, k-1)= 0$ and
$\mathbf{b}_1\cdot \mathbf{w}_k= 0$.
Since $\mathbf{b}_0 \cdot \mathbf{w}_k= 0$, we have
$\mathbf{u}_{n-1} \cdot
\mathbf{w}_k= 0$ for $1 \leq k \leq n-3$.
Further,
\begin{align*}
\mathbf{u}_{n-1} \cdot \mathbf{w}_{n-2}&=
a_1^{(n)}(\mathbf{b}_0 \cdot
\mathbf{w}_{n-2}) + a_0^{(n)} (\mathbf{b}_1 \cdot \mathbf{w}_{n-2})\\
&=a_1^{(n)}F(n-1, n-3) + a_0^{(n)} \{F(n-1, n-2) - b_0 F(n, n-3)\}\\
&=a_0^{(n)} F(n-1, n-2) = 1.
\end{align*}
Also,
\begin{align*}
\mathbf{u}_{n-1} \cdot \mathbf{w}_{n-1} &= a_1^{(n)} F(n-1, n-2) +
a_0^{(n)} \{F(n-1, n-1)- b_0 F(n, n-2)\}\\
&= a_1^{(n)} + a_0^{(n)}(-(2n-3))= 2n+1 -(2n-3) = 4.
\end{align*}
Finally,
\begin{align*}
\mathbf{u}_{n-1} \cdot \mathbf{w}_n&=a_1^{(n)}F(n-1, n-1) +
a_0^{(n)} \{F(n-1, n) - b_0 F(n, n-1)\}\\
&=H_2^{(n)} - a_2^{(n)}=- a_2^{(n)},\ {\rm by\ (11.15)}.
\end{align*}
This proves (11.16)
for $j=1$.
Now we assume that $2 \leq j \leq n-1$ and compute
$\mathbf{u}_{n-j} \cdot
\mathbf{w}_k$, $1 \leq k \leq n$. Then
\begin{align*}
\mathbf{u}_{n-j} \cdot \mathbf{w}_k
&=\sum_{i=0}^{j} a_{j-i}^{(n)} (
\mathbf{b}_i \cdot \mathbf{w}_k)\\
&= a_j^{(n)} (\mathbf{b}_0 \cdot \mathbf{w}_k)
+ \sum_{i=1}^{j} a_{j-i}^{(n)} (\mathbf{b}_i \cdot \mathbf{w}_k)\\
&=a_j^{(n)}F(n-1, k-1) + \sum_{i=1}^{j}a_{j-i}^{(n)} \{F(n-1, k+i-1) \\
&\ \ \ \ -\sum_{\ell=0}^{i-1} b_{i-1-\ell}F(n, k-1+ \ell)\}\\
&= \sum_{i=0}^{j} a_{j-i}^{(n)} F(n-1, k+i-1) -
\sum_{i=1}^{j}a_{j-i}^{(n)} \sum_{\ell=0}^{i-1} b_{i-1-\ell} F(n,
k-1+\ell).
\end{align*}
Therefore, the coefficient of $F(n, k-1+q)$, $0 \leq q \leq i-1$, is equal to
\begin{align*}
\sum_{i=1}^{j} a_{j-i}^{(n)} b_{i-1-q}
&=\sum_{i=q+1}^{j}a_{j-i}^{(n)} b_{i-1-q}\\
&=a_{j-q-1}^{(n-1)}\ {\rm by}\ (11.9)(1),\ {\rm and\ hence}
\end{align*}
\begin{equation}
\mathbf{u}_{n-j} \cdot \mathbf{w}_k=\sum_{i=0}^{j} a_{j-i}^{(n)}
F(n-1, k+i-1) - \sum_{q=0}^{j-1}a_{j-q-1}^{(n-1)}F(n, k-1+q).
\end{equation}
If $1 \leq k \leq n-j-2$, then $k+i-1 \leq k+j-1 \leq n-3$ and also $k-1+q
\leq k+j-2 \leq n-4$,
and hence, $\mathbf{u}_{n-j} \cdot \mathbf{w}_k = 0$.
If $k = n-j-1$, then $\mathbf{u}_{n-j} \cdot \mathbf{w}_{n-j-1}=
a_0^{(n)}F(n-1, n-2)=a_0^{(n)}= 1$.
Further,
$\mathbf{u}_{n-j} \cdot \mathbf{w}_{n-j}=a_0^{(n)} F(n-1, n-1) + a_1^{(n)}
F(n-1, n-2)=-(2n-3) + 2n+1= 4$.
Now suppose $n-j+1 \leq k \leq n-1$. Then by (11.9),
\begin{align*}
\mathbf{u}_{n-j} \cdot \mathbf{w}_k
&=\sum_{i=0}^{j}a_{j-i}^{(n)} F(n-1,k+i-1)
- \sum_{q=0}^{j-1}a_{j-q-1}^{(n-1)} F(n,k-1+q)\\
&= \sum_{i=n-k-1}^{j}a_{j-i}^{(n)} F(n-1, k+i-1) -
\sum_{q=n-k}^{j-1}a_{j-q-1}^{(n-1)} F(n,k-1+q).
\end{align*}
That is exactly $H_{k-(n-j-1)}^{(n)}$ and hence $\mathbf{u}_{n-j} \cdot
\mathbf{w}_k= 0$ for $n-j+1 \leq k \leq n-1$.
Finally, a similar computation shows that
\begin{align*}
\mathbf{u}_{n-j} \cdot \mathbf{w}_n&=\sum_{i=0}^{j} a_{j-i}^{(n)} F(n-1,
n+i-1) - \sum_{q=0}^{j-1}a_{j-1-q}^{(n-1)} F(n,n+q-1)\\
&=H_{j+1}^{(n)} - a_{j+1}^{(n)} F(n-1,n-2)\\
&=- a_{j+1}^{(n)}\\
&= -c_{n-j-1}^{(n)}.
\end{align*}
This proves (11.16) and a proof of Proposition \ref{prop:a3}
is now complete.
\fbox{}
\noindent
{\bf Acknowledgements. }
The first author is
partially supported by MEXT, Grant-in-Aid for
Young Scientists (B) 18740035,
and the second author is
partially supported by NSERC Grant~A~4034.
The authors thank Daniel Silver and Joe Repka
for their invaluable comments.
\end{document}
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\begin{document}
\title[]{Equilibration on average in quantum processes with finite temporal resolution}
\author{Pedro Figueroa--Romero}
\email[]{[email protected]}
\author{Kavan Modi}
\author{Felix A. Pollock}
\affiliation{School of Physics \& Astronomy, Monash University, Victoria 3800, Australia}
\date{\today}
\begin{abstract}
We characterize the conditions under which a multi-time quantum process with a finite temporal resolution can be approximately described by an equilibrium one. By providing a generalization of the notion of equilibration on average, where a system remains closed to a fixed equilibrium for most times, to one which can be operationally assessed at multiple times, we place an upper-bound on a new observable distinguishability measure comparing a multi-time process with a finite temporal resolution against a fixed equilibrium one. While the same conditions on single-time equilibration, such as a large occupation of energy levels in the initial state remain necessary, we obtain genuine multi-time contributions depending on the temporal resolution of the process and the amount of disturbance of the observer's operations on it.
\end{abstract}
\keywords{Suggested keywords}
\maketitle
A fundamental question at the core of statistical mechanics is that of how equilibrium arises from purely quantum mechanical laws in closed systems. This phenomenon is generically known as \emph{equilibration} or \emph{thermalization}, where in the latter case the system relaxes to a thermal state. The dissipative nature of equilibration, however, is at odds with the unitary nature of quantum mechanics. There are three main approaches to resolving this conundrum: \emph{typicality}~\cite{Popescu2006,GogolinPureQStat, gemmer2009, Goldstein_2010, Gogolin, Garnerone_2013}, which argues that small subsystems of a composite are in thermal equilibrium for almost all pure states of the whole; \emph{dynamical equilibration on average}~\cite{Tasaki_1998, Reimann_2008, PhysRevE.79.061103, ShortSystemsAndSub, ShortFinite, XXEisert}, which demonstrates that time-dependent quantities of quantum systems evolve towards fixed values and stay close to them for most times, even if they eventually deviate greatly from it; and the \emph{eigenstate thermalization hypothesis}~\cite{Deutsch1991,Srednicki, Srednicki_1999,PhysRevE.87.012118,Rigol2008, Turner2018}, which argues that the expectation values of a `physical observable' at long times are indistinguishable for an isolated system from a thermal one.
What these approaches have in common, is that they look at the statistical properties of the state of the system at long times; however, finding a system in or close to an equilibrium state does not necessarily imply all observable properties of the system have equilibrated. In particular, when measurements are coarse (i.e. only a subpart is measured), it may be that temporal correlations due to a sequence of observations may contain signatures indicating whether the system is in equilibrium or not. The recent studies of out-of-time-order correlation functions use multi-time statistics to distinguish between thermalised systems and coherent complex systems~\cite{Kitaev}. However, it may be that these multi-time statistics also equilibrate in general; that is, they are most often found close to some average value.\textsuperscript{\footnote{The out-of-time-order correlations require propagating the system back and forth in time. Here we only go forward in time.}}
In this manuscript, we focus on the case of finite temporal resolution for the dynamical equilibration of quantum processes where multiple operations are applied in sequence. We present sufficient conditions for general multi-time observations to relax close to their equilibrium values when the corresponding operations are implemented with an imperfect, or \emph{fuzzy}, clock (or, equivalently, on a system with uniformly fluctuating energies). In particular, we place an upper bound on how distinguishable the statistics of such observations are from those made in equilibrium.
\begin{figure}
\caption{We characterize the attainability of quantum process equilibration, i.e., under what conditions a $k$-step process $\overline{\Upsilon}
\label{Fig: processes}
\end{figure}
We first briefly recapitulate the landmark results of Refs.~\cite{ShortFinite,Reimann_2008}, in which the attainability of observable equilibration on average for a single observation is characterized mainly by two factors: the energy eigenstates of the system having a large overlap with the initial state, and the \emph{scale} set by the operator norm of the observable being measured. Building on the works of Refs.~\cite{ShortFinite,Reimann_2008}, we first ask, for the case of a single measurement, how different is an equilibrium process from an out-of-equilibrium one, when the available clock is fuzzy. We then generalise to considering observations over multiple points in time; here, temporal correlations become relevant. While the original conditions for equilibration to occur still hold, we obtain additional conditions related to the temporal resolution of the observations and how much these disturb the intermediate states. Our results hold for Hamiltonian dynamics of the total system, while the measurements are allowed to be general quantum operations that are coarse and may only act on a sub-part of the whole system.
\section{Equilibration on average}\label{sec: equilibration on avg}
In the past decade, the program of equilibration has focused on upper bounding fluctuations of observable expectation values around equilibrium, from which conclusions about equilibration of the state of the system itself have been drawn~\cite{ShortSystemsAndSub, Gogolin}. The basic mechanism behind equilibration is that of dephasing~\cite{Reimann_2008, Oliveira_2018}, and equilibration will occur as long as the initial state, following a perturbation, has an overlap with many energy eigenstates of the Hamiltonian driving the dynamics. The only further assumption is that there are not too many degenerate energy gaps~\cite{PhysRevA.98.022135}, ensuring that the majority of the system plays a dynamical role~\cite{PhysRevE.79.061103}. Specifically, observable equilibration in the sense of Ref.~\cite{ShortFinite} considers time-independent Hamiltonian dynamics given by a unitary operator $U=\exp\{-i H t\}$, with results depending on energetic properties of the Hamiltonian $H$, such as the number of distinct energy levels $\mathfrak{D}\leq{d}$ and the maximum number of energy gaps $N(\epsilon)$ in an energy window of width $\epsilon>0$.
While the choice of an equilibrium state is arbitrary, an intuitive candidate is the infinite-time-averaged subsystem state
\begin{gather}
\omega := \lim_{T\to\infty}\overline{\rho}^T \qquad \mbox{with} \qquad \overline{X}^T :=\frac{1}{T}\int_0^T X(t)\,dt
\end{gather}
where $\rho$ stands for the initial state of the whole system, time-averaged over a finite time window of width $T$. Notably, this corresponds to a dephasing of the initial state with respect to $H$, i.e. $\omega=\mc{D}(\rho)$ where we define
\begin{gather}
\mc{D}(\cdot):=\sum_nP_n(\cdot)P_n,
\end{gather}
with $P_n$ a projector onto the $n$th eigenspace of $H=\sum_n E_n P_n$. In fact, we can similarly describe general finite-time-averaged evolution by the map
\begin{gather}
\begin{split}
&\mc{G}_T(\cdot):=\sum_{n,m}G_{nm}^{(T)}P_n(\cdot)P_m \quad \mbox{with} \\ &G_{nm}^{(T)}:=\overline{\exp[it(E_m-E_n)]}^T
\end{split}\label{eq: G maps}
\end{gather}
Whenever it is clear that the averaging window is $T$, we will simply denote these by $\mc{G}$ and $G_{nm}$.
In this minimal setting, the authors of Ref.~\cite{ShortFinite} prove an important result on observable equilibration by showing the closeness between the evolved state $\rho(t)$ and $\omega$. Specifically, they upper-bound the temporal fluctuations of the expectation value of a general operator $A$ around equilibrium within a finite-time window as
\begin{gather}
\overline{|\tr[A(\rho(t) - \omega)]|^2}^T \leq \frac{\|A\|^2 N(\epsilon)f(\epsilon{T})}{d_\text{eff}(\rho)},
\end{gather}
where $f(\epsilon{T}) = 1+8\log_2 (\mathfrak{D})/\epsilon{T}$ and $\|\cdot\|$ denotes largest singular value; crucially, the so-called inverse effective dimension (or inverse participation ratio) of the initial state, $d_\text{eff}^{-1}(\rho) := \sum_n[\tr(P_n\rho)]^2$, quantifies the number of energy levels contributing significantly to the dynamics of the initial state $\rho$.
In general, we have the hierarchy $1\leq{d}_\text{eff}(\rho)\leq\mathfrak{D}\leq{d}$, and this result implies that equilibration is attained for large $d_\text{eff}$. It has been argued, on physical grounds, that the effective dimension takes a large value in realistic situations~\cite{Gogolin,Reimann_2008}, increasing exponentially in the number of constituents of generic many-body systems~\cite{XXEisert}, and it has been proven that it takes a large value for local Hamiltonian systems whenever correlations in the initial state decay rapidly~\cite{PhysRevLett.118.140601}. The temporal fluctuations of the expectation values of $A$ around equilibrium constitute a meaningful quantifier of equilibration: a small variance relates to the expectation value of $A$ concentrating around its mean~\textsuperscript{\footnote{Strictly, the full statistics should then display equilibration.}}.
This behaviour of long-time fluctuations around equilibrium has been studied both analytically and numerically in various physical models~\cite{PhysRevLett.109.247205,PhysRevE.87.012106, PhysRevE.88.032913, PhysRevE.89.022101, PhysRevB.101.174312}, as well as for the more restrictive case of thermalization~\cite{Rigol2008,Mori_2018,Gluza_2019,PhysRevLett.123.200604}. Similarly, related questions such as an absence of equilibration~\cite{PhysRevB.76.052203, Nandkishore, Hess, PhysRevLett.120.080603}, or the robustness of equilibration and further relaxation after a perturbation have been investigated~\cite{Robinson1973,PhysRevLett.118.130601,PhysRevLett.118.140601, PhysRevA.98.022135, PhysRevE.98.062103}. Here, we take a related approach towards investigating the behaviour of quantum quantum process when the interrogations are fuzzy in time. Doing so gives focus on an operationally meaningful scenario where we show that observations that have a finite temporal resolution make it hard to tell an out-of-equilibrium process from one that is in equilibrium.
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\section{Equilibration on average}\label{sec: equilibration on avg}
In the past decade, the program of equilibration has focused on upper bounding fluctuations of observable expectation values around equilibrium, from which conclusions about equilibration of the state of the system itself have been drawn~\cite{ShortSystemsAndSub, Gogolin}. The basic mechanism behind equilibration is that of dephasing~\cite{Reimann_2008, Oliveira_2018}, and equilibration will occur as long as the initial state, following a perturbation, has an overlap with many energy eigenstates of the Hamiltonian driving the dynamics. The only further assumption is that there are not too many degenerate energy gaps~\cite{PhysRevA.98.022135}, ensuring that the majority of the system plays a dynamical role~\cite{PhysRevE.79.061103}. Specifically, observable equilibration in the sense of Ref.~\cite{ShortFinite} considers time-independent Hamiltonian dynamics given by a unitary operator $U=\exp\{-i H t\}$, with results depending on energetic properties of the Hamiltonian $H$, such as the number of distinct energy levels $\mathfrak{D}\leq{d}$ and the maximum number of energy gaps $N(\epsilon)$ in an energy window of width $\epsilon>0$.
While the choice of an equilibrium state is arbitrary, an intuitive candidate is the infinite-time-averaged subsystem state
\begin{gather}
\omega := \lim_{T\to\infty}\overline{\rho}^T \qquad \mbox{with} \qquad \overline{X}^T :=\frac{1}{T}\int_0^T X(t)\,dt
\end{gather}
where $\rho$ stands for the initial state of the whole system, time-averaged over a finite time window of width $T$. Notably, this corresponds to a dephasing of the initial state with respect to $H$, i.e. $\omega=\mc{D}(\rho)$ where we define
\begin{gather}
\mc{D}(\cdot):=\sum_nP_n(\cdot)P_n,
\end{gather}
with $P_n$ a projector onto the $n$th eigenspace of $H=\sum_n E_n P_n$. In fact, we can similarly describe general finite-time-averaged evolution by the map
\begin{gather}
\begin{split}
&\mc{G}_T(\cdot):=\sum_{n,m}G_{nm}^{(T)}P_n(\cdot)P_m \quad \mbox{with} \\ &G_{nm}^{(T)}:=\overline{\exp[it(E_m-E_n)]}^T
\end{split}\label{eq: G maps}
\end{gather}
Whenever it is clear that the averaging window is $T$, we will simply denote these by $\mc{G}$ and $G_{nm}$.
In this minimal setting, the authors of Ref.~\cite{ShortFinite} prove an important result on observable equilibration by showing the closeness between the evolved state $\rho(t)$ and $\omega$. Specifically, they upper-bound the temporal fluctuations of the expectation value of a general operator $A$ around equilibrium within a finite-time window as
\begin{gather}
\overline{|\tr[A(\rho(t) - \omega)]|^2}^T \leq \frac{\|A\|^2 N(\epsilon)f(\epsilon{T})}{d_\text{eff}(\rho)},
\end{gather}
where $f(\epsilon{T}) = 1+8\log_2 (\mathfrak{D})/\epsilon{T}$ and $\|\cdot\|$ denotes largest singular value; crucially, the so-called inverse effective dimension (or inverse participation ratio) of the initial state, $d_\text{eff}^{-1}(\rho) := \sum_n[\tr(P_n\rho)]^2$, quantifies the number of energy levels contributing significantly to the dynamics of the initial state $\rho$.
In general, we have the hierarchy $1\leq{d}_\text{eff}(\rho)\leq\mathfrak{D}\leq{d}$, and this result implies that equilibration is attained for large $d_\text{eff}$. It has been argued, on physical grounds, that the effective dimension takes a large value in realistic situations~\cite{Gogolin,Reimann_2008}, increasing exponentially in the number of constituents of generic many-body systems~\cite{XXEisert}, and it has been proven that it takes a large value for local Hamiltonian systems whenever correlations in the initial state decay rapidly~\cite{PhysRevLett.118.140601}. The temporal fluctuations of the expectation values of $A$ around equilibrium constitute a meaningful quantifier of equilibration: a small variance relates to the expectation value of $A$ concentrating around its mean~\textsuperscript{\footnote{Strictly, the full statistics should then display equilibration.}}.
This behaviour of long-time fluctuations around equilibrium has been studied both analytically and numerically in various physical models~\cite{PhysRevLett.109.247205,PhysRevE.87.012106, PhysRevE.88.032913, PhysRevE.89.022101, PhysRevB.101.174312}, as well as for the more restrictive case of thermalization~\cite{Rigol2008,Mori_2018,Gluza_2019,PhysRevLett.123.200604}. Similarly, related questions such as an absence of equilibration~\cite{PhysRevB.76.052203, Nandkishore, Hess, PhysRevLett.120.080603}, or the robustness of equilibration and further relaxation after a perturbation have been investigated~\cite{Robinson1973,PhysRevLett.118.130601,PhysRevLett.118.140601, PhysRevA.98.022135, PhysRevE.98.062103}. Here, we take a related approach towards investigating the behaviour of quantum quantum process when the interrogations are fuzzy in time. Doing so gives focus on an operationally meaningful scenario where we show that observations that have a finite temporal resolution make it hard to tell an out-of-equilibrium process from one that is in equilibrium.
\section{Equilibration due to finite temporal resolution}
Motivated by the above result, we consider the operationally relevant implications of limited resolution in time. Firstly, we focus on the dynamics of a $d_S$-dimensional subpart $\mathsf{S}$ of a $d_Ed_S$-dimensional system $\mathsf{SE}$, and we refer to subsystem equilibration as the relaxation of $\mathsf{S}$ towards some steady state, while the whole $\mathsf{SE}$ evolves unitarily with a general time-independent Hamiltonian $H$; our results can then naturally reduce to closed-system equilibration if coarse operations on the whole $\mathsf{SE}$ are considered. We then ask how different an evolving quantum state appears from equilibrium when measured at a time that can vary in each realisation, being randomly drawn from some distribution that quantifies the \emph{fuzziness} associated with finite temporal resolution.
Specifically, by a finite-temporal resolution observation we mean an observable $A$ (either on subsystem $\mathsf{S}$ or acting coarsely on $\mathsf{SE}$) measured after a time $t>0$ sampled from a probability distribution with density function $\mathscr{P}_{T}$, i.e. which is such that $\int_0^\infty\,dt\,\mathscr{P}_T(t)=1$. Here the parameter $T$ represents the \emph{uncertainty} or \emph{fuzziness} of the distribution; for example, it could be associated with the variance of the distribution. With this definition, we may generalize the time-average over a time-window $T$ by
\begin{gather}
\overline{X}^{\mathscr{P}_T} := \int_{0}^{\infty}dt\,\mathscr{P}_T(t)\,X.
\end{gather}
In particular, we require that the distributions $\mathscr{P}_T$ are such that the finite-time averaging map gives the dephasing map in the infinite-time limit, $\lim_{T\to\infty}\mc{G}=\mc{D}$, or equivalently, such that $\lim_{T\to\infty}G_{nm}=\delta_{nm}$; this also renders the equilibrium state $\omega=\lim_{T\to\infty}\overline{\rho(t)}^T$ to be independent of the particular choice of distribution.
The average distinguishability by means of an observable $A$ between the equilibrium and non-equilibrium cases can be quantified as $|\tr[A(\overline{\rho}^{\mathscr{P}_T}-\omega)]|$. This can be upper-bounded by
\begin{gather}
\left|\left<A\right>_{\overline{\rho}^{\mathscr{P}_T}-\omega}\right| \leq \mathscr{S}\, \|A\|\|\rho-\omega\|_2,
\label{eq: main standard case}
\end{gather}
where $\left<X\right>_\sigma := \tr[X\sigma]$ and $\mathscr{S}:=\max_{n\neq{m}}|G_{nm}|$. Here $\|\sigma\|_2=\sqrt{\tr(\sigma\sigma^\dg)}$ and $\|\rho-\omega\|_2^2\leq1-(d_Ed_S)^{-1}$ is the difference in purity of the full state $\rho$ with respect to that of the equilibrium $\omega$.
\noindent\textit{Proof.} Given that $\left|\left<A\right>_{\overline{\rho}^{\mathscr{P}_T}-\omega}\right|=|\tr[A(\mc{G}-\mc{D})(\rho)]|$ and $\tr[X\sigma]\leq\|X\|\|\sigma\|_2$, Eq.~\eqref{eq: main standard case} follows because
\begin{align}
\left\| \left(\mc{G} - \mc{D}\right)(\rho)\right\|_2^2 =&\tr\left|\sum_{n \neq m} {G}_{nm} P_n\rho P_m\right|^2\nonumber\\
=& \sum_{\substack{n \neq m \\ n^\prime \neq m^\prime}} G_{nm}G_{m^\prime{n}^\prime}\tr\left[ P_n\rho P_mP_{m^\prime}\rho P_{n^\prime}\right]\nonumber\\
=&\sum_{n \neq m } |G_{nm}|^2\tr\left[ P_n \rho P_m\rho\right]\nonumber\\
\leq& \max_{n\neq{m}}|G_{nm}|^2\sum_{n \neq m}\tr[P_n\rho P_m \rho]\nonumber\\
=&\max_{n\neq{m}}|G_{nm}|^2 \ \tr(\rho^2-\omega^2)\nonumber\\
=&\|\rho-\omega\|_2^2 \ \max_{n\neq{m}}|G_{nm}|^2, \label{eq:singlestep}
\end{align}
where we used $\tr(\rho^2-\omega^2) = \|\rho-\omega\|_2^2$, as $\tr(\rho\,\omega) = \tr(\omega^2)$.
In general $\|\omega\|_2^2\leq{d}_\text{eff}^{-1}(\rho)$, with equality both for pure $\rho$ or when the Hamiltonian is non-degenerate; both quantities relate to how spread the initial state $\rho$ is in the energy eigenbasis.
In particular, when the fuzziness $T$ corresponds to that of the uniform distribution over an interval $[0,T]$, the probability density function is $\mathscr{P}_T=T^{-1}$, as in the results of Ref.~\cite{ShortFinite} outlined above, and we get $|G_{nm}|=|\mathrm{sin}(T\mc{E}_{nm})/T\mc{E}_{nm}|$ with $\mc{E}_{nm} := (E_n-E_m)/2$. The bound in Eq.~\eqref{eq: main standard case} then tells us that the evolved state $\rho(t)$ will differ from the equilibrium $\omega$ when measured at a given time with a temporal-resolution $T$ at most with proportion $|T\mc{E}_{nm}|^{-1}$ for the smallest energy gap $\mc{E}_{nm}$, with a scale set by the size of the observable $A$ and how different the initial state $\rho$ is from the equilibrium $\omega$.
One, however, might not stop at a single observation but continue gathering data to assess how close the system remains to equilibrium with respect to a set of possible operations, $\{\mc{A}_i\}$, as we suggestively depict in Fig.~\ref{Fig: processes}. The reason for fuzziness in the initial time is that we do not know when the process actually began. However, one question we can ask is whether, by making a sequence of measurements, we are able to overcome the fuzziness of the initial interval. These operations can correspond to any possible experimental intervention, which can be correlated with any other interventions previously made, through an ancillary system. In this case the information between time-steps propagated through the environment and the disturbance introduced by the operations might become relevant. However, the subsequent measurements will also suffer from some level of fuzziness and this must be accounted for. We now precisely establish the description for multi-time quantum processes in such generality, followed by a generalisation of Eq.~\eqref{eq: main standard case} through an upper bound to the distinguishability between a finite-time resolution process and an equilibrium one.
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\section{Multi-time quantum processes}
Consider an initial state $\rho$ of the joint $\mathsf{SE}$ system unitarily evolving through a time-independent Hamiltonian dynamics until, at time $t_0$, an operation $\mc{A}_0$ is made on $\mathsf{S}$ along with an ancilla $\mathsf{\Gamma}$, which is initially uncorrelated in state $\gamma$. We denote the full initial state by $\varrho:=\rho\otimes\gamma$. After the first operation, the environment and system evolve unitarily again for a time $t_1$ until another operation $\mc{A}_1$ is made on $\mathsf{S\Gamma}$, and so on for $k$ time-steps. The joint expectation value of the series of operations is given by
\begin{gather}
\langle{\mc{A}_k,\ldots,\mc{A}_0}\rangle := \tr[\mc{A}_k\,\mc{U}_k \cdots \mc{A}_0\,\mc{U}_0\,(\varrho)],
\label{eq: exp val}
\end{gather}
where $\mc{U}_\ell(\cdot)=\ex^{-iH_\ell{t}_\ell}(\cdot)\,\ex^{iH_\ell{t}_\ell}$ acts on $\mathsf{SE}$, while by an operation we explicitly mean $\mathcal{A}_\ell(\cdot) := \sum_\mu a_{\ell_\mu} K_{\ell_\mu}(\cdot)K_{\ell_\mu}^\dg$, with $\sum_\mu K_{\ell_\mu}^\dg\,K_{\ell_\mu}\leq \mbb{1}$, which acts solely on $\mathsf{S\Gamma}$; here $K_{\ell_\mu}$ are Kraus operators, potentially corresponding to measurement outcomes, and $a_{\ell_\mu}$ are the corresponding outcome weights. The Hamiltonians $H_\ell$ are in general different at each step. The ancillary space $\mathsf{\Gamma}$ can be interpreted as a quantum memory device, and might carry information about previous interactions with the system.
The information about the intrinsic dynamical process, i.e., the initial $\mathsf{SE}$ state $\rho$ and the joint unitary evolutions $\mc{U}_i$ with their respective timescales at each step, can be encoded in a positive semi-definite tensor $\Upsilon$, and similarly, the sequence of operations $\{\mc{A}_i\}$ can be encoded in a tensor of the form $\Lambda$, as depicted in Figure~\ref{Fig: processes} and detailed in Appendix~\ref{appendix: process tensor}. This simplifies the joint expectation value in Eq.~\eqref{eq: exp val} as the inner product
\begin{equation}
\langle\Lambda\rangle_\Upsilon := \tr[\Lambda\Upsilon] = \tr[\mc{A}_k\,\mc{U}_k \cdots \mc{A}_0\,\mc{U}_0\,(\varrho)]
\end{equation}
which can be seen as a generalisation of the Born rule to multi-time step quantum processes~\cite{Costa}. Here, $\Upsilon$ becomes an unnormalized many-body density operator, and $\Lambda$ an observable. Temporal correlations, or memory, in operations are carried through space $\mathsf{\Gamma}$; any $\Lambda$ can be represented as a sequence of uncorrelated operations on a joint $\mathsf{S\Gamma}$ system. Both classically correlated operations, where the measurement basis is conditioned on past outcomes, and coherent quantum correlated measurements can be represented in this way~\cite{PhysRevA.99.042108}. The case of infinite memory and the case of completely uncorrelated operations are then extreme limits of this general setting. Formally, $\Upsilon$ is the Choi state~\cite{watrous_2018} of a quantum process, containing all its accessible dynamical information~\cite{ Markovorder1, PhysRevA.99.042108} and is the quantum generalisation of a stochastic process~\cite{Quolmogorov,processtensor, processtensor2, OperationalQDynamics}.
We finally notice that when a process ends at the first intervention $\mc{A}_0$ we have $\Upsilon(t_0) =\rho_S(t_0)$, becoming the corresponding quantum state, so that all of the previous results (single measurement) apply for such case.
\begin{figure}
\caption{We consider equilibration for quantum processes with a fuzzy clock, by which we mean that each Hamiltonian evolution is time-averaged ($\mc{G}
\label{fig: fuzziness}
\end{figure}
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