text
stringlengths 571
40.6k
| label
int64 0
1
|
|---|---|
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study algebraic aspects of the equivariant quantum cohomology algebra of the flag manifolds. We introduce and study the quantum double Schubert polynomials \(\widetilde{\mathfrak{S}}_w(x,~y)\), which are the Lascoux-Schützenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy identity. We define also quantum Schubert polynomials \(\widetilde{\mathfrak{S}}_w(x)\) as the Gram-Schmidt orthogonalization of some set of monomials with respect to the scalar product, defined by the Grothendieck residue. Using the quantum Cauchy identity, we prove that \(\widetilde{\mathfrak{S}}_w(x)=\widetilde{\mathfrak{S}}_w(x,y)|_{y=0}\) and as a corollary we obtain a simple formula for the quantum Schubert polynomials \(\widetilde{\mathfrak{S}}_w(x)=\partial_{ww_0}^{(y)}\widetilde{\mathfrak{S}}_{w_0}(x,~y)|_{y=0}\). We also prove the higher genus analog of Vafa-Intriligator's formula for the flag manifolds and study the quantum residues generating function. We introduce the Ehresmann-Bruhat graph on the symmetric group and formulate the equivariant quantum Pieri rule. quantum double Schubert polynomials; Ehresmann-Bruhat graph; quantum Pieri's rule; cohomology; flag manifold Kirillov, A. N.; Maeno, T.: Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-intriligator formula. Discrete math. 217, No. 1-3, 191-223 (2000) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main aim of the present article is to define and study polynomials that the authors propose as type \(B\), \(C\) and \(D\) double Schubert polynomials. For the general linear group the corresponding objects are the double Schubert polynomials of Lascoux and Schützenberger. These type \(A\) polynomials possess a series of remarkable properties and the authors propose a theory with as many of the analogous properties as possible. They succeed in obtaining several properties which are desirable both from the geometric and combinatorial points of view.
When restricted to maximal Grassmannian elements of the Weyl group, the single versions of the polynomials are the \(\widetilde P\)- and \(\widetilde Q\)-polynomials of \textit{P. Pragacz} and \textit{J. Ratajski} [J. Reine Angew. Math. 476, 143--189 (1996; Zbl 0847.14029)]. The latter polynomials play, in some sense, the role in types \(B\), \(C\) and \(D\) analogous to that of Schur's \(S\)-functions in type \(A\). The utility of the \(\widetilde P\)- and \(\widetilde Q\)-polynomials in the description of Schubert calculus and degeneracy loci was studied by \textit{P. Pragacz} and \textit{J. Ratajski} [Compos. Math. 107, No. 1, 11--87 (1997; Zbl 0916.14026)], and according to the authors [see Compos. Math. 140, No. 2, 482--500 (2004; Zbl 1077.14083) and J. Reine Angew. Math. 516, 207--223 (1999; Zbl 0934.14018)] the multiplication of \(\widetilde Q\)-polynomials describes both the arithmetic and quantum Schubert calculus on the Lagrangian Grassmannian. Thus the double Schubert polynomials in the present article are closely related to natural families of representing polynomials.
In many cases the authors obtain an analogue of the determinantal formula for Schubert cycles in Grassmannians and they answer a question of \textit{W. Fulton} and \textit{P. Pragacz} [Schubert varieties and degeneracy loci. Lect. Notes Math. 1689 (1998; Zbl 0913.14016)]. The formulas generalize those obtained by Pragacz and Ratajski [loc. cit.].
The main ingredients in the proofs are the geometric work of \textit{W. Fulton} [Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044) and Isr. Math. Conf. Proc. 9, 241--262 (1996; Zbl 0862.14032)] and \textit{W. Graham} [J. Differ. Geom. 45, 471--487 (1997; Zbl 0935.14015)] and the algebraic tools developed by \textit{A. Lascoux} and \textit{P. Pragacz} [Adv. Math. 140, No. 1, 1--43 (1998; Zbl 0951.14035) and Mich. Math. J. 48, Spec. Vol., 417--441 (2000; Zbl 1003.05106)]. determinantal formula; Weyl groups; Grassmannians; Lagrangian; Schubert cycles; Chern classes A. Kresch and H. Tamvakis, ''Double Schubert Polynomials and Degeneracy Loci for the Classical Groups,'' Ann. Inst. Fourier 52(6), 1681--1727 (2002). Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Double Schubert polynomials and degeneracy loci for the classical groups. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce the quantum multi-Schur functions, quantum factorial Schur functions and quantum Macdonald polynomials. We prove that for restricted vexillary permutations, the quantum double Schubert polynomial coincides with some quantum multi-Schur function and prove a quantum analog of the Nägelsbach-Kostka and Jacobi-Trudi formulae for the quantum double Schubert polynomials in the case of Grassmannian permutations. We prove also an analog of the Giambelli and the Billey-Jockusch-Stanley formula for quantum Schubert polynomials. Finally we formulate two conjectures about the structure of quantum double and quantum Schubert polynomials for 321-avoiding permutations. Anatol N. Kirillov, Quantum Schubert polynomials and quantum Schur functions, Internat. J. Algebra Comput. 9 (1999), no. 3-4, 385 -- 404. Dedicated to the memory of Marcel-Paul Schützenberger. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Quantum Schubert polynomials and quantum Schur functions. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove that twisted versions of Schubert polynomials defined by \(\widetilde{\mathfrak{S}}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}\) and \(\widetilde{\mathfrak{S}}_{ws_i} = (s_i+\partial_i)\widetilde{\mathfrak{S}}_w\) are monomial positive and give a combinatorial formula for their coefficients. In doing so, we reprove and extend a previous result about positivity of skew divided difference operators and show how it implies the Pieri rule for Schubert polynomials. We also give positive formulas for double versions of the \(\widetilde{\mathfrak{S}}_w\) as well as their localizations. combinatorial formula; positivity of skew divided difference operators; Pieri rule for Schubert polynomials Combinatorial aspects of algebraic geometry, Classical problems, Schubert calculus Twisted Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials \(\mathfrak{S}_w\) were first introduced by \textit{I. N. Bernstein} et al. [Russ. Math. Surv. 28, No. 3, 1--26 (1973; Zbl 0289.57024)] as certain polynomial representatives of cohomology classes of Schubert cycles \(X_w\) in flag varieties. They were extensively studied by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 294, 447--450 (1982; Zbl 0495.14031)] using an explicit definition in terms of difference operators \({\partial}_w\). Subsequently, a combinatorial expression for Schubert polynomials as the generating polynomial for compatible sequences for reduced expressions of a permutation \(w\) was discovered by \textit{S. C. Billey} et al. [J. Algebr. Comb. 2, No. 4, 345--374 (1993; Zbl 0790.05093)]. In the special case of the Grassmannian subvariety, Schubert polynomials are Schur polynomials, which also arise as the irreducible characters for the general linear group.
The Stanley symmetric functions \(F_w\) were introduced by \textit{R. P. Stanley} [Eur. J. Comb. 5, 359--372 (1984; Zbl 0587.20002)] in the pursuit of enumerations of the reduced expressions of permutations, in particular of the long permutation \(w_0\). They are defined combinatorially as the generating functions of reduced factorizations of permutations. Stanley symmetric functions are the stable limit of Schubert polynomials
\[
F_w (x_1, x_2, \ldots ) = \lim_{m\rightarrow \infty} \mathfrak{S}_{1^m \times w}(x_1, x_2, \ldots, x_{n+m}).
\]
\textit{P. Edelman} and \textit{C. Greene} [Adv. Math. 63, 42--99 (1987; Zbl 0616.05005)] showed that the coefficients of the Schur expansion of Stanley symmetric functions are nonnegative integer coefficients. Demazure modules for the general linear group are closely related to Schubert classes for the cohomology of the flag manifold. In certain cases these modules are irreducible polynomial representations, and so the Demazure characters also contain the Schur polynomials as a special case. Lascoux and Schützenberger stated that Schubert polynomials are nonnegative sums of Demazure characters.
In this paper the authors prove the converse to limit identity above by showing that Schubert polynomials are Demazure truncations of Stanley symmetric functions. Specifically, they show that the combinatorial objects underlying the Schubert polynomials, namely the compatible sequences, exhibit a Demazure crystal truncation of the full Stanley crystal of Morse and Schilling. They prove this, in which they give an explicit Demazure crystal structure on semi-standard key tableaux, which coincide with semi-skyline augmented fillings. Also they show that the crystal operators on reduced factorizations intertwine with (weak) Edelman-Greene insertion, proves their main result. Schubert polynomials; Demazure characters; Stanley symmetric functions; crystal bases Classical problems, Schubert calculus, Combinatorial aspects of representation theory, Permutations, words, matrices, Symmetric functions and generalizations, Group actions on combinatorial structures, Quantum groups (quantized function algebras) and their representations A Demazure crystal construction for Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials are explicit representatives for Schubert classes in the cohomology ring of a flag variety. Those of type \(A_n\) were introduced by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [Polynômes de Schubert, C. R. Acad. Sci., Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031)]. The \(K\)-theory of flag varieties, which is the next level of complexity after singular cohomology in encoding the topological structure of these varieties, leads to a generalization of the Schubert calculus. The analogs of Schubert polynomials, as representatives for the \(K\)-theory classes determined by the structure sheaves of Schubert varieties, are the Grothendieck polynomials which where introduced by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [Symmetry and flag manifolds, Lect. Notes Math. 996, 118-144 (1983; Zbl 0542.14031)]. In the paper under review Grothendieck polynomials indexed by Grassmannian permutations are studied. Transition matrices between these polynomials and Schur polynomials are given. Moreover, a Pieri formula for the Grothendieck polynomials indexed by Grassmannian permutations is given. Grothendieck polynomials; Schur polynomials; Pieri formula; Schubert polynomials; Schubert classes; Grassmannian permutations Lenart, C., Combinatorial aspects of the \(K\)-theory of Grassmannians, Ann. Comb., 4, 1, 67-82, (2000) Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Combinatorial aspects of the \(K\)-theory of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review deals with type \(B\) analogues of the Schubert polynomials of Lascoux and Schützenberger. Let \(W=W_n\) be the Coxeter group of type \(A_n\) (i.e. the symmetric group \(S_{n+1}\)) with the natural action of \(W\) on \({\mathbb C}[x_1,\ldots,x_{n+1}]\). Let \(I_W\) be the ideal generated by the \(W\)-symmetric polynomials without constant terms. The Schubert polynomials \(X_w\) are laballed with the elements \(w\in W\) and are a distingushed linear basis of \({\mathbb C}[x_1,\ldots,x_{n+1}]/I_W\).
The authors choose five properties in the ``ordinary'' \(A\) case: (0) \(X_w\) is homogeneous of degree the length \(l(w)\) of \(w\) with respect to the natural set of generators \(s_i\) of \(W\). (1) \(X_w\) can be recursively defined by divided difference operators by \(\partial_i(X_{ws_i})=X_w\) if \(l(ws_i)=l(w)+1\). (2) For any \(u,v\in W\) and for a sufficiently large \(m\), in the multiplication of \(X_u\) and \(X_v\) one can get rid of the unpleasant ``mod \(I_W\)'', i.e. \(X_uX_v=\sum_{w\in W_m}c_{uv}^wX_w\). (3) The polynomials \(X_w\) are with nonnegative integer coefficients. (4) The Schubert polynomials are stable with respect to the natural embedding \(W_n\subset W_m\), \(n<m\), and the corresponding projection
\[
{\mathbb C}[x_1,\ldots,x_{m+1}]\longrightarrow {\mathbb C}[x_1,\ldots,x_{n+1}]
\]
sending the extra variables to 0.
The authors show that it is impossible to transfer all these properties to the case of the hyperoctahedral group \(B_n\), i.e. the group of symmetries of the \(n\)-dimensional cube. They require the properties (0) and (4) and construct \(B_n\)-Schubert polynomials satisfying also the conditions (1)-(2) and (2)-(3). The constructions are based on the exponential solution of the Yang-Baxer equation in the nil-Coxeter algebra . The authors show that the two kinds of Schubert polynomials are related by a certain ``change of variables''. Finally, they construct a family of polynomials \(X_w\) and conjecture that they satisfy the conditions (0), (1), (3) and (4). Recently the conjecture was solved affirmatively by Tao Kai Lam. Other interesting results involving, e.g., Stanley symmetric functions of type \(B\), are also obtained in this paper. Schubert polynomials; Yang-Baxer equation; symmetric functions; Coxeter groups Fomin, S.; Kirillov, A. N., Combinatorial \(B_n\)-analogues of Schubert polynomials, Trans. Am. Math. Soc., 348, 3591-3620, (1996) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Combinatorial \(B_ n\)-analogues of Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove the Cauchy type identities for the universal double Schubert polynomials recently introduced by \textit{W. Fulton} [Duke Math. J. 96, 575--594 (1999; Zbl 0891.14022)]. As a corollary, the determinantal formula for some specializations of the universal double Schubert polynomials corresponding to the Grassmannian permutations are obtained. We also introduce and study the universal Schur functions and a multiparameter deformation of Schubert polynomials. universal Schur functions A. N. Kirillov, Cauchy identities for universal Schubert polynomials , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 283 (2001), 123--139., 260. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Cauchy identities for universal Schubert polynomials | 1 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The coefficients of the Kazhdan-Lusztig polynomials \(P_{v,w}(q)\) are nonnegative integers that are upper semicontinuous relative to Bruhat order. Conjecturally, the same properties hold for \(h\)-polynomials \(H_{v,w}(q)\) of local rings of Schubert varieties. This suggests a parallel between the two families of polynomials. We prove our conjectures for Grassmannians, and more generally, covexillary Schubert varieties in complete flag varieties, by deriving a combinatorial formula for \(H_{v,w}(q)\). We introduce drift configurations to formulate a new and compatible combinatorial rule for \(P_{v,w}(q)\). From our rules we deduce, for these cases, the coefficient-wise inequality \(P_{v,w}(q) \preceq H_{v,w}(q)\). Kazhdan-Lusztig polynomials; Hilbert series; Schubert varieties Li L., Yong A.: Kazhdan-Lusztig polynomials and drift configurations. Algebra Number Theory 5(5), 595--626 (2011) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Hecke algebras and their representations Kazhdan-Lusztig polynomials and drift configurations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We investigate the long-standing problem of finding a combinatorial rule for the Schubert structure constants in the \(K\)-theory of flag varieties (in type \(A\)). The Grothendieck polynomials of \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 295, 629--633 (1982; Zbl 0542.14030)] serve as polynomial representatives for \(K\)-theoretic Schubert classes; however no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in \(n\) variables) which we call glide polynomials, and give a positive combinatorial formula for the expansion of a Grothendieck polynomial in this basis. We then provide a positive combinatorial Littlewood-Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the \(\beta\)-Grothendieck polynomials of \textit{S. Fomin} and \textit{A. N. Kirillov} [``Grothendieck polynomials and the Yang-Baxter equation'', in: Proceedings of the sixth conference in formal power series and algebraic combinatorics, DIMACS. Piscataway, NJ. 183--189 (1994)], representing classes in connective \(K\)-theory, and we state our results in this more general context. glide polynomials; combinatorial Littlewood-Richardson rule Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical real and complex (co)homology in algebraic geometry Decompositions of Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct an integrable colored vertex model whose partition function is a double Grothendieck polynomial and relate it to bumpless pipe dreams. This gives a new proof of recent results of \textit{A. Weigandt} [J. Comb. Theory, Ser. A 182, Article ID 105470, 52 p. (2021; Zbl 1475.05172)]. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of \textit{K. Motegi} and \textit{K. Sakai} [J. Phys. A, Math. Theor. 46, No. 35, Article ID 355201, 26 p. (2013; Zbl 1278.82042)] to give a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. We then obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by \textit{P. J. McNamara} [Electron. J. Comb. 13, No. 1, Research paper R71, 40 p. (2006; Zbl 1099.05078)]. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström-Gessel-Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations. Grothendieck polynomial; colored lattice model; vexillary permutation Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Double Grothendieck polynomials and colored lattice models | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Kazhdan and Lusztig have introduced the so-called \(P\)-polynomials \(P_{x,y}(q)\) for each pair \((x,y)\) of elements of a Coxeter group \(W\) with \(x\prec y\), where \(\prec\) denotes the Bruhat order of \(W\). The polynomial \(P_{x,y}(q)\) is a measure for the singularity of the Schubert variety \(V_y\) at the generic point of \(V_x\) in the sense that \(V_y\) is smooth along the generic point of \(V_x\) if and only if \(P_{x,y}(q)=1\).
In general it is very hard to calculate the polynomials \(P_{x,y}(q)\). The authors give some explicit formulas for \(P_{x,y}(q)\) in the case \(W\) is the symmetric group \(S_n\) and \(y\) is a particular permutation associated to any flag variety, while \(x\) is arbitrary. Kazhdan-Lusztig polynomials; \(R\)-polynomials; Schubert cells; Coxeter group; \(P\)-polynomials; symmetric group; singularity of a Schubert variety Shapiro, B.; Shapiro, M.; Vainshtein, A., Kazhdan-Lusztig polynomials for certain varieties of incomplete flags, \textit{Discrete Math.}, 180, 345-355, (1998) Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects) Kazhdan-Lusztig polynomials for certain varieties of incomplete flags | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, the author proves a tableau formula for the eta polynomials of \textit{A. S. Buch} et al. [``A Giambelli formula for even orthogonal Grassmannians'', Preprint, \url{arXiv:1109.6669}] and the Stanley symmetric functions which correspond to Grassmannian elements of the Weyl group \(\widetilde{W}_n\) of type \(D_n\).
The formula for the eta polynomial \(H_\lambda(x,y)\) is written as a sum of monomials over certain fillings of the Young diagram of \(\lambda\) called typed \(k^\prime\)-bitableaux. The proof is the result of various reduction formulas which are then combined to obtain a main tableau formula for \(H_\lambda(x , y)\). In the last section, the author relates this theory to the type D Schubert polynomials and Stanley symmetric functions in [\textit{S. Billey} and \textit{M. Haiman}, J. Am. Math. Soc. 8, No. 2, 443--482 (1995; Zbl 0832.05098); \textit{T. K. Lam}, B and D analogues of stable Schubert polynomials and related insertion algorithms. Cambridge, MA: Massachusetts Institute of Technology (PhD Thesis) (1995)], and study the skew elements of \(\widetilde{W}_{n+1}\). Stanley symmetric functions; type D Schubert polynomials Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Symmetric functions and generalizations A tableau formula for eta polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give new formulas for Grothendieck polynomials of two types. One type expresses any specialization of a Grothendieck polynomial in at least two sets of variables as a linear combination of products of Grothendieck polynomials in each set of variables, with coefficients Schubert structure constants for Grothendieck polynomials. The other type is in terms of chains in the Bruhat order. We compare this second type to other constructions of Grothendieck polynomials within the more general context of double Grothendieck polynomials and the closely related \({\mathcal H}\)-polynomials. Our methods are based upon the geometry of permutation patterns. \beginbarticle \bauthor\binitsC. \bsnmLenart, \bauthor\binitsS. \bsnmRobinson and \bauthor\binitsF. \bsnmSottile, \batitleGrothendieck polynomials via permutation patterns and chains in the Bruhat order, \bjtitleAmer. J. Math. \bvolume128 (\byear2006), no. \bissue4, page 805-\blpage848. \endbarticle \OrigBibText Cristian Lenart, Shawn Robinson, and Frank Sottile, Grothendieck polynomials via permutation patterns and chains in the Bruhat order , Amer. J. Math. 128 (2006), no. 4, 805-848. \endOrigBibText \bptokstructpyb \endbibitem Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Grothendieck polynomials via permutation patterns and chains in the Bruhat order | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The cohomology ring of a complete flag variety has a basis of Schubert classes labelled by elements of the symmetric group, in which the structure constants can be described combinatorially. In a polynomial presentation of the cohomology ring, this basis corresponds to Schubert polynomials.
Similar statements are known for the equivariant cohomology ring, and for the so-called ``quantum'' deformation of the ordinary (non-equivariant) cohomology. The relevant polynomials are double Schubert polynomials and quantum Schubert polynomials, respectively. It the latter case, the structure constants encode genus \(0\) Gromov-Witten invariants of the flag variety.
In this paper, it is shown the the quantum double Schubert polynomials introduced by Kirillov and Maeno, and by Ciocan-Fontanine and Fulton, represent equivariant quantum Schubert classes of the complete flag variety. Furthermore, the result is generalized to partial flag varieties and parabolic quantum double Schubert polynomials. This affords in particular a combinatorial description of equivariant Gromov-Witten invariants.
The discussion of the maximally extended results is somewhat difficult to follow. The main result has also been obtained independently by \textit{D. Anderson} and \textit{L. Chen} [Adv. Math. 254, 300--330 (2014; Zbl 1287.14025)]. Lam, T.; Shimozono, M.: Quantum double Schubert polynomials represent Schubert classes Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum double Schubert polynomials represent Schubert classes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The cohomology or \(K\)-theory fundamental classes of Schubert varieties can be expressed as values of some universal polynomials. In cohomology these are called Schubert polynomials and in \(K\)-theory they are called Grothendieck polynomials. Both kinds are usually defined by recursion where the recursion step is a divided difference operation.
Analogous results are expected in extraordinary cohomology theories if the notion of fundamental class can be defined in the theory and it is independent of some choices (resolution). It turns out that these conditions are satisfied for Schubert varieties exactly if the formal group law of the theory is multiplicative.
In the present paper the author studies the fundamental classes of Schubert varieties in connective \(K\)-theory. He expresses them as values of polynomials that are the generalizations of Schubert and Grothendieck polynomials for connective \(K\)-theory. Connective \(K\)-theory is the universal extraordinary cohomology theory with multiplicative formal group law. The relation with the double beta polynomials of Fomin-Kirillov is discussed. Thom-Porteous formula; Schubert calculus; connective \(K\)-theory Hudson, T., A thom-porteous formula for connective \(K\)-theory using algebraic cobordism, J. K-Theory: K-Theory Appl. Algebra Geom Topol., 14, 343-369, (2014), URL http://journals.cambridge.org/article_S1865243314000221 Algebraic cycles, Classical problems, Schubert calculus, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds A Thom-Porteous formula for connective \(K\)-theory using algebraic cobordism | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the double Grothendieck polynomials of Kirillov-Naruse for the symplectic and odd orthogonal Grassmannians. These functions are explicitly written as Pfaffian sum form and are identified with the stable limits of fundamental classes of the Schubert varieties in torus equivariant connective \(K\)-theory of these isotropic Grassmannians. We also provide a combinatorial description of the ring formally spanned be the double Grothendieck polynomials. equivariant \(K\)-theory; isotropic Grassmannians; Schubert class; Pfaffian Symmetric functions and generalizations, Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Equivariant \(K\)-theory Double Grothendieck polynomials for symplectic and odd orthogonal Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the spirit of Alain Lascoux, the authors propose the use of Schubert polynomials for (certain) computations with polynomials in several variables. The idea comes from situations like doing computations with symmetric functions: There, computations are (usually) not done with monomials, but with a basis adapted to the specific problem that we are dealing with, such as Schur functions, for example.
Most of the paper is devoted to survey the background and basic facts about Schubert polynomials. When we regard the complete ring of polynomials in \(x_1,x_2,\dots,x_n\) as a ring over the ring of symmetric polynomials in \(x_1,x_2,\dots,x_n\), then the Schubert polynomials indexed by permutations in \(S_n\) (the symmetric group on \(n\) elements) constitute a linear basis. Similarly, the ring of polynomials in \(x_1,x_2,\dots,x_n\) with coefficients that are polynomials in \(y_1,y_2,\dots,y_n\) has as a linear basis the double Schubert polynomials. In order to use Schubert polynomials efficiently for computations in such rings, one of the first things we need is a rule for multiplying Schubert polynomials. No general formula for multiplying Schubert polynomials has been found yet (in contrast to Schur functions, where we have the Littlewood-Richardson rule). At least, at the very basic level, there is Monk's formula for the multiplication of a Schubert polynomial in \(x_1,x_2,\dots,x_n\) by one of the variables. However, this formula (possibly) involves Schubert polynomials which are indexed by permutations in \(S_{n+1}\) (and not just \(S_n\)). The authors show how to modify the formula so that one obtains, within the ring of polynomials in \(x_1,x_2,\dots,x_n\), regarded as a ring over the symmetric polynomials, expansions consisting of Schubert polynomials indexed by permutations in \(S_n\). A ``Monk's formula'' for double Schubert polynomials is proved as well. Schubert polynomials; symmetric functions; Monk's formula; divided differences Kohnert, A.; Veigneau, S.: Using Schubert basis to compute with multivariate polynomials. Adv. appl. Math. 19, 45-60 (1997) Symmetric functions and generalizations, Polynomials, factorization in commutative rings, Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds Using Schubert basis to compute with multivariate polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent results of \textit{A. Weigandt} [J. Comb. Theory, Ser. A 182, Article ID 105470, 52 p. (2021; Zbl 1475.05172)] relating double Grothendieck polynomias with bumpless pipe dreams. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of \textit{K. Motegi} and \textit{K. Sakai} [J. Phys. A, Math. Theor. 46, No. 35, Article ID 355201, 26 p. (2013; Zbl 1278.82042)] to give a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. Taking the stable limit of double Grothendieck polynomials, we obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by \textit{P. J. McNamara} [Electron. J. Comb. 13, No. 1, Research paper R71, 40 p. (2006; Zbl 1099.05078)]. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström-Gessel-Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations. integrable colored six-vertex model; Lindström-Gessel-Viennot lemma Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Double Grothendieck polynomials and colored lattice models | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper under review is concerned with the small quantum Schubert Calculus on flag varieties, namely homogeneous spaces of the form \(G/B\), where \(G\) is a connected simply connected complex Lie group and \(B\) is a Borel subgroup of it. Each homogeneous space carries a natural cellular decomposition and so, by general results of algebraic topology, the homology classes of the closure of the affine cells provide a basis of the homology over the integers, whose elements are said to be Schubert cycles. Poincaré duality holds in \(G/B\), because of its smoothness, and so the Schubert cocycles, Poincaré dual of the Schubert cycles, form a basis of the cohomology ring of the flag variety.
Schubert calculus of \(G/B\) amounts to knowing the \textsl{Schubert structure constants}, according to the authors' terminology, i.e., the structural constants of the cohomology algebra with respect to the basis of Schubert (co)cycles. In the case when \(G=SL(n+1, {\mathbb C})\) and \(B\) is the Borel subgroup of the triangular matrices, the picture is very well understood, as one falls in the usual intersection theory of the Grassmann variety: the structure constants are traditionally known as Littlewood--Richardson coefficients.
The two main theorems concern the small quantum cohomology \(QH^*(G/B)\) of flag varieties, i.e., the determination of what the authors call the \textsl{quantum Schubert constants}. Here \(QH^*(G/B)\) is a deformation of \(H^*(G/B)\): the support is \(H^*(G/B)\otimes_{\mathbb Z}{\mathbb C}[[t]]\) while the product structure is obtained by ``correcting'' the classical product by means of the appropriate Gromow-Witten invariants, which basically count number of rational curves in \(G/B\) having suitable incidence properties with respect to some given configuration of Schubert varieties.
The first main theorem computes the quantum Schubert structural constants of \(G/B\) by means of a certain function involving some combinatorially defined quantities, which are rational functions in simple roots. Its precise and detailed explanation in a review would not be more helpful for the interested reader. The second theorem computes, through a very explicit formula, the structural constants of the Pontryagin products in \(H^T_*(\Omega K)\), which is the equivariant (Borel-Moore) homology, with respect to the action of a \(n\)-dimensional torus \(T\), of the based loop group of the maximal compact subgroup \(K\) of \(G\).
The beginning of the paper consists in a careful introduction followed by a review of general knowledges about Kac--Moody algebras and a set up of notation (Section 2). Section 3 is devoted to deduce an explicit formula for the Pontryagin product on \(H^T_* (\Omega K)\) while Section 4 uses the formula to prove the Main Theorem. The sequence of instructive examples of Section 5 give the measure of how is the formula effective. The appendix is finally devoted to provide the proofs of some properties stated in Section 4. Quantum Schubert Calculus of homogeneous spaces; Pontryagin product, loop groups, Equivariant homology and cohomology of homogeneous spaces, Gromov-Witten invariants Leung, N. C.; Li, C., Gromov-Witten invariants for \(G / B\) and Pontryagin product for {\(\omega\)}\textit{K}, Trans. Amer. Math. Soc., 5, 2567-2599, (2012) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Infinite-dimensional Lie groups and their Lie algebras: general properties Gromov-Witten invariants for \(G/B\) and Pontryagin product for \(\Omega K\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Schubert polynomials give explicit polynomial representatives for the Schubert classes in the cohomology ring of the complete flag variety, with the goal of facilitating computations of intersection numbers. \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 294, 447--450 (1982; Zbl 0495.14031)] first defined Schubert polynomials indexed by permutations in terms of divided difference operators, and later \textit{S. C. Billey} et al. [J. Algebr. Comb. 2, No. 4, 345--374 (1993; Zbl 0790.05093)] and \textit{S. Fomin} and \textit{R. P. Stanley} [Adv. Math. 103, No. 2, 196--207 (1994; Zbl 0809.05091)] gave direct monomial expansions. \textit{N. Bergeron} and \textit{S. Billey} [Exp. Math. 2, No. 4, 257--269 (1993; Zbl 0803.05054)] reformulated this again to give a beautiful combinatorial definition of Schubert polynomials as generating functions for \(RC\)-graphs, often called pipe dreams.
In this paper, the authors introduce a new tool to aid in the study of Schubert polynomials.They define two new families of polynomials they call the monomial slide polynomials and fundamental slide polynomials. Both monomial and fundamental slide polynomials are combinatorially indexed by weak compositions, and both families form a basis of the polynomial ring. Moreover, the Schubert polynomials expand positively into the fundamental slide basis, which in turn expands positively into the monomial slide basis. While there are other bases that refine Schubert polynomials, most notably key polynomials, it has two main properties that make it a compelling addition to the theory of Schubert calculus. First, these polynomials exhibit a similar stability to that of Schubert polynomials, and so they facilitate a deeper understanding of the stable limit of Schubert polynomials, which, as originally shown by Macdonald, are Stanley symmetric functions. Second, and in sharp contrast to key polynomials, their bases themselves have positive structure constants, and so their Littlewood-Richardson rule for the fundamental slide expansion of a product of Schubert polynomials takes one step closer to giving a combinatorial formula for Schubert structure constants. Schubert polynomials; Stanley symmetric functions; pipe dreams; reduced decompositions; quasisymmetric functions S. Assaf and D. Searles. ''Schubert polynomials, slide polynomials, Stanley symmetric func tions and quasi-Yamanouchi pipe dreams''. Adv. Math. 306 (2017), pp. 89--122.DOI. Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper places itself within the rich framework of the literature regarding quantum cohomology of homogeneous varieties produced by the authors themselves. This new piece of mathematics regards certain \textsl{Quantum Giambelli} formulas for isotropic Grassmannians. If \(V\) is a vector space equipped with a non degenerate bilinear form \(\eta\), one can consider the Grassmannian \(X:=IG(k,V)\) of isotropic subspaces of \(V\) of a fixed dimension \(k\), namely the variety of those points \([W]\in G(k,V)\) such that the restriction of \(\eta\) to \(W\) is trivial (i.e. \(\eta_{|W\times W}=0\)).
If the bilinear form is skew-symmetric, the dimension of \(V\) is even and one is then concerned with symplectic vector spaces. As well known, the cohomology ring \(H^*(X,{\mathbb{Z}})\) is generated as a \({\mathbb{Z}}\)-algebra by certain special Schubert cycles and it is also a well known fact that such cycles generate the quantum cohomology of \(X\) as well. The latter is a deformation of the usual cohomology encoding the Gromov-Witten invariants which count, roughly speaking, numbers of maps of a given degree from the projective line to \(X\). The authors find and prove quantum Giambelli's formulas expressing an arbitrary Schubert class in the small quantum cohomology ring of \(X\) as a polynomial in the special Schubert classes alluded above.
The two main theorems of the article (concerning Giambelli's formulas) are analogous to those proven for the quantum cohomology of the orthogonal and Lagrangian Grassmannians in [J. Algebr. Geom. 12, No. 4, 777--810 (2003; Zbl 1051.53070)] and [Compos. Math. 140, No. 2, 482--500 (2004; Zbl 1077.14083)], by \textit{A. Kresch} and \textit{H. Tamvakis}. The proof are however quite different, due to the fact that for non maximal isotropic Grassmannians, the explicit recursion used in the quoted references is no longer available. The latter is replaced, however, by another kind of recursion, neatly steted and proved in Proposition 3.
The reviewed paper is for all people interested in the combinatorial aspects of cohomology theories (quantum, equivariant, quantum-equivariant) of homogeneous varieties. quantum Schubert Calculus; isotropic Grassmannians; Giambelli's Formulas Buch, AS; Kresch, A; Tamvakis, H, Quantum Giambelli formulas for isotropic Grassmannians, Math. Ann., 354, 801-812, (2012) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Quantum Giambelli formulas for isotropic Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Grothendieck polynomials, introduced by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [Lect. Notes Math. 996, 118--144 (1983; Zbl 0542.14031)], are certain \(K\)-theory representatives for Schubert varieties. Symplectic Grothendieck polynomials, described more recently by \textit{B. Wyser} and \textit{A. Yong} [Transform. Groups 22, No. 1, 267--290 (2017; Zbl 1400.14130)], represent the \(K\)-theory classes of orbit closures for the complex symplectic group acting on the complete flag variety. We prove a transition formula for symplectic Grothendieck polynomials and study their stable limits. We show that each of the \(K\)-theoretic Schur \(P\)-functions of Ikeda and Naruse arises from a limiting procedure applied to symplectic Grothendieck polynomials representing certain ``Grassmannian'' orbit closures. Grassmannians, Schubert varieties, flag manifolds, Grothendieck groups, \(K\)-theory, etc., Symmetric functions and generalizations, Combinatorial aspects of algebraic geometry On some properties of symplectic Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a new proof that three families of polynomials coincide: the double Schubert polynomials of \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [Lett. Math. Phys. 10, 111--124 (1985; Zbl 0586.20007)] defined by divided difference operators, the pipe dream polynomials of \textit{N. Bergeron} and \textit{S. Billey} [Exp. Math. 2, No. 4, 257--269 (1993; Zbl 0803.05054)], and the equivariant cohomology classes of matrix Schubert varieties. All three families are shown to satisfy a ``co transition formula'' which we explain to be some extent projectively dual to Lascoux' transition formula. We comment on the \(K\)-theoretic extensions. Classical problems, Schubert calculus, Representations of finite symmetric groups, Combinatorial aspects of algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry Schubert polynomials, pipe dreams, equivariant classes, and a co-transition formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials First the authors give the description of the double Schubert polynomials of types \(B,C\) and \(D\) using Schur \(P\) and Schur \(Q\)-functions in the rings \(R_{\infty}\) and \(R'_{\infty}\), and the actions of divided difference operators on these rings. After then they are giving the basic properties of these super-symmetric polynomials. They are also giving the ways how to calculate double Schubert polynomials.
The double Schubert polynomials represent the torus equivariant Schubert classes in the equivariant cohomology of flag manifolds \(G/B\) where \(G\) is semi-simple Lie group and \(B\) is its Borel subgroup. In this paper the authors explain for the case of the type \(C_n\). But I could not see the product formulas for these classes(or double Schubert polynomials). double Schubert polynomials; excited Young diagrams; Schur P-functions; Schur Q-functions; divided difference operators; torus equivarant cohomology T. Ikeda, H. Naruse, Double Schubert polynomials of classical type and Excited Young diagrams, Kôkyûroku Bessatsu B11 (2009). Classical problems, Schubert calculus, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Double Schubert polynomials of classical type and excited Young diagrams | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review, the authors give an explicit formula for the degree of the Grothendieck polynomial \(\mathfrak{G}_w\) of a Grassmannian permutation \(w\) in the symmetric group \(\mathfrak{S}_n\). Their method uses a formula of [\textit{C. Lenart}, Ann. Comb. 4, No. 1, 67--82 (2000; Zbl 0958.05128)] that expresses \(\mathfrak{G}_w\) in terms of Schur polynomials. The authors then use their degree formula to give an explicit formula for the Castelnuovo-Mumford regularity of (the homogeneous coordinate ring of) the Grassmannian matrix Schubert variety \(X_w\) associated to \(w\). The authors also give a counterexample to a conjectured formula, as well as a corrected formula, for the regularity of (the affine coordinate rings of) standard open patches of certain Grassmannian Schubert varieties appearing in [\textit{M. Kummini} et al., Pac. J. Math. 279, No. 1--2, 299--328 (2015; Zbl 1342.14103)]. They further give a conjectured formula for the regularity of standard open patches of arbitrary Grassmannian Schubert varieties. Castelnuovo-Mumford regularity; Grothendieck polynomial; Schubert variety Linkage, complete intersections and determinantal ideals, Classical problems, Schubert calculus, Combinatorial aspects of commutative algebra Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this fundamental and beautiful article the author introduces universal Schubert polynomials that specialize to all previously known Schubert polynomials: those of Lascoux and Schützenberger, the quantum Schubert polynomials of Fomin, Gelfand, and Postnikov, and the quantum Schubert polynomials for partial flag varieties of Ciocan-Fontanine. Also double versions of these polynomials are given, that generalize the previously known double Schubert polynomials of Lascoux, MacDonald, Kirillov and Maeno, and those of Ciocan-Fontanine and Fulton.
The universal Schubert polynomials describe degeneracy loci of maps of vector bundles, in a more general setting than that of the author's beautiful earlier article [\textit{W. Fulton}, Duke Math. J. 65, 381-420 (1992; Zbl 0788.14044)].
The setting is a sequence of maps of locally free \(\mathcal O_X\)-modules
\[
F_1\to F_2\to \cdots \to F_n \to E_n \to \cdots \to E_2\to E_1
\]
on a scheme \(X\). In contrast to the mentioned article (loc. cit.) the maps \(F_i \to F_{i+1}\) do not have to be injective and the maps \(E_{i+1} \to E_i\) do not have to be surjective. For each \(w\) in the symmetric group \(S_{n+1}\), there is a degeneracy locus
\[
\Omega_w =\{x\in X\mid \text{rank}(F_q(x) \to E_p(x)) \leq r_w(p,q) \text{ for all } 1\leq p, q\leq n\},
\]
where \(r_w(p,q)\) is the number of \(i\leq p\) such that \(w(i)\leq q\). Such degeneracy loci are described by the double form \({\mathfrak S}_w(c,d)\) of universal Schubert polynomials evaluated at the Chern classes of all the bundles involved.
The classical approaches of \textit{Demazure}, or \textit{Bernstein, Gel'fand}, and \textit{Gel'fand} do not work in this case. Instead a locus in a flag bundle is found that maps to a given degeneracy locus \(\Omega_w\), such that one has injections and surjections of the bundles involved, and such that the results of the article mentioned above can be applied. Then the formula is pushed forward to get a formula for \(\Omega_w\). universal Schubert polynomials; quantum Schubert polynomials; partial flag varieties; double Schubert polynomials; degeneracy loci; Chern classes Fulton W. (1999). Universal Schubert polynomials. Duke Math. J. 96(3): 575--594 Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Determinantal varieties, Characteristic classes and numbers in differential topology Universal Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The so-called Schubert polynomials are significant in combinatorics and in geometry.
There are two combinatorial ways of writing Schubert polynomials as sums of products of linear factors. One is parameterized by ``pipe dreams'' [\textit{A. Knutson} and \textit{E. Miller}, Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)], the other is parameterized by ``bumpless pipe dreams'' [\textit{A. Lascoux}, ``Chern and Yang through ice'', Preprint; \textit{T. Lam} et al., Compos. Math. 157, No. 5, 883--962 (2021; Zbl 07358686)].
In geometry Schubert polynomials are fundamental classes of matrix Schubert varieties [\textit{L. M. Fehér} and \textit{R. Rimányi}, Cent. Eur. J. Math. 1, No. 4, 418--434 (2003; Zbl 1038.57008); Knutson-Miller, loc. cit.]. Under favorable (e.g. Gröbner) degenerations the fundamental class of a variety does not change. Hence if a matrix Schubert variety is degenerated to a union of linear spaces, then its fundamental class is written as a sum of product of linear factors. Such geometric interpretation for the ``pipe dream'' formula is known, via the anti-diagonal Gröbner degeneration.
The paper under review searches for such a geometric interpretation of the ``bumpless pipe dream'' formula for Schubert polynomials. Namely, the authors conjecture that the ``bumpless pipe dream'' formula corresponds naturally to any diagonal Gröbner degeneration. They prove this conjecture for a large class of permutations they call banner permutations. Basic steps of their recursive arguments rely on Lascoux-Schützenberger transitions. Schubert polynomial; bumpless pipe dream; matrix Schubert variety; Gröbner bases Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Gröbner geometry of Schubert polynomials through ice | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials One of the main concerns of this multifaceted paper is that of dealing with the quantum cohomology of flag varieties parametrizing inclusions of subspaces of a given vector space, as well as to supply an innovative description of the product structure of the affine Grassmannian \(\mathrm{Gr}\) associated to the group \(\mathrm{SL}_n({\mathbb C})\). Just not to loose the less experienced reader, we recall that the \(\mathrm{Gr}\) we alluded above is in fact the quotient of the group of all \(n\times n\) unimodular square matrices with entries in the ring of Laurent series modulo the action of the subgroup of unimodular square matrices with entries in the ring of formal power series with complex coefficients. The new sharp description that the authors offer for the cohomology \(H^*(\mathrm{Gr})\) and homology \(H_*(\mathrm{Gr})\), whose details are richly displayed in Section 5, definitely shed light of the intimate relationship of the subject, the so-called affine Schubert calculus, with the rich, celebrated and still mysterious theory of Macdonald polynomials. The latter is about some of the richest objects in mathematics. As the authors themselves declare in a enthusiastically inspiring introduction, Macdonald polynomials do not occur just in combinatorics, but also in the theory of double affine Hecke algebras, quantum relativistic systems, diagonal harmonics and Hilbert schemes of points in the plane.
The origin of the noble story told by the authors in the paper, has to do with a basic and a fundamental question. Nearly every professional mathematician is aware that symmetric polynomials admit a basis of Schur polynomials, parametrized by partitions of non-negative integers. Although we shall not recall here their definition, Macdonald polynomials may be seen as symmetric polynomials depending on two extra parameters, say \(t\) and \(q\), and it is then natural to wonder about the transition matrix relating them with the more familiar Schur polynomials. The still open conjecture is that the entries of the transition matrix, a generalization of the so called Kostka-Foulkes polynomial, are polynomials with non-negative integer coefficients. In other words, Macdonald polynomials relate positively to Schur polynomials, a conjecture that inspired many more researches whose output has been the dramatic emerging of the relationship with the affine Schubert calculus. The authors so come to cope with the problem of a more flexible description of the homology and cohomology of \(\mathrm{Gr}\) by introducing a clever new combinatorial tool, which is of crucial importance in all the paper, called Affine Bruhat Countertableaux (ABC). Their generating functions form a basis of \(H^*(\mathrm{Gr})\) and everything leads to a refinement of the Kotska-Foulkes polynomials. The authors deal also with the problem of providing a closer description of the constants structure of the quantum cohomology of flag varieties, where Gromov-Witten invariants related with the art of counting rational curves in homogeneous varieties. There are many more beautiful and interesting features, in this paper, that deserve to be discussed, but this at the price of giving a more detailed account of the fine technical combinatorial tools masterly employed by the authors. This cannot be evidently done in a review, but we can end it by quickly describing the organization. Let us start from the abstract: it already contains the juice of the article and say the reader what it can be found inside. The introduction is simply as beautiful as exciting and is enhanced by the second section where there is a useful interesting account of the related literature. The preliminaries are collected in Section 3: here the reader can be made aware with selected tools from the theory of symmetric functions, explain the basic vocabulary related with Ferrer diagrams, horizontal strips, addable corners, extremal cells and so on. Section 4 enters into the deep core of the paper, being devoted to the affine Pieri's rule, described in terms of sophisticated but versatile combinatorics. The explicit representative of Schubert classes is provided in this section, where the ABC order is also introduced. More relations with Macdonald polynomials are collected in Section 6. The rich reference list is still preceded by an appendix where the output of a Sage routine is displayed to check a conjecture on the equality of two kinds of symmetric functions and by section 7, eventually devoted to the quantum cohomology of the flags. Macdonald polynomials; Hall-Littlewood polynomials; affine Schubert calculus; quantum Schubert calculus; type-A affine Weyl group; affine Grassmannian; Gromov-Witten invariants; Bruhat order; weak \(k\)-Pieri rule; \(k\)-tableaux; affine Bruhat counter-tableaux Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Enumerative combinatorics Quantum and affine Schubert calculus and Macdonald polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G = G(m, N)\) denote the Grassmannian of \(m\)-dimensional subspaces of \(\mathbb C_N\). To each integer partition \(\lambda = (\lambda_1, \ldots, \lambda_m)\) whose Young diagram is contained in an \(m \times (N- m)\) rectangle, they associate a Schubert class \(\sigma_{\lambda}\) in the cohomology ring of \(G\). The special Schubert classes \(\sigma_1, \ldots, \sigma_{N-m}\) are the Chern classes of the universal quotient bundle \(\mathcal Q\) over \(G(m, N)\); they generate the graded cohomology ring \(H^\ast(G,\mathbb Z)\). The classical Giambelli formula is an explicit expression for \(\sigma_{\lambda}\) as a polynomial in the special classes.
In this paper the authors prove: Let \(X\) be a symplectic or odd orthogonal Grassmannian. They prove a Giambelli formula which expresses an arbitrary Schubert class in \(H^\ast(X,\mathbb Z)\) as a polynomial in certain special Schubert classes. This polynomial will be called a theta polynomial, is defined using raising operators, and we study its image in the ring of Billey-Haiman Schubert polynomials. Giambelli formula; isotropic Grassmannians; raising operators; theta polynomials; Schubert polynomials Buch, A.; Kresch, A.; Tamvakis, H., \textit{A Giambelli formula for isotropic Grassmannians}, Selecta Math. (N.S.), 23, 869-914, (2017) Classical problems, Schubert calculus, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds A Giambelli formula for isotropic Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0719.00018.]
This paper, as a continuation of [\textit{M. Kashiwara}, The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 407-433 (1990; Zbl 0727.17013)], completes the proof of the Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebras. The proof consists of two parts: (1) the algebraic part --- the correspondence between \({\mathcal D}\)-modules on the flag variety and representations of the Kac-Moody Lie algebra, (2) the topological part --- the description of geometry of Schubert varieties in terms of the Kazhdan-Lusztig polynomials. The algebraic part is already established in the paper cited above and the paper under review is devoted to the topological part. There are two points in the proof except which the proof is similar to the finite dimensional case. The first one is the usage of the theory of mixed Hodge modules and the second one is the interpretation of the inverse Kazhdan-Lusztig polynomials as the coefficients of certain elements in the dual of the Hecke-Iwahori algebra.
Let \({\mathfrak h}\) be the Cartan subalgebra of a symmetrizable Kac-Moody Lie algebra and \(W\) the Weyl group. For \(w\in W\) define the action on \({\mathfrak h}^*\) by \(w\cdot\lambda=w(\lambda+\rho)-\rho\). Let \(P_{z,w}(q)\) be the Kazhdan-Lusztig polynomial and \(Q_{z,w}(q)\) the inverse Kazhdan- Lusztig polynomial. They are related by
\[
\sum_ w(-1)^{\ell(w)- \ell(y)}Q_{y,w}(q)P_{w,z}(q)=\delta_{y,z}.
\]
The main result of the paper is the following. For a dominant integral weight \(\lambda\in{\mathfrak h}^*\), one has
\[
ch L(w\cdot\lambda)=\sum_ z(-1)^{\ell(z)- \ell(w)}Q_{w,z}(1)ch M(z\cdot\lambda),
\]
or equivalently \(ch M(w\cdot\lambda)=\sum_ zP_{w,z}(1)ch L(z\cdot\lambda)\). Kazhdan-Lusztig conjecture; symmetrizable Kac-Moody Lie algebras; \({\mathcal D}\)-modules; flag variety; representations; geometry of Schubert varieties; Kazhdan-Lusztig polynomials; mixed Hodge modules O.J. Ganor, \textit{Supersymmetric interactions of a six-dimensional self-dual tensor and fixed-shape second quantized strings}, \textit{Phys. Rev.}\textbf{D 97} (2018) 041901 [arXiv:1710.06880] [INSPIRE]. Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebra. II: Intersection cohomologies of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by \textit{A. Knutson} and \textit{E. Miller} [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, \textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)] obtained alternative combinatorics for the class of ``vexillary'' matrix Schubert varieties. We initiate a study of general diagonal degenerations, relating them to a neglected formula of \textit{A. Lascoux} [``Chern and Yang through ice'', Preprint] in terms of the 6-vertex ice model (recently rediscovered by \textit{T. Lam} et al. [Compos. Math. 157, No. 5, 883--962 (2021; Zbl 07358686)] in the guise of ``bumpless pipe dreams''). Schubert polynomial; bumpless pipe dream; matrix Schubert variety Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Gröbner geometry of Schubert polynomials through ice | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials are the building blocks of several cohomological degeneracy locus formulas. Their role in \(K\)-theory is played by so-called Grothendieck polynomials. In particular, ``stable'' versions of Schubert and Grothendieck polynomials turn up naturally, for example in degeneracy locus problems associated with certain quiver representations.
The paper under review studies the expansion of stable Grothendieck polynomials in the basis of stable Grothendieck polynomials for partitions. This generalizes the result of Fomin-Green in the cohomological setting, where the analogues of stable Grothendick polynomials for partitions are the Schur polynomials. The existence of such a finite, integer linear combination expansion was proved by Buch, and the sign of the coefficients were determined by Lascoux. In this paper the authors give a new, non-recursive combinatorial rule for the coefficients. Namely, they prove that the coefficients, up to explicit sign, count the number of increasing tableau of a given shape, with an associated word having explicit combinatorial properties stemming from the combinatorics of the 0-Hecke monoid. The main ingredient of the proof is a generalized, so-called Hecke insertion algorithm.
The main application showed in the paper is a \(K\)-theoretic analogue of the factor sequence formula of Buch-Fulton for the cohomological quiver polynomials (of equioriented type A). Grothendieck polynomials; \(K\)-theory; 0-Hecke monoid; insertion algorithm; factor sequence formula Buch, A.; Kresch, A.; Shimozono, M.; Tamvakis, H.; Yong, A., Stable Grothendieck polynomials and \textit{K}-theoretic factor sequences, Math. Ann., 340, 359-382, (2008) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], \(K\)-theory of schemes, Symmetric functions and generalizations Stable Grothendieck polynomials and \(K\)-theoretic factor sequences | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The very interesting paper under review continues the investigation begun by the same authors in the article [Int. Math. Res. Not. 2012, No. 16, 3706--3722 (2012; Zbl 1252.14038)]. Its main goal is to deduce quantum Pieri rules for the quantum cohomology of generalized flag varieties \(G/P\), where \(G\) is a complex Lie group of type A B, C or D and \(P\) is a maximal parabolic subgroup. The case when \(G\) is of type \(A\) corresponds to the classical Grassmann variety \(G(k,n)\) parameterizing \(k\)-dimensional vector subspaces of \({\mathbb C}^n\).
For the convenience of a more general audience of readers, recall that the cohomology ring \(H^*(G(k,n), {\mathbb Z})\) is a free \({\mathbb Z}\)-module generated by the Schubert cycles \(\sigma_\lambda\), which are the cohomology classes of the Schubert varieties associated to a given complete flag of \({\mathbb C}^n\): they are parameterized by the partitions \(\lambda\) whose Young diagram is contained in a \(k\times (n-k)\) rectangle of \(k(n-k)\) boxes. The special Schubert cycles \(\sigma_1,\ldots, \sigma_k\) generate \(H^*(G(k,n), {\mathbb Z})\) as a \({\mathbb Z}\)-algebra, and classical Pieri's rule compute the coefficients \(a_\mu\) of the expansion of the cup-product \(\sigma_i\cup \sigma_\lambda\) as a \({\mathbb Z}\)-linear combination \(\sum_\mu a_\mu\sigma_\mu\), where \(\mu\) runs over all partitions of weight equal to the weight of \(\lambda\) added by \(i\).
To deduce quantum Pieri's rule for the mentioned kind of flag varieties \(G/P\), the authors exploit the principle that relevant genus zero three points Gromov--Witten invariants coincide with certain classical intersection numbers. This is almost obvious in the case of the small quantum cohomology of the usual Grassmannians \(G(k,n)\): any product of two classes can be seen as the classical product in a Grassmannian of sufficiently high dimension. For instance the equality \(\sigma_1^4=2\sigma_{2,2}+2t\sigma_0\), holding in \(QH^*(G(2,4))\cong H^*(G(2,4),{\mathbb Z})\otimes{\mathbb Z}[t]\), is a consequence of the equality \(\sigma_1^4=2\sigma_{2,2}+3\sigma_{3,1}+\sigma_{4}\) holding in \(H^*(G(2,6))\) modulo the identification \(\sigma_{3,1}\mapsto t\sigma_0\) and \(\sigma_4\mapsto -t\sigma_0\) -- see e.g. [\textit{L. Gatto}, Asian J. Math. 9, No. 3, 315--322 (2005; Zbl 1099.14045)].
If \(G\) is a group of Lie type B, C, or D, the homogeneous space \(G/P\) may be seen as the variety of subspaces which are isotropic with respect to some given bilinear form (i.e. symplectic, symmetric, \dots) which suggests the notation \(IG(k,n)\) for the \(k\)-dimensional isotropic subspaces of \({\mathbb C}^n\). One of the main result of the paper, exposed in the definitely fine introduction, is concerned with the quantum Pieri rule for tautological subbundles of \(IG(k,2n)\), which is stated in a typical combinatorial language. Section 3 and 4 of the paper are concerned to prove many and many Pieri's rules for Grassmannians of type B, C, D, emphasizing their classical aspects. In the very useful appendix the authors reprove the well-known quantum Pieri rules for Grassmannians of type A (i.e. complex Grassmannians), obtained by \textit{A. Bertram} [Adv. Math. 128, No. 2, 289--305, Art. No. AI971627 (1997; Zbl 0945.14031)] and \textit{A. S. Buch} [Compos. Math. 137, No. 2, 227--235 (2003; Zbl 1050.14053)], to exemplify the methods developed along the paper. The article, whose main achievement consists in recognizing the classical roots of the small quantum cohomology of the homogeneous varieties, reconciling it with the ordinary cohomology, concludes itself with a useful and comprehensive bibliography which allows the interested reader to draw his own path to get into this fascinating subject. quantum Pieri rules; isotropic Grassmannians; Gromov-Witten invariants; quantum cohomology Leung, NC; Li, C, Quantum Pieri rules for tautological subbundles, Adv. Math., 248, 279-307, (2013) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum Pieri rules for tautological subbundles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of \(G/B\). We enhance a result of Fulton and Woodward by showing that the minimal monomial in the quantum parameters that occurs in the quantum product of two Schubert classes has a simple interpretation in terms of directed paths in this graph.
We define path Schubert polynomials, which are quantum cohomology analogs of skew Schubert polynomials recently introduced by \textit{C. Lenart} and \textit{F. Sottile} [Proc. Am. Math. Soc. 131, 3319--3328 (2003; Zbl 1033.05097)]. They are given by sums over paths in the quantum Bruhat graph of type \(A\). The 3-point Gromov-Witten invariants for the flag manifold are expressed in terms of these polynomials. This construction gives a combinatorial description for the set of all monomials in the quantum parameters that occur in the quantum product of two Schubert classes. quantum Bruhat graph; Bruhat order; quantum cohomology; Schubert classes; path Schubert polynomials Postnikov, A., Quantum Bruhat graph and Schubert polynomials. Proc. Amer. Math. Soc., 133 (2005), 699--709. Symmetric functions and generalizations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum Bruhat graph and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main goal of the present article is to give explicit formulas for the fundamental classes of Schubert subschemes in Lagrangian and orthogonal grassmannians of maximal isotropic subbundles. This is accomplished by the usual method of desingularization and pushing down classes from the desinguarization to the original space. In general the approach gives an efficient algorithm for computing formulas for Lagrangian and orthogonal Schubert classes. In some cases, like for one or two Schubert conditions, closed formulas are obtained. The classes are expressed in terms of \(\widetilde Q\)- and \(\widetilde P\)-polynomials introduced by the authors. These are variants of Schur's \(Q\)- and \(P\)-polynomials.
The geometric situation is technically more complicated than the well known situation for Schubert cycles in grassmannians. Computations of the class of the relative diagonal and orthogonality properties of the \(\widetilde Q\)- and \(\widetilde P\)-polynomials are used to overcome the additional difficulties. Frequent references are made to the work of the first author. Schubert subschemes; orthogonal Grassmannians; Lagrangian P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; \textit{Q}-polynomial approach, Compos. Math. 107 (1997), no. 1, 11-87. Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Formulas for Lagrangian and orthogonal degeneracy loci; \(\tilde Q\)-polynomial approach | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Hecke-Grothendieck polynomials were introduced by \textit{A. N. Kirillov} [SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016; Zbl 1334.05176)] as a common generalization of Schubert polynomials, dual \(\alpha\)-Grothendieck polynomials, Di Francesco-Zinn-Justin polynomials, etc. Kirillov conjectured that the coefficients of every generalized Hecke-Grothendieck polynomial are nonnegative combinations of certain parameters. Here we prove a weak version of Kirillov's conjecture, that is, under certain conditions, every Hecke-Grothendieck polynomial has only nonnegative integer coefficients. In particular, the proof of this weak version of Kirillov's conjecture serves as a unified proof for the fact that all the Schubert polynomials, dual \(\alpha\)-Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials have only nonnegative coefficients. Hecke-Grothendieck polynomial; Gröbner-Shirshov basis; positivity Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds A weak version of Kirillov's conjecture on Hecke-Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In cohomology theory fundamental classes \(S_w\) of Schubert varieties \(\Omega_w\) can be interpreted as universal degeneracy locus formulas. The classes \(S_w\)-called double Schubert polynomials--can be obtained by an explicit description of \(S_{w_0}\) and a recursion.
Fundamental classes exist and are well defined in \(K\)-theory as well, and similar results hold as the ones described above.
Analogous results are expected in extraordinary cohomology theories if the notion of fundamental class is defined in the theory and is independent of the choices (resolution). It turns out that these theories are exactly the one with multiplicative group laws.
The author calculates these classes for connective \(K\)-theory, and hence finds an interpretation of Fomin-Kirillov's double \(\beta\)-polynomials. As a corollary Thom-Porteous formulas are obtained for connective \(K\)-theory. connective \(K\)-theory; Schubert calculus K. Hayano, Modification rule of monodromies in \(R_{2}\) -move, preprint, Algebraic cycles, Classical problems, Schubert calculus, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) Modification rule of monodromies in an \(R_{2}\)-move | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author develop a general formalism for equivariant Schubert calculus of Grassmannians consisting of a basis theorem, Pieri formula and Giambelli formula in the previous works. In this paper he present an extract of of the theory containing the essential features of the ring. In particular he emphasize the importance of Goresky-Kottwitz- MavPherson (GKM) conditions.
The formalism in this paper are influenced by the combinatorial formalism given by Knutson and Tau for equivariant cohomology of Grassmannians and of use of factorial Schur polynomials on equivariant quantum cohomology of Grassmannians. factorial Schur polynomials; equivariant quantum cohomology; GKM conditions; equivariant Schubert calculus Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus A relation between symmetric polynomials and the algebra of classes, motivated by equivariant Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This monograph includes much of the author's previous work on Schur and Schubert polynomials and generalizations, and considerable background material. \textit{G. P. Thomas} [Frames, Young tableaux, and Baxter sequences; Adv. Math. 26, 275-289 (1977; Zbl 0375.05005) and Further results on Baxter sequences and generalized Schur functions; Lect. Notes Math. 579, 155-167 (1977; Zbl 0364.05007)] constructed the Schur functions \(S_\lambda\) combinatorially from the set of standard Young tableaux of shape \(\lambda\), using algebraic ``mixed Baxter-multiplication operators'' on the algebra of polynomials. The author [Sequences of symmetric polynomials and combinatorial properties of tableaux; Adv. Math. 134, No. 1, 46-89 (1998; Zbl 0902.05078)] generalizes this construction, providing an effective construction of Q-Schur, Hall-Littlewood, Jack, and Macdonald polynomials, all of which are generalizations of the Schur functions. The Baxter construction can also be applied to the Schubert polynomials; see \textit{R. Winkel} [A combinatorial bijection between standard Young tableaux and reduced words of Grassmannian permutations; Sémin. Lothar. Comb. 36, B36h (1996; Zbl 0886.05115) and Schubert functions and the number of reduced words of permutations; Sémin. Lothar. Comb. 39, B39a (1997; Zbl 0886.05119)]. Recursive methods for construction of the Schubert polynomials are given, and used to prove their basic properties; see \textit{R. Winkel} [Recursive and combinatorial properties of Schubert polynomials; Sémin. Lothar. Comb. 38, B38c (1996; Zbl 0886.05111)]. The original construction of the Schubert polynomials of type \(A_n\) by \textit{A. Lascoux} and \textit{M. P. Schützenberger} [Polynômes de Schubert, C. R. Acad. Sci. Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031)] by divided differences, and also the construction by recursive structures, are generalized to give constructions of the cases \(B_n\), \(C_n\), and \(D_n\); see \textit{R. Winkel} [Schubert polynomials of types A--D; Manuscr. Math. 100, No. 1, 55-79 (1999; Zbl 0936.05088)]. The weak Bruhat order on Coxeter groups gives a partial order which can be partitioned into poset-isomorphic parts. This is used to give a combintorial computation of the Poincaré polynomials of the finite and some affine Coxeter groups, and a non-recursive computation of standard reduced words for signed and unsigned permutations; see \textit{R. Winkel} [A combinatorial derivation of the Poincaré polynomials of the finite irreducible Coxeter groups; Discrete Math., to appear]. Expansions of Schubert polynomials into standard elementary monomials are constructed combinatorially; see \textit{R. Winkel} [On the expansion of Schur and Schubert polynomials into standard elementary monomials; Adv. Math. 136, No. 2, 224-250 (1998; Zbl 0920.05069)]. Baxter operator; Lehmer code; Schubert polynomials; Schur functions; Young tableaux; Macdonald polynomials; weak Brunat order on Coxeter groups; Poincaré polynomials; reduced words Research exposition (monographs, survey articles) pertaining to combinatorics, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] On algebraic and combinatorial properties of Schur and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We present a general theory of Schubert polynomials, which are explicit representatives for Schubert classes in the cohomology ring of a flag variety with certain combinatorial properties. The starting point for this theory is a construction of Schubert classes in the cohomology ring of the flag variety of any semi-simple complex Lie group by Bernstein- Gelfand-Gelfand and Demazure. For the groups \(\text{SL}(n, \mathbb{C})\), Lascoux and Schützenberger made the crucial observation that one particular choice of representative of the top cohomology class yields Schubert polynomials simultaneously for all \(n\). In the present work we replicate the theory of \(\text{SL}(n, \mathbb{C})\) Schubert polynomials for the other infinite families of classical Lie groups and their flag varieties---the orthogonal groups \(\text{SO}(2n, \mathbb{C})\) and \(\text{SO}(2n+ 1,\mathbb{C})\) and the symplectic groups \(\text{Sp}(2n, \mathbb{C})\). We define Schubert polynomials to be elements in an inverse limit, which can be calculated as the unique solution of an infinite system of divided difference equations. The solution is derived using two equivalent formulas; one is an analog of the Billey-Jockusch-Stanley formula, while the other expresses our polynomials in terms of \(\text{SL}(n)\) Schubert polynomials and Schur \(Q\)- or \(P\)-functions. Our second formula involves the `shifted Edelman-Greene correspondences' and analogs of the Stanley symmetric functions. The Schubert polynomials form a \(\mathbb{Z}\)-basis for the ring in which they are defined. The non-negative integer coefficients that appear when they are multiplied give intersection multiplicities for Schubert varieties directly, without the need to reduce the product modulo an ideal. Schur functions; Schubert polynomials; Schubert classes; cohomology ring; flag variety; Lie group; orthogonal groups; symplectic groups; divided difference equations; Billey-Jockusch-Stanley formula; Stanley symmetric functions; Schubert varieties Billey, S.; Haiman, M., \textit{Schubert polynomials for the classical groups}, J. Amer. Math. Soc., 8, 443-482, (1995) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Schubert polynomials for the classical groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(W\) be a Weyl group and let \({\Sigma}_{w}, w\in W\) be a Schubert variety so that they are additive basis of homology of flag varieties and as dual theory let \({\Sigma}^{w}, w\in W\) be the Schubert differential form (or class). These classes are additive basis for dual cohomology. A classical problem of Schubert calculus is to define explicit classes \({\Sigma}^{w}, w\in W\) to represent Schubert varieties in cohomology rings of a partial flag variety. In equivariant cohomology this problem reduces to finding the polynomials \({\Sigma}^{w}[v], v,w\in W\) which are nonzero only if \([v] \geq [w]\) in Bruhat order. For more general spaces the uniqueness or even existence of generalized Schubert classes named \textbf{flow-up classes} is not guaranteed.
There is a combinatorial construction of these classes by Goresky, Kottwitz and MacPherson in [\textit{M. Goresky} et al., Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)] called \textbf{GKM theory}. The equivariant cohomology is computed directly from the Bruhat graph \({\Gamma}_W\) of the Weyl group \(W\). The Schubert classes are constructed by divided difference cohomology operators
\[
{\partial}_i : {\Sigma}^{w}(u) \rightarrow \frac{1}{{\alpha}_i}({\Sigma}^{w}(u) - s_i {\Sigma}^{w}(s_i u)),
\]
where \({\alpha}_i\) is a simple root and \(s_i\) is the reflection in the Weyl group \(W\) associated with the simple root \({\alpha}_i\). These operators were first introduced by Bernstein, Gelfand and Gelfand [\textit{I. N. Bernstein} et al., Usp. Mat. Nauk 28, No. 3(171), 3--26 (1973; Zbl 0286.57025)] and by \textit{M. Demazure} [Invent. Math. 21, 287--301 (1973; Zbl 0269.22010)] for ordinary cohomology. \textit{B. Kostant} and \textit{S. Kumar} [Adv. Math. 62, 187--237 (1986; Zbl 0641.17008)] generalized them to equivariant cohomology. These type operators are called BGG-operators. More recently, employing GKM theory Tymoczko uses a left action of \(W\) and defines new divided difference operators in [\textit{J. S. Tymoczko}, Am. J. Math. 130, No. 5, 1171--1194 (2008; Zbl 1200.14103)]. Flow-up classes for \(G/B\) are unique, so this construction agrees with the earlier work.
In this paper, the author defines GKM rings for certain subgraphs of the Bruhat graph as a combinatorial analog of equivariant cohomology. As with the Bruhat graph these rings construct the equivariant cohomology of algebraic varieties called the regular semisimple Hessenberg varieties. Two important examples of regular semisimple Hessenberg varieties are the complete flag variety \(G/B\) and the toric variety associated to the Coxeter complex.
Also he generalizes the divided difference operator for \(G/B\) to what is call the highest root Hessenberg variety. This result is a model first step toward defining bases which would allow us to investigate the representation on the cohomology (ordinary and equivariant). As a last result he announces that for the highest root Hessenberg variety the Shareshian and Wachs conjecture is true. Bernstein-Gelfand-Gelfand operator; divided difference operator; Schubert calculus; flag variety; Bruhat graph; equivariant cohomology; flow-up classes; Hessenberg variety N. J. Teff. ''The Hessenberg representation''. PhD thesis. University of Iowa, 2013. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Discrete subgroups of Lie groups A divided difference operator for the highest root Hessenberg variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The problem of computing products of Schubert classes in the cohomology ring can be formulated as the problem of expanding skew Schur polynomials into the basis of ordinary Schur polynomials. In contrast, the problem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to its basis of Schubert structure sheaves is not equivalent to expanding skew stable Grothendieck polynomials into the basis of ordinary stable Grothendiecks. Instead, we show that the appropriate \(K\)-theoretic analogy is through the expansion of skew reverse plane partitions into the basis of polynomials which are Hopf-dual to stable Grothendieck polynomials. We combinatorially prove this expansion is determined by Yamanouchi set-valued tableaux. A by-product of our results is a dual approach proof for Buch's \(K\)-theoretic Littlewood-Richardson rule for the product of stable Grothendieck polynomials. Grothendieck polynomials; Littlewood-Richardson rule; tableaux Li, H., Morse, J., Shields, P.: Structure constants for \(K\)-theory of Grassmannians revisited (2016) (preprint). arXiv:1601.04509 Combinatorial aspects of representation theory, Classical problems, Schubert calculus Structure constants for \(K\)-theory of Grassmannians, revisited | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{W. Kraśkiewicz} and \textit{P. Pragacz} [C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011)] defined certain modules \(\mathcal{S}_w (w\in S_\infty)\), which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product of KP modules always has a KP filtration, i.e. a filtration whose each successive quotients are isomorphic to KP modules. In this paper we explicitly construct such filtrations for certain special cases of these tensor product modules, namely \(\mathcal{S}_w \otimes S^d(K^i)\) and \(\mathcal{S}_w \otimes \bigwedge^d(K^i)\), corresponding to Pieri and dual Pieri rules for Schubert polynomials. Schubert polynomials; Kraśkiewicz-Pragacz modules Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Solvable, nilpotent (super)algebras, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Determinantal varieties Kraśkiewicz-Pragacz modules and Pieri and dual Pieri rules for Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper relates a formula for quiver varieties found by \textit{A. S. Buch} and \textit{W. Fulton} [Invent. Math. 135, 665-687 (1999; Zbl 0942.14027)] to Stanley symmetric functions. A Stanley symmetric function is a symmetric homogeneous power series in infinitely many variables used to determine how many reduced words a given permutation has. The connection between the two is found using Schubert polynomials. Let \(w\in S_{m}.\) A Stanley symmetric function has been shown to be a limit of Schubert polynomials [\textit{A. Lascoux} and \textit{M.-P. Schützenberger}, C. R. Acad. Sci., Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031) and \textit{I. G. MacDonald}, Lond. Math. Soc. Lect. Note Ser. 166, 73-99 (1991; Zbl 0784.05061)]: Such a polynomial can be written using Schur functions as a basis---the coefficients are denoted \(\alpha _{w\lambda }\) where \(\lambda \) is a partition of the length of \(w\). The quiver variety formula given in the author's previous work specializes to the double Schubert polynomial for the permutation \(w.\) The coefficients of this double Schubert polynomial are denoted \(c_{w}(a,b,\lambda),\) where \(a\) and \(b\) are exponents. The \(c_{w}(a,b,\lambda)\) are special cases of Littlewood-Richardson coefficients. The author shows that the constant term \(c_{w}(0,0,\lambda)\) corresponds to Stanley's coefficient \(\alpha _{w^{-1}\lambda },\) thereby directly relating the double Schubert polynomial and the symmetric function. The first half of the paper is a review of the ideas found in the author's previous paper cited above.
After proving that \(c_{w}(0,0,\lambda)=\alpha _{w^{-1}\lambda },\) the results are related to the Littlewood-Richardson conjecture, a conjecture which states that generalized Littlewood-Richardson coefficients are nonnegative. quiver varieties; Stanley symmetric functions; Schubert polynomials; Schur functions; Littlewood-Richardson conjecture; Littlewood-Richardson coefficients A.S. Buch, ''Stanley symmetric functions and quiver varieties,'' J. Algebra 235 (2001), 243--260. Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Stanley symmetric functions and quiver varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a new operator formula for Grothendieck polynomials that generalizes Magyar's Demazure operator formula [\textit{P. Magyar}, Comment. Math. Helv. 73, No. 4, 603--636 (1998; Zbl 0951.14036)] for Schubert polynomials. Our proofs are purely combinatorial, contrasting with the geometric and representation theoretic tools used by Magyar. We apply our formula to prove a necessary divisibility condition for a monomial to appear in a given Grothendieck polynomial. Schubert polynomials; Demazure operator formula Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds An orthodontia formula for Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(w \in S_n\) be a permutation, \(X_w\) the corresponding Schubert variety, and \(P_{id,w}(q)\) the corresponding Kazhdan-Lusztig polynomial. It is known that \(P_{id,w}(1)\) equals \(1\) if and only if \(X_w\) is smooth. (Since Kazhdan-Lusztig polynomials have non--negative integer coefficients and constant term \(1\), \(P_{id,w}(1)=1\) is equivalent to \(P_{id,w}(q)=1\)). In this paper, the author gives necessary and sufficient conditions for \(P_{id,w}(1)=2\): he shows that this is equivalent to a geometric condition, namely that the singular locus of \(X_w\) has exactly one irreducible component, plus a combinatorial one, namely that the permutation \(w\) avoids a list of six patterns. He further shows that when \(P_{id,w}(1)=2\), \(P_{id,w}(q)=1+q^h\) where \(h\) is computed combinatorially from \(w\).
An appendix by Sara Billey and Jonathan Weed gives a characterization of \(P_{id,w}(1)=2\) purely in terms of pattern avoidance, the number of patterns used being \(66\). Schubert variety; Kazhdan-Lusztig Woo, A, Permutations with Kazhdan-Lusztig polynomial \(P_{Id, w}(q)=1+q^h\), with an appendix by sara billey and jonathan weed, Electron. J. Combin., 16, r10, (2009) Classical problems, Schubert calculus, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects) Permutations with Kazhdan-Lusztig polynomial \(P_{id,w}(q)=1+q^{h}\). With an appendix by Sara Billey and Jonathan Weed | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors present the theory of Schur and Schubert polynomials, revisited from the point of view of generalized Thom polynomials. The Schur and Schubert polynomials are realized as first obstructions of certain fiber bundles. After presenting some results about the Thom polynomials for group actions, the authors obtain the calculation of these polynomials via the method of restriction equations. Then they obtain a new definition for the Schur and Schubert polynomials by applying the general method to compute them as Thom polynomials. They also redefine the double Schubert polynomials and the Kempf-Laksov-Schur polynomials Schur polynomials; Schubert polynomials; Thom polynomials; method of restriction equations Fehér, L.; Rimányi, R., Schur and Schubert polynomials as thom polynomials--cohomology of moduli spaces, Cent. eur. J. math., 1, 4, 418-434, (2003) Singularities of differentiable mappings in differential topology, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Enumerative problems (combinatorial problems) in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Schur and Schubert polynomials as Thom polynomials -- cohomology of moduli spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using a formula of \textit{S. C. Billey}, \textit{W. Jockusch} and \textit{R. P. Stanley} [Some combinatorial properties of Schubert polynomials, J. Algebr. Comb. 2, No. 4, 345-374 (1993; Zbl 0790.05093)], \textit{S. Fomin} and \textit{A. N. Kirillov} [Yang-Baxter equation, symmetric functions, and Schubert polynomials, Proceedings of the conference on power series and algebraic combinatorics, Firenze (1993)] have introduced a new set of diagrams that encode the Schubert polynomials. In this paper, these objects are called rc-graphs. Here, two variants of an algorithm for constructing the set of all rc-graphs for a given permutation are defined and proved. This construction makes many of the identities known for Schubert polynomials more apparent, and yields new ones. In particular, we find a new proof of Monk's rule using an insertion algorithm on rc- graphs. This insertion rule is a generalization of the Schensted insertion for tableaux. We find two conjectures of analogs of Pieri's rule for multiplying Schubert polynomials. The authors also extend the algorithm to generate the double Schubert polynomials. Schubert polynomials; rc-graphs; Monk's rule; Pieri's rule N. Bergeron and S. Billey. ''RC-graphs and Schubert polynomials''. Experiment. Math. 2 (1993), pp. 257--269.DOI. Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Enumerative combinatorics, Combinatorial identities, bijective combinatorics, Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds rc-graphs and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a formula for double Schubert and Grothendieck polynomials, specialized to two re-arrangements of the same set of variables. Our formula generalizes the usual formulas for Schubert and Grothendieck polynomials in terms of RC-graphs, and it gives immediate proofs of many other important properties of these polynomials. double Schubert polynomials Buch, Anders S.; Rimányi, Richárd, Specializations of Grothendieck polynomials, C. R. Math. Acad. Sci. Paris, 339, 1, 1-4, (2004) Classical problems, Schubert calculus, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Specializations of Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For part I see \textit{P. Pragacz} and \textit{A. Weber}, Fundam. Math. 195, No.\,1, 85--95 (2007; Zbl 1146.05049).]
We study Thom polynomials associated with Lagrange singularities. We expand them in the basis of \(\widetilde{Q}\)-functions. This basis plays a key role in the Schubert calculus of isotropic Grassmannians. We prove that the \(\widetilde{Q}\)-function expansions of the Thom polynomials of Lagrange singularities always have nonnegative coefficients. This is an analog of a result on the Thom polynomials of mapping singularities and Schur \(S\)-functions, established formerly by the last two authors. Lagrange singularities; Thom polynomials; \(\widetilde{Q}\)-functions; jets; numerical positivity; Schubert calculus; isotropic Grassmanians M. Mikosz, P. Pragacz and A. Weber, Positivity of Thom polynomials II: the Lagrange singularities, Fund. Math. 202 (2009), 65-79. Symmetric functions and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Classical problems, Schubert calculus, Homology of classifying spaces and characteristic classes in algebraic topology, Singularities of differentiable mappings in differential topology Positivity of Thom polynomials~II: the Lagrange singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials are explicit representatives for Schubert classes in the cohomology ring of a flag variety. Those of type \(A_n\) were introduced by \textit{A. Lascoux} and \textit{M. P. Schürzenberger} [Polynomes de Schubert, C. R. Acad. Sci. Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031)]. \textit{S. Billey} and \textit{M. Haiman} [Schubert polynomials for the classical groups, J. Am. Math. Soc. 8, No. 2, 443-482 (1995; Zbl 0832.05098)] extended the theory of \(A_n\)-Schubert polynomials for the groups of type \(B_n, C_n\) and \(D_n\) and their flag varieties using combinatorial methods. The starting point for the theory of Schubert polynomials is the observation of \textit{I. N. Bernstein, I. M. Gelfand} and \textit{S. I. Gelfand }[Schubert cells and cohomology of the spaces \(G/P\), Russ. Math. Surveys 28, No. 3, 1-26 (1973; Zbl 0286.57025)] that all Schubert classes can be computed by applying a sequence of divided difference operators to the cohomology class of highest codimension (the Schubert class of a point). For the type \(A_n\) Lascoux and Schürzenberger found a particular polynomial to represent the top cohomology class which yields Schubert polynomials that represent the Schubert classes simultaneously for all \(n\), the top polynomial. Billey and Haiman [loc. cit.] described the top polynomials of type \(B_n, C_n\) and \(D_n\). In the paper under review the author follows closely the original algebraic approach of Lascoux and Schürzenberger in type \(A_n\). He is able to present simple formulas for the top polynomials of type \(C_n\) and \(D_n\). He uses creation operators for \(Q\)-Schur and \(P\)-Schur functions which also allows him in types \(B_n, C_n\) and \(D_n\) to give: (1) formulas for the easy computation with all divided differences, (2) recursive structures, and (3) simplified derivations of basic properties. Schubert polynomial; Schur function; top polynomials; classical groups; Schubert variety Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Combinatorial aspects of representation theory Schubert polynomials of types A-D | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We extend the work of Fomin and Greene on noncommutative Schur functions by defining noncommutative analogs of Schubert polynomials. If the variables satisfy certain relations (essentially the same as those needed in the theory of noncommutative Schur functions), we prove a Pieri-type formula and a Cauchy identity for our noncommutative polynomials. Our results imply the conjecture of Fomin and Kirillov concerning the expansion of an arbitrary Grothendieck polynomial on the basis of Schubert polynomials; we also present a combinatorial interpretation for the coefficients of the expansion. We conclude with some open problems related to it. Schur functions; Schubert polynomials; conjecture of Fomin and Kirillov; Grothendieck polynomial Lenart, C., Noncommutative Schubert calculus and Grothendieck polynomials.Adv. Math., 143 (1999), 159--183. Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Noncommutative Schubert calculus and Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the Littlewood-Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant \(K\)-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang-Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood-Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by \textit{A. Knutson} and \textit{T. Tao} [Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)] and \textit{R. Vakil} [Ann. Math. (2) 164, No. 2, 371--422 (2006; Zbl 1163.05337)]. Grassmannian permutations Combinatorial aspects of algebraic geometry, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Groups acting on specific manifolds Littlewood-Richardson coefficients for Grothendieck polynomials from integrability | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper contributes to the study of the small quantum cohomology ring of partial flag manifolds over \({\mathbb C}\).
The classical Schubert calculus provides a description of the classical cohomology ring of a partial flag manifold in terms of generators and relations. The generators are determined by Chern classes of tautological vector bundles. A central role is played by the Giambelli formula, which expresses the Schubert classes as polynomials in the generators. These results have been extended to the quantum cohomology ring by \textit{I. Ciocan-Fontanine} [Duke Math. J. 98, No. 3, 485--524 (1999; Zbl 0969.14039)] and \textit{S. Fomin, S. Gelfand} and \textit{A. Postnikov} [J. Am. Math. Soc. 10, No. 3, 565--596 (1997; Zbl 0912.14018)].
In the paper under review, simplified and more natural proofs for these results are provided. The key idea is to study the geometry of the relationship between quantum Schubert polynomials and \textit{W. Fulton}'s universal Schubert polynomials, which solve a certain degeneracy locus problem [Duke Math. J. 96, No. 3, 575--594 (1999; Zbl 0981.14022)]. The proofs are based on calculations in the cohomology of so-called hyperquot schemes. These schemes compactify moduli spaces of maps from \({\mathbb P}^ 1\) to the flag manifold in a different way than Kontsevich's stable map spaces do. Gromov-Witten invariants; quot schemes; Schubert polynomial; degeneracy locus L. Chen : Quantum cohomology of flag manifolds, Adv. Math. 174 (2003), no. 1, 1-34. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Quantum cohomology of flag manifolds. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(k\) be a positive integer and \(OG = OG(n-k, 2n)\) be the Grassmannian that parametrizes isotropic subspaces of dimension \(n-k\) in the vector space \(\mathbb C^{2n}\), equipped with an orthogonal form. The eta polynomials \(H_{\lambda}(c)\) of Buch, Kresch, and the author are Giambelli polynomials that represent the Schubert classes in the cohomology ring of \(OG\).
In this paper using Young raising operators the author defines double eta polynomials \(H_{\lambda}(c|t)\), which represent the equivariant Schubert classes in the equivariant cohomology ring \({H^\ast}_T (OG)\), where \(T\) is a maximal torus of the complex even orthogonal group. Eta polynomials; double eta polynomials; Giambelli polynomials; Young raising operators; Schubert calculus; equivariant cohomology Tamvakis, H., \textit{double eta polynomials and equivariant Giambelli formulas}, J. Lond. Math. Soc. (2), 94, 209-229, (2016) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Double eta polynomials and equivariant Giambelli formulas | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author considers the Schubert polynomials \(X_{\pi}\in {\mathbb{Z}}[x_1,x_2,\ldots]\) associated with the permutations \(\pi\) contained in the symmetric groups \(S_n\). There are many possible ways to introduce the Schubert polynomials via divided difference operators, recursive generation without divided differences based on the Monk rule and the Bruhat order on permutations, via nil-Coxeter relations, the formula of Billey-Jockusch-Stanley, via sums of mixed shift and multiplication operators, via balanced labeled tableaux, via configurations of labeled pseudo-lines, via flagged Schur modules associated to a Rothe diagram, etc. The theme of the paper is the combinatorial generation of Schubert polynomials via sets of box diagrams. The main reasons to expect a combinatorial rule in terms of box diagrams are: the coefficients in \(X_{\pi}\) are non-negative integers and should count some discrete objects; in the special case of Grassmannian permutations the Schubert polynomial is equal to a Schur function in a finite number of variables and the well-known combinatorial properties of Schur functions should extend to Schubert polynomials. The main result of the paper is the proof of a very elegant and easily applicable combinatorial rule for the generation of Schubert polynomials conjectured in 1990 in the thesis by Kohnert. A similar type of combinatorial rule was given by Bergeron. As an intermediate step in the proof of the Kohnert conjecture the author also obtains a simplified proof of the Bergeron rule. In particular, he shows that the Bergeron rule may be also simplified to a version which is very similar to the recent combinatorial rule of Magyar proved with algebro-geometric methods. The author obtains a direct combinatorial proof of the Magyar rule as well. This shows that there is an algebro-geometric meaning of the Bergeron rule. On the other hand it makes apparent the possibility to give fully combinatorial proofs of other results concerning, e.g., the fact that the Schubert polynomial is the character of the flagged Schur module associated to a Rothe diagram. box diagrams; diagram rules; tableaux; Schubert polynomials; symmetric functions; Schur functions R. Winkel. ''Diagram rules for the generation of Schubert polynomials''. J. Combin. Theory Ser. A 86 (1999), pp. 14--48.DOI. Symmetric functions and generalizations, Polynomials, factorization in commutative rings, Grassmannians, Schubert varieties, flag manifolds Diagram rules for the generation of Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a flag variety \(\text{Fl}_n\), the classes of structure sheaves of Schubert varieties form an integral basis in the Grothendieck ring. A major open problem in the modern Schubert calculus is to determine the \(K\)-theory Schubert structure constants, which express the product of two Schubert classes in terms of this basis.
The authors derive explicit Pieri-type formulae in the Grothendieck ring of a flag variety, which generalize both the \(K\)-theory Monk formula [see \textit{C.~Lenart}, J. Pure Appl. Algebra 179, No. 1--2, 137--158 (2003; Zbl 1063.14060)] and the cohomology Pieri formula [see \textit{F.~Sottile}, Ann. Inst. Fourier 46, No. 1, 89--110 (1996; Zbl 0837.14041)]. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special classes are indexed by cycles of the form \((k-p+1, k-p+2,\dots,k+1)\) or \((k+p, k+p-1, \dots ,k)\), and are pulled back from the projection of \(\text{Fl}_n\) to the Grassmannian of \(k\)-planes.
The formula is expressed in terms of certain labelled chains in the \(k\)-Bruhat order of the symmetric group, and the multiplicities in it are certain binomial coefficients. The proof exploits algebraic-combinatorial setting of Grothendieck polynomials and a Monk-like formula for multiplying a Grothendieck polynomial by a variable. Grothendieck polynomial; Schubert variety; Bruhat order; Pieri's formula DOI: 10.1090/S0002-9947-06-04043-8 Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics, Geometric applications of topological \(K\)-theory A Pieri-type formula for the \({K}\)-theory of a flag manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study multiplication of any Schubert polynomial \(\mathfrak{S}_w\) by a Schur polynomial \(s_{\lambda}\) (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive explicit nonnegative combinatorial expressions for the expansion coefficients for certain special partitions \(\lambda\), including hooks and the \(2\times 2\) box. We also prove combinatorially the existence of such nonnegative expansion when the Young diagram of \(\lambda\) is a hook plus a box at the \((2,2)\) corner. We achieve this by evaluating Schubert polynomials at the Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the nonnegativity conjecture of Fomin and Kirillov.{
}This approach works in the more general setup of the (small) quantum cohomology ring of the complex flag manifold and the corresponding (3-point) Gromov-Witten invariants. We provide an algebro-combinatorial proof of the nonnegativity of the Gromov-Witten invariants in these cases, and present combinatorial expressions for these coefficients. Schubert polynomials; symmetric functions; Fomin-Kirillov algebra Mészáros, Karola; Panova, Greta; Postnikov, Alexander, Schur times Schubert via the Fomin-Kirillov algebra, Electron. J. Combin., 21, 1, Paper 1.39, 22 pp., (2014) Symmetric functions and generalizations, Classical problems, Schubert calculus Schur times Schubert via the Fomin-Kirillov algebra | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of this article is to supply simpler proofs of the main theorems about the (small) quantum cohomology ring of a Grassmann variety. This includes \textit{A. Bertram's} quantum version of the Pieri and Giambelli formulas [Adv. Math. 128, 289--305 (1997; Zbl 0945.14031)]. In contrast to Bertram's proofs, which require the use of quot schemes, the presented proofs in this article stay with more elementary algebraic geometric methods and do not use any moduli space techniques. Essentially everything is based only on the definition of the Gromov-Witten invariants.
The author shows that the quantum Pieri formula is a consequence of the classical Pieri formula. Furthermore, he shows that the quantum Giambelli formula follows immediately from the quantum Pieri formula together with the classical Giambelli formula and associativity of the quantum cohomology. Also a proof is given of the Grassmannian case of a formula of Fulton and Woodward for the minimal \(q\)-power which appears in a quantum product of two Schubert classes. A proof of a simple version of the rim-hook algorithm is supplied. Finally, the presentation in terms of generators and relations of the quantum cohomology of Grassmannians is obtained.
The idea of the author is to start from the simple fact that if a rational curve of degree \(d\) passes through a Schubert variety in the Grassmannian \(\text{Gr}(l,\mathbb C^n)\), then the linear span of the points of this curve gives rise to a point in \(\text{Gr}(l+d,\mathbb C^n)\) which must lie in a related Schubert variety. This idea can be used to conclude in many cases that no curves pass trough three Schubert varieties in general position because the intersection of the related Schubert varieties is empty. In particular, the quantum Giambelli formula can be deduced by knowing that certain Gromov-Witten invariants are zero, which follows because the codimensions of the related Schubert varieties add up to more than the dimension of \(\text{Gr}(l+d,\mathbb C^n)\). quantum cohomology; Gromov-Witten invariants; Grassmann varieties; Pieri formula; Giambelli formula A. Buch. ''Quantum cohomology of Grassmannians''. Comp. Math. 137 (2003), pp. 227--235. DOI. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Quantum cohomology of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author shows how to associate to any polynomial \(P\), of degree \(d\), with non-negative integer coefficients and constant term equal to 1, a pair of elements \(y_P\) and \(w_P\) in the symmetric group \(S_n\) where \(n=d+P(1)+1\). Then he proves that \(P\) is indeed the Kazhdan-Lusztig polynomial of those two elements, by reducing the problem to the case when \(P-1\) is a monomial and by using intersection cohomology of Schubert varieties. Kazhdan-Lusztig polynomials; Schubert varieties; intersection cohomology P. Polo, Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups, \textit{Repre-} \textit{sent. Theory}, 3 (1999), 90--104.Zbl 0968.14029 MR 1698201 Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Representations of finite symmetric groups, Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors introduce a new set of combinatorially defined nonsymmetric functions whose symmetrizations are Molev's dual Schur functions. \textit{A. I. Molev} [Electron. J. Comb. 16, No. 1, Research Paper R13, 44 p. (2009; Zbl 1182.05128)] described some properties of dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions and a multiplication rule for the dual Schur functions. Schur functions are an old subject and much is known about them. They are studied in relation to many different subjects from a number of different points of view. In this work they follow the Lascoux-Schützenberger approach [\textit{A. Lascoux} and \textit{M.-P. Schützenberger}, C. R. Acad. Sci., Paris, Sér. I 294, 447--450 (1982; Zbl 0495.14031)], viewing Schur functions as (symmetric) special cases of Schubert polynomials. From this point of view, it is natural to ask how one can define a larger set of nonsymmetric functions, which will include Molev's dual Schur functions as their symmetric counterparts. This theme is the main focus of their work. On the algebraic geometry side, they obtain a duality formula for the Schubert classes in Grassmannians in terms of rational Schubert (key) polynomials. Also they point out that a dominant rational Schubert polynomial can be described as a configuration of lines as in the work of \textit{S. Fomin} and \textit{A. N. Kirillov} [Discrete Math. 153, No. 1--3, 123--143 (1996; Zbl 0852.05078)]. dual Schur functions; Schubert polynomials; configuration of lines Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Rational Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials \({\mathfrak S}_ \sigma(x_ 1,x_ 2,\dots)\) indexed by permutations have been introduced and investigated by \textit{I. N. Bernstein}, \textit{I. M. Gel'fand} and \textit{S. I. Gel'fand} [Russ. Math. Surveys 28, No. 3, 1-26 (1973; Zbl 0286.57025)], \textit{M. Demazure} [Ann. Sci. École Norm. Sup., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009)], and by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031)]; see also their paper [Symmetry and flag manifolds, Lect. Notes in Math. 996, 118-144 (1983; Zbl 0542.14031)].
In this paper the theory of Schubert polynomials is recovered using the nilCoxeter algebra \({\mathfrak C}_ n\) with the identity element \(e\), given by its generators and defining relations as the \(K\)-algebra
\[
\begin{multlined} {\mathfrak C}_ n=\Bigl\langle u_ 1,\dots, u_{n-1}\mid u^ 2_ i= 0\;(i\in I_{n-1}),\;u_ i u_ j= u_ j u_ i\\ (\text{for }| i- j|\geq 2),\text{ and } u_ i u_{i+1} u_ i= u_{i+1} u_ i u_{i+1} (\text{for } i\in I_{n-2})\Bigr\rangle\end{multlined}
\]
over any commutative ring \(K\); here \(I_ n= \{1,2,\dots, n\}\). This algebra can be faithfully represented by the algebra of operators generated by \(\Phi_ i\) \((i\in I_{n-1})\),
\[
\Phi_ i(\sigma)= \begin{cases} \sigma\tau_ i &\text{if } \ell(\sigma\tau_ i)= \ell(\sigma)+1;\\ 0 & \text{otherwise}.\end{cases}
\]
Here, \(\sigma\) is any permutation in the symmetric group \({\mathcal S}_ n\) defined on \(I_ n\), \(\tau_ i\) \((i\in I_{n-1})\) is the `adjacent' transposition \((i,i+1)\), and \(\ell(\sigma)\) is the length of \(\sigma\in {\mathcal S}_ n\) defined as the minimal \(p\) such that \(\sigma= \tau_{a_ 1}\cdot\tau_{a_ 2}\cdot\dots\cdot \tau_{a_ p}\) for some \(a_ j\in I_{n-1}\). A sequence \(a= (a_ 1,\dots, a_ p)\), \(a_ j\in I_{n-1}\) is called a reduced decomposition of \(\sigma\) if \(p= \ell(\sigma)\). \(R(\sigma)\) denote the set of all reduced decompositions for \(\sigma\). For any reduced decomposition \(a= (a_ 1,\dots, a_ p)\) let us identify the monomial \(u_{a_ 1} u_{a_ 2}\cdots u_{a_ k}\) in \({\mathfrak C}_ n\) with \(\tau_{a_ 1} \cdot \tau_{a_ 2}\cdot\dots\cdot \tau_{a_ k}\) in \({\mathcal S}_ n\); the defining relations for \({\mathfrak C}_ n\) guarantee the correctness of such notation, and we see that \({\mathcal S}_ n\) gives a \(K\)-basis for \({\mathfrak C}_ n\). As usual, denote by \(\langle f,\sigma\rangle\) the coefficient of \(\sigma\in {\mathcal S}_ n\) in the \(K\)- expression for \(f\in {\mathfrak C}_ n\). Further, denote
\[
A_ i(x)= (e+ xu_{n-1})\cdot (e+ xu_{n-2})\cdot\dots\cdot (e+ xu_ i)
\]
for any \(i\in I_{n-1}\), \(\bar x= (x_ 1,\dots, x_{n-1})\), \({\mathfrak S}(\bar x)= A_ 1(x_ 1)\cdot A_ 2(x_ 2)\cdot \dots\cdot A_{n-1}(x_{n- 1})\) and let \({\mathfrak S}_ \sigma(\bar x)= \langle{\mathfrak S}(\bar x),\sigma\rangle\). Among the results of this paper is Theorem 2.2 saying that \({\mathfrak S}_ \sigma(\bar x)\) is a Schubert polynomial. The authors prove also (Lemma 2.3) that in the case of \(\text{char } K= 0\),
\[
{\mathfrak S}_ \sigma(1,\dots, 1)= {1\over p!} \sum_{(a_ 1,\dots, a_ p)\in R(\sigma)} a_ 1\cdots a_ p.
\]
Also proved is the \(q\)-analogue of this last formula conjectured by \textit{I. Macdonald} [Notes on Schubert polynomials, LACIM, Université du Québec, Montréal (1991)]:
\[
{\mathfrak S}_ \sigma(1,q,\dots, q^{n-2})= {1\over [1]\cdot[2]\cdot\dots\cdot [p]}\sum_{(a_ 1,\dots, a_ p)\in R(\sigma)} [a_ 1]\cdot\dots\cdot [a_ p]q^{\sum_{\{i\mid a_ i\leq a_{i+1}\}}} i,
\]
where \([t]= 1+ q+\cdots+ q^{t-1}\). Schubert polynomials; nilCoxeter algebra; reduced decomposition S. Fomin and R. P. Stanley. ''Schubert polynomials and the NilCoxeter algebra''. Adv. Math. 103(1994), pp. 196--207.DOI. Symmetric functions and generalizations, Representations of finite symmetric groups, Grassmannians, Schubert varieties, flag manifolds, Combinatorial identities, bijective combinatorics Schubert polynomials and the nilCoxeter algebra | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the back stable Schubert calculus of the infinite flag variety. Our main results are:
\begin{itemize}
\item a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part;
\item a novel definition of double and triple Stanley symmetric functions;
\item a proof of the positivity of double Edelman-Greene coefficients generalizing the results of Edelman-Greene and Lascoux-Schützenberger;
\item the definition of a new class of \textit{bumpless} pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman-Greene insertion algorithm;
\item the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case;
\item equivariant Pieri rules for the homology of the infinite Grassmannian;
\item homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.
\end{itemize} flag variety; Grassmannian; Schubert polynomial; Schur polynomial Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Back stable Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0759.00012.]
Let \(k_{q}[G]\) be the \(q\)-version of the algebra of functions on the group \(G=SL(N)\) constructed by the FRT-method, where \(q\) is an indeterminate. Let \(P_{d}\) be a maximal parabolic subgroup of \(G\) (obtained by omitting the simple root \(\alpha_{d}\) and \(l_{d,n}=\{(i_{1},\dots,i_{d}): 1\leq i_{1}<\dots<i_{d}\leq n\}\); it is known that \(l_{d,n}\) is in bijective correspondence with the set of Schubert varieties in \(G/P_{d}\), say \(X(\tau)\leftrightarrow\tau\in l_{n,d}\). Let \(B\) be the Borel subgroup of \(G\) consisting of upper diagonal matrices and let \(k_{q}[B]\) be the quantum Hopf algebra of functions on \(B\). Let \(w\) be an element of the Weyl group and let \(X(w)\subset G/B\) be the corresponding Schubert variety. The authors define the quantum algebras \(k_{q}[G/P_{d}]\), \(k_{q}[G/B]\), \(k_{q}[X(\tau)]\), \(k_{q}[X(w)]\); the first two are subcomodules of \(k_{q}[G]\), the last two are quotients of (respectively) the first two and are \(k_{q}[B]\)-comodules. Each of these four algebras has, in the classical case, a basis consisting of standard monomials -- compatible with canonical \(\mathbb{Z}\) or \(\mathbb{Z}^{\ell}\)-gradations. The authors prove the existence of such basis and gradations in the quantum case. Schubert varieties; quantum Hopf algebra of functions; quantum algebras; basis; gradations Lakshmibai, V.; Reshetikhin, N. Y.: Quantum deformation of sln/B and its Schubert varieties. Special functions, ICM-90 satellite conference Proceedings (1991) Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act Quantum deformations of \(SL_ n/B\) and its Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The goal of the present paper is to extend the mitosis algorithm, originally developed by \textit{A. Knutson} and \textit{E. Miller} [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] for the case of Schubert polynomials, to the case of Grothendieck polynomials. In addition we will also use this algorithm to construct a short combinatorial proof of Fomin-Kirillov's formula for the coefficients of Grothendieck polynomials. Grothendieck polynomials; pipe dreams; mitosis algorithm Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Linkage, complete intersections and determinantal ideals Mitosis algorithm for Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is an expository paper of some recent computations of Gromov-Witten invariants of a flag manifold, based on the quadratic algebra approach. The author emphasizes the role of quantum Schubert polynomials that play in this case the role similar to the usual Schubert polynomials in the classical theory (the Bernstein-Gelfand-Gelfand theorem). More detailed accounts are quoted in the references, while this exposition can be seen as a nice and friendly invitation to the subject. Gromov-Witten invariants Sergey Fomin, Lecture notes on quantum cohomology of the flag manifold, Publ. Inst. Math. (Beograd) (N.S.) 66(80) (1999), 91 -- 100. Geometric combinatorics (Kotor, 1998). Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Grassmannians, Schubert varieties, flag manifolds Lecture notes on quantum cohomology of the flag manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{N. Fan} and \textit{P. Guo} [Sci. China, Math. 65, No. 6, 1319--1330 (2022; Zbl 1490.05272)] proved a combinatorial formula for the Schubert polynomial of any permutation which avoids the patterns 1423 and 1432. The authors use this formula to prove the nonnegativity of an inclusion-exclusion-inspired formula for these Schubert polynomials, in which the coefficients are given by combinatorial expressions in the Rothe diagrams of the permutations. \textit{Y. Gao} [Eur. J. Comb. 94, Article ID 103291, 12 p. (2021; Zbl 1462.05358)] conjectured that the principal specialization \({\mathfrak S}_w(1,\dots,1)\) of the Schubert polynomial is a sum of positive constants \(c_{\mathrm {perm}(v)}\) over the permutations corresponding to all subwords \(v\) of \(w\). The inclusion-exclusion formula specializes in the Möbius inversion of the conjectured formula, and thus proves the conjecture for permutations avoiding 1432 and 1423. There are generalizations of the main lemmas which lead to conjectures and a possible framework for similar results for arbitrary permutations. Schubert polynomial; principal specialization; nonnegative linear combination Symmetric functions and generalizations, Permutations, words, matrices, Classical problems, Schubert calculus Inclusion-exclusion on Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove that if \(X\) is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring \(\mathrm{QH}(X)\) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring \(H^*(X)\) that multiplies with nonnegative structure constants. This implies that the (three point, genus zero) Gromov-Witten invariants of \(X\) are uniquely determined by Witten's presentation of \(\mathrm{QH}(X)\) and the fact that they are nonnegative. We conjecture that the same is true for any flag variety \(X=G/P\) of simply laced Lie type. For the variety of complete flags in \(\mathbb{C}^n\), this conjecture is equivalent to Fomin, Gelfand, and Postnikov's conjecture that the quantum Schubert polynomials of type A are uniquely determined by positivity properties. Our proof for Grassmannians answers a question of Fulton. quantum cohomology; Grassmannians; positivity; Gromov-Witten invariant; Schubert basis; quantum Schubert polynomials; flag varieties; symmetric functions; Seidel representation Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Positivity determines the quantum cohomology of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper of exceptional importance, the authors have been able to ``explain'' many combinatorial phenomena that are associated to a Weyl group in different contexts. The key discovery is that of a set of certain polynomials \(P_{x,y}\) in one variable with integral coefficients associated to a pair \((x,y)\) of elements of a Coxeter group \(W\). These polynomials are used extensively to (1) construct certain representations of the Hecke algebra of \(W\) thereby obtaining important information on representations of \(W\), (2) give a formula (conjecturally) for the multiplicities in the Jordan-Hölder series of Verma modules or equivalently for the formal characters of irreducible highest weight modules (extending the celebrated Weyl character formula to ``nondominant'' weights), (3) give a complete description for the inclusion-relations between various primitive ideals in the enveloping algebra of a complex semisimple Lie algebra \(\mathfrak g\); in case \(\mathfrak g\) is of type \(A_n\), this is further tied up with the dimensions of certain representations of the corresponding Weyl group (via the ``Jantzen conjecture'' -- cf. a paper by \textit{A. Joseph} [Lect. Notes Math. 728, 116--135 (1979; Zbl 0422.17004)], (4) give a measure of the failure of local Poincaré duality in the geometry of Schubert cells in flag varieties. (In a later paper [\textit{D. Kazhdan} and \textit{G. Lusztig}, Proc. Symp. Pure Math. 36, 185--203 (1980; Zbl 0461.14015)] the authors give a more precise interpretation of the coefficients of \(P_{x,y}\) in terms of a certain cohomology theory called ``middle intersection cohomology'' associated with the geometry of Schubert cells.)
Considering the various topics involved and their importance, it seems worthwhile to give a detailed review in order to give some idea of the wealth of information contained in this paper.
We first describe the combinatorial setup involved. Let \((W,S)\) be a Coxeter group. Let \(\mathbb Z[q^{1/2},q^{-1/2}]\) be the ring of Laurent polynomials in the indeterminate \(q^{1/2}\) over \(\mathbb Z\). Let \(\mathcal H\) be the free \(\mathbb Z[q^{1/2},q^{-1/2}]\)-module with \(\{T_y\mid y\in W\}\) as a basis; the multiplication in \(\mathcal H\) is given by: for \(s\in S\), \(y\in W\), \(T_s\cdot T_y=T_{sy}\) if \(l(sy)\geq l(y)\) and \(T_s\cdot T_y=(q-1)T_y+q\cdot T_{sy}\) if \(l(sy)\leq l(y)\). (Classically, one considers \(\mathbb Z[q]\)-coefficients only; the algebra \(\mathcal H\) thus obtained, called the Hecke algebra of \(W\), is isomorphic to the space of intertwining operators on the ``\(1_B^G\)''-representation of a finite Chevalley group with \(W\) as the Weyl group.)
It can be seen that \(T_y\) is invertible in \(\mathcal H\) and so \(\mathcal H\) has an involution \({}^-\) under which \(T_y\) goes to \(T_{y^{-1}}^{-1}\) and \(q^{1/2}\) goes to \(q^{-1/2}\). The main discovery of the paper can now be stated as the theorem: For any \(y\in W\), there is a unique element \(C_y\in\mathcal H\) such that (i) \(\overline C_y=C_y\) and (ii) \(C_y=\sum_{x\in W}(-1)^{l(x)+l(y)}\cdot(q^{1/2})^{l(y)}\cdot q^{-l(x)}\overline P_{x,y}\cdot T_x\), where \(P_{x,y}\in\mathbb Z[q]\) with \(P_{y,y}=1\) and \(\deg P_{x,y}\leq(l(y)-l(x)-1)/2\) if \(x\mathop{<}\limits_{\neq}y\) (\(\leq\) is the Bruhat ordering on \(W\)) and \(P_{x,y}=0\) otherwise.
The authors give an inductive formula for \(P_{x,y}\); however, no closed formula is available as yet. It is conjectured that the coefficients of \(P_{x,y}\) are nonnegative; the authors have proved it in the case of Weyl groups and affine Weyl groups by showing them to be dimensions of certain cohomology groups [cf. the authors, op. cit.].
We now describe the various applications of the polynomials \(P_{x,y}\).
(1) Representations of \(\mathcal H\): In order to obtain certain representations of \(H\), the authors introduce the notion of a \(W\)-graph as follows: It is a graph \(\Gamma\) without loops such that to each vertex \(x\in\Gamma\) is associated a subset \(I_x\) of \(S\) (the set of simple reflections in \(W\)) and to each edge \((x,y)\) is associated a nonzero integer \(\mu(x,y)\) which is required to satisfy certain compatibility conditions. (These conditions ensure that one can define a representation of \(\mathcal H\).) Define a preorder \(x\leq_\Gamma y\) on vertices of \(\Gamma\) by: \(x\leq_\Gamma y\) if there exist \(x=x_0,x_1,\dots,x_n=y\in\Gamma\) such that for all \(i\), \((x_i,x_{i+1})\) is an edge of \(\Gamma\) with \(I_{x_i}\not\subset I_{x_{i+1}}\). Let \(\sim\) be the equivalence relation associated with \(\leq_\Gamma\) (i.e. \(x\sim y\) if \(x\leq_\Gamma y\) and \(y\leq_\Gamma x\)). Then each equivalence class considered as a full subgraph of \(\Gamma\) and the assignments ``\(I_x\) and \(\mu(x,y)\)'' coming from \(\Gamma\) is a \(W\)-graph itself and thus one gets many representations of \(\mathcal H\) (e.g., the ``reflection'' representation of \(\mathcal H\) can be obtained in this way). The authors construct a \(W\)-graph from the polynomials \(P_{x,y}\) in the following way: The set \(W\) is the set of vertices and \((x,y)\) is an edge if either \(x\mathop{<}\limits_{\neq}y\) with \(\deg P_{x,y}=(l(y)-l(x)-1)/2\) or \(y<x\) with \(\deg P_{y,x}=(l(x)-l(y)-1)/2\) (one denotes such pairs by \(x\prec y\) or \(y\prec x\) as the case may be). For \(x\in W\), assign \(I_x=L_x=\{s\in S\mid l(sx)\leq l(x)\}\) and for an edge \((x,y)\), assign \(\mu(x,y)=\) leading coefficient of \(P_{x,y}\) (or \(P_{y,x}\) as the case may be). The authors show that this gives a \(W\)-graph and the corresponding representation is in fact the left-regular representation of \(\mathcal H\). The equivalence classes of this \(W\)-graph are called left cells (``left'' because the set \(L_x\) involves multiplication on the left by the elements of \(S\)). One has a similar \(W\)-graph by considering multiplication on the right and a \(W\times W^0\)-graph (\(W^0\) is the opposite group) by considering the left and right multiplications simultaneously. The equivalence classes are called right cells and two-sided cells, respectively. It turns out that the configuration of these cells forms important combinatorial data from which much information can be obtained. In case \(W=S_n\), the representations of \(W\) obtained from left cells of \(W\) by specializing \(q=1\) cover all complex representations equipped with a distinguished basis.
(2) Characters of highest weight representations of a complex semisimple Lie algebra \(\mathfrak g\): Fix a Cartan subalgebra \(\mathfrak h\) and a set of simple roots \(\Pi\) for the root system of \((\mathfrak g,\mathfrak h)\). Let \((W,S)\) be the corresponding Coxeter system. For \(x\in W\), let \(M_x\) be the Verma module with highest weight \(x\rho-\rho\) (\(\rho\) is the half-sum of positive roots) and let \(L_x\) denote the (unique) irreducible quotient of \(M_x\). For \(x,y\in W\), let \(\text{mtp}(x,y)\) be the multiplicity with which \(L_y\) occurs in a Jordan-Hölder series of \(M_x\). It is then known that \(\text{mtp}(x,y)\neq 0\) if and only if \(x\leq y\) (\(\leq\) being the Bruhat ordering). The problem of determining \(\text{mtp}(x,y)\) has been considered by several people (e.g., [\textit{J. Lepowsky} and the reviewer, J. Algebra 49, 512--524 (1977; Zbl 0381.17004); \textit{J. C. Jantzen}, Moduln mit einem höchsten Gewicht. Berlin-Heidelberg-New York: Springer-Verlag (1979; Zbl 0426.17001)]). The authors conjecture that \(\text{mtp}(x,y)=P_{x,y}(1)\). (This conjecture has been proved recently by Brylinski and Kashiwara and, independently, by Beilinson and Bernstein.) This has an equivalent formulation in terms of the formal character of \(L_x\). (Recall: If \(x=\text{id}\) then one has the Weyl character formula for a finite-dimensional representation of \(\mathfrak g\).) In his talk at the AMS Santa Cruz Conference on Finite Groups [\textit{G. Lusztig}, Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 313--317 (1980; Zbl 0453.20005)] the second author proposed a modular analogue analogue the above conjecture from which the character formula of a rational irreducible representation of a Chevalley group \(G\) over an algebraically closed field of positive characteristic can be obtained.
(3) Theory of primitive ideals in the enveloping algebra \(U(\mathfrak g)\) of a complex semisimple Lie algebra \(\mathfrak g\): A two-sided ideal \(I\) in \(U(\mathfrak g)\) is said to be ``primitive'' if it is the annihilator of an irreducible \(\mathfrak g\)-module. One is then interested in the space \(X\) of primitive ideals (equipped with the Jacobson topology). As every irreducible \(\mathfrak g\)-module has a central character, one gets a map \(X\rightarrow Z(\hat{\mathfrak g})\) (\(=\text{Hom}_{\mathbb C-\text{alg}}(Z(\mathfrak g),{\mathbb C})\) where \(Z(\mathfrak g)\) is the centre of \(U(\mathfrak g)\)). By a well-known theorem of Harish-Chandra \(Z(\hat{\mathfrak g})\simeq\mathfrak h^\ast|_W\). Let \(\Lambda\) be an equivalence class. One is interested in the fibre \(X_\Lambda\) over \(\Lambda\). Let \(\lambda\in\Lambda\) and \(J(\lambda)=\text{Ann}\,L(\lambda)\), where \(L(\lambda)\) is the irreducible \(\mathfrak g\)-module with highest weight \(\lambda\). Then \(J(\lambda)\in X_\lambda\) and in fact a deep result of Duflo asserts that every element of \(X_\Lambda\) is of this form. There are various results known about the structure of fibres, ``similarity'' of two fibres, etc. The most important problem is to determine the inclusion relation between two elements of the same fibre. Known results by Borho, Duflo, Jantzen, Joseph, Vogan, etc. give sufficient conditions for it. However, since the conjecture ``\(\text{mtp}(x,y)=P_{x,y}(1)\)'' is proved to be true, the left cells of \(W\) determine the inclusion completely. To be more precise, let \(\lambda\) be an antidominant integral element. Let \(\Lambda\) be the equivalence class to which it belongs. Then every element of \(X_\Lambda\) is of the form \(J(x\cdot\lambda)\) where \(x\cdot\lambda=x(\lambda+\rho)-\rho\). Then \(J(x\cdot\lambda)\subseteq J(y\cdot\lambda)\) if and only if \(y\leq_Lx\). (The inclusion relations in other fibres can be written down with the help of a ``suitable'' subgroup \(W\) which corresponds to a sub-root-system.) It may be recalled that the primitive spectrum \(X\) is related to nilpotent orbits in \(\mathfrak g\).
(4) Geometry of Schubert cells in flag varieties: Let \(G\) be a complex semisimple algebraic group, \(B\) a Borel subgroup and \(G/B\) be the corresponding flag variety. \(B\) acts on \(G/B\) and one has the Bruhat decomposition \(G/B=\bigcup_{x\in W}BxB\). For \(y\in W\), let \(X(y)=\overline{ByB}=\bigcup_{x\leq y}BxB\) (\(\leq\) is the Bruhat ordering). Let \(e_x\) be the point \(xB\in G/B\). Then one is interested in determining the nature of the singularity at \(e_x\) when considered as a point of \(X(y)\;(x\leq y)\). The authors show that the condition ``\(P_{x',y}\equiv 1\) for all \(x\leq x'\leq y\)'' is related to the singularity of \(e_x\) in \(X(y)\). A closer tie-up is given in a later paper [the authors, loc. cit.].
Considering the central role played by the polynomials \(P_{x,y}\) in above-mentioned contexts, it is desirable to have an explicit knowledge of their coefficients in terms of certain combinatorial data. The relation \(x\prec y\) is another key notion which should be investigated further. Weyl groups; Coxeter groups; representations of Hecke algebras; Jordan-Hölder series of Verma modules; irreducible highest weight modules; Weyl character formula; primitive ideals in enveloping algebras; complex semisimple Lie algebras; local Poincaré duality; geometry of Schubert cells; flag varieties; intersection cohomology; Laurent polynomials; intertwining operators; finite Chevalley groups; affine Weyl groups; cohomology groups; simple reflections; highest weight representations; Cartan subalgebras D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, \textit{Invent.} \textit{Math.}, 53 (1979), no. 2, 165--184.Zbl 0499.20035 MR 560412 Reflection and Coxeter groups (group-theoretic aspects), Representation theory for linear algebraic groups, Hecke algebras and their representations, Universal enveloping (super)algebras, Linear algebraic groups over arbitrary fields, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Group actions on varieties or schemes (quotients) Representations of Coxeter groups and Hecke algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0511.00010.]
In this article the authors study the cohomology and Grothendieck ring of flag manifolds. It starts with a quick and elegant introduction to Schubert functions using symmetry operators. The origin of this approach is works of Demazure and Gelfand-Gelfand-Bernstein. The authors give a remarkable formula for some of the symmetrizing operators in terms of Vandermonde determinants that generalizes Jacobi's expression for Schur functions, Weyl's character formula for linear groups and Bott's theorem for line bundles on flag manifolds. Then, extending work of Ehresman, Chevalley and Monk they introduce the Ehresmanoeder and treat formulas for muliplication of Schubert polynomials, like the Pieri formula. The main reason for studying the Schubert polynomials is that they give a basis for the cohomology ring of the flag manifold. The authors go on studying this ring and show how to express an element of the ring in terms of Schubert polynomials and then give an effective way of expressing polynomials in terms of elementary symmetric polynomials. The latter method gives a strengthening of straightening methods of Bott and Rota. - The authors also show how their methods can be used to express the projective degree of a Schubert cycle (Schubert polynomial) in terms of the number of paths in the Ehresmanoeder.
Finally they indicate how their methods can give interesting information about (i) The representation of the linear and symmetric groups. (ii) The enumerative geometry of flag manifolds. (iii) Root systems and Coxeter groups. (vi) Determinants. (v) Ferrers diagrams and Young tableaux. - They show that their methods give analogous results in the cohomology ring and the Grothendieck ring of a flag manifold. Grothendieck polynomials; projective degree of Schubert cycles; flag manifolds; symmetrizing operators; Ehresmanoeder; Schubert polynomials; Pieri formula; enumerative geometry; Root systems; Coxeter groups; Young tableaux; cohomology ring; Grothendieck ring Lascoux, Alain and Schützenberger, Marcel-Paul, Symmetry and flag manifolds, Invariant Theory ({M}ontecatini, 1982), Lecture Notes in Math., 996, 118-144, (1983), Springer, Berlin Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Cohomology theory for linear algebraic groups, Homogeneous spaces and generalizations, Representation theory for linear algebraic groups Symmetry and flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This text grew out of an advanced course that the author taught at the Fourier Institute (Grenoble, France) during the 1995-96 academic year. The first part (chapter I) is of purely algebraic-combinatorial nature and provides a modern, very thorough introduction to the classical topic of symmetric functions, with a special emphasis on the combinatorics of Schur polynomials. The various properties of the family of Schur polynomials, which have proved to be a powerful tool in the representation theory of groups, are described by means of the combinatorial operations on Young diagrams (or Young tableaux) and the insertion algorithm of D. Knuth, and the corresponding various (classical and more recent) identities such as Pieri's formulae, Giambelli's formula, the rule of Littlewood-Richardson, and others are derived in the course of the discussion. This introductory chapter also contains a section on Kostka-Foulkes polynomials, including the proof of the Foulkes conjecture by A. Lascoux and M.-P. Schützenberger (1978), and some comments on the action of the symmetric group on the set of (semi-standard) Young diagrams.
The second part (chapter II) is devoted to the study of the so-called Schubert polynomials, which were introduced by A. Lascoux and M.-P. Schützenberger about twenty years ago. These polynomials, defined in terms of divided differences, are closely related to the Bruhat order on symmetric groups as well as to certain Hecke algebras of these groups, to the celebrated Yang-Baxter equation, and to certain Schur functions. All this plentiful, fascinating and fairly recent material is thoroughly covered by chapter II, together with numerous combinatorial applications. The third part (chapter III) is, in contrast, of purely geometric nature. Its main topic is the algebro-geometric, enumerative study of Schubert varieties inside Grassmannians and flag manifolds. The classical fact that the homology classes of Schubert varieties can be represented by Schur polynomials, and also by the more recently introduced Schubert polynomials, makes it possible to interpret most of the combinatorial results of the previous chapters in the language of algebraic geometry. Moreover, since Schubert varieties are also universal models for certain degeneracy loci of vector bundle morphisms, Schur or Schubert polynomials can be used to compute the homology classes of those degeneracy loci in terms of the characteristic classes of the involved vector bundles. This method is described and illustrated in the course of chapter III, whereat the basic concepts and facts from algebraic geometry (e.g.: Grassmannians, Schubert varieties, Chern classes of vector bundles, flag varieties, degeneracy loci of vector bundle maps) are briefly explained for the purpose of self-containedness of the text.
In addition, the author has included a concise appendix to the text, in which the fundamental notions from algebraic topology (i.e., singular homology and cohomology groups, fundamental classes and Poincaré duality, and some concepts from algebro-geometric intersection theory) are neatly compiled. As for the geometric part of this excellent text, the reader should be referred to the recent, more detailed, and really brilliant lecture notes ``Schubert varieties and degeneracy loci'' by \textit{W. Fulton} and \textit{P. Pragacz} [Lect. Notes Math. 1689 (1998)]. The text under review comes with numerous further-leading exercises and remarks, and with a rich bibliography, which makes the study of it very profitable. symmetric functions; Schur polynomials; Young diagrams; Schubert polynomials; Schubert varieties; degeneracy loci of vector bundle morphisms; Grassmannians; Chern classes; flag varieties Manivel, Laurent, Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence, 2-85629-066-3, Cours Spécialisés [Specialized Courses] 3, vi+179 pp., (1998), Société Mathématique de France, Paris Grassmannians, Schubert varieties, flag manifolds, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Research exposition (monographs, survey articles) pertaining to combinatorics, Symmetric functions and generalizations, Representations of finite symmetric groups, Combinatorial aspects of representation theory Symmetric functions, Schubert polynomials and degeneracy loci | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe the torus-equivariant cohomology of weighted partial flag orbifolds \({\text{w}}\Sigma\) of type A. We establish counterparts of several results known for the partial flag variety that collectively constitute what we refer to as ``Schubert Calculus on \({\text{w}}\Sigma \)''. For the weighed Schubert classes in \({\text{w}}\Sigma \), we give the Chevalley's formula. In addition, we define the weighted analogue of double Schubert polynomials and give the corresponding Chevalley-Monk's formula. weighted flag varieties; equivariant cohomology; Schubert classes; double Schubert polynomials Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) The equivariant cohomology of weighted flag orbifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials There is a remarkable formula for the principal specialization of a type A Schubert polynomial as a weighted sum over reduced words. Taking appropriate limits transforms this to an identity for the backstable Schubert polynomials recently introduced by \textit{T. Lam} et al. [Compos. Math. 157, No. 5, 883--962 (2021; Zbl 07358686)]. In this paper, the authors identify some apparently new analogues of Macdonald's identity for the principal specializations of Schubert polynomials in other classical types B, C, and D and also derive some more general identities for Grothendieck polynomials. The methods used are based on the algebraic techniques of \textit{S. Fomin} and \textit{R. P. Stanley} [Adv. Math. 103, No. 2, 196--207 (1994; Zbl 0809.05091)]. Schubert polynomials; Grothendieck polynomials; Coxeter systems; reduced words Symmetric functions and generalizations, Classical problems, Schubert calculus Principal specializations of Schubert polynomials in classical types | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we introduce a family of symmetric polynomials by specializing the factorial Schur polynomials. We show that it represents the weighted Schubert classes of the cohomology of the weighted Grassmannian introduced by Corti-Reid, and we regard it as an analogue of the Schur polynomials. Furthermore, we prove that these polynomials are the characters of certain representations, and hence, we give an interpretation of the Schubert structure constants of the weighted Grassmannians as the (rational) multiplicities of the tensor products of the representations. We also derive two determinantal formulas for the weighted Schubert classes: One is in terms of the special weighted Schubert classes, and the other is in terms of the Chern classes of the tautological orbi-bundles. Schur polynomial; weighted Grassmannian; orbifold; Schubert calculus; representation Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Combinatorial aspects of representation theory, Topology and geometry of orbifolds, Classical problems, Schubert calculus Schur polynomials and weighted Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Authors' abstract: We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold \(G/B\). The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. In the second half of the paper, we study the classical flag manifold and discuss related combinatorial objects: flagged Schur polynomials, \(312\)-avoiding permutations, generalized Gelfand-Tsetlin polytopes, the inverse Schubert-Kostka matrix, parking functions, and binary trees. flag manifold; Schubert varieties; Bruhat order; saturated chains; harmonic polynomials; Grothendieck ring; Demazure modules; Schubert polynomials; flagged Schur polynomials; 312-avoiding permutations; Kempf elements; vexillary permutations; Gelfand-Tsetlin polytope; toric degeneration; parking functions; binary trees Postnikov, A.; Stanley, R., Chains in the Bruhat order, J. Algebr. Comb., 46, 133-174, (2009) Grassmannians, Schubert varieties, flag manifolds, Permutations, words, matrices, Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Chains in the Bruhat order | 0 |
End of preview. Expand
in Dataset Viewer.
- Downloads last month
- 70