Roland Bacher
We relate properties of weighted flags (or multiflags) of type AD to statistics of the corresponding Weyl groups. For type A, we recover the Mahonian statistics on symmetric groups. Finally, we sketch briefly an easy extension incorporating statistics for so-called Euler-polynomials.
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The generating functions of the major index and of the flag-major index, with each of the one-dimensional characters over the symmetric and hyperoctahedral group, respectively, have simple product formulas. In this paper, we give a factorial-type formula for the generating function of the D-major index with sign over the Weyl groups of type D. This completes a picture which is now known for all the classical Weyl groups.
We study the generalized Galois numbers which count flags of length r in N-dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically Gaussian normally distributed as N becomes large. Furthermore, we interpret the generalized Galois numbers as weighted inversion statistics on the descent classes of the symmetric group on N elements and identify their asymptotic limit as the Mahonian inversion statistic when r approaches infinity. Finally, we apply our statements to derive further statistical aspects of generalized Rogers-Szegoe polynomi…
Signed Mahonian Polynomials on Derangements in Classical Weyl Groups
本文推导了经典Weyl群（对称群、超八面体群和偶符号置换群）中错排上的（带符号）Mahonian多项式的公式，并应用这些公式得到了具有偶数长度的错排数量的枚举公式。
We study the distribution of the major index with sign on some parabolic quotients of the symmetric group, extending and generalizing simultaneously results Gessel-Simion and Adin-Gessel-Roichman, and on some special trees that we call rakes. We further consider and compute the distribution of the flag-major index on some parabolic quotients of wreath products and other related groups. All these distributions turn out to have very simple factorization formulas.
We study the generating polynomial of the flag major index with each one-dimensional character, called signed Mahonian polynomial, over the colored permutation group, the wreath product of a cyclic group with the symmetric group. Using the insertion lemma of Han and Haglund-Loehr-Remmel and a signed extension established by Eu et al., we derive the signed Mahonian polynomial over the quotients of parabolic subgroups of the colored permutation group, for a variety of systems of coset representatives in terms of subsequence restrictions. This generalizes the related work over parabolic quotients…
Signed Mahonian Polynomials on Colored Derangements
研究带色错位上的符号马霍尼多项式，推导偶长度彩色错位计数公式及偶奇长度错位数量差公式。
A new extension of the major index, defined in terms of Coxeter elements, is introduced. For the classical Weyl groups of type $B$, it is equidistributed with length. For more general wreath products it appears in an explicit formula for the Hilbert series of the (diagonal action) invariant algebra.
Markov chains on Weyl groups from the geometry of the flag variety
本文研究了旗空间 $G/B$ 上的 Burnside 过程，通过 Bruhat 分解在对称群上构建 Markov 链，给出实用算法并确定其极限行为，最终推广至有限 Chevalley 群，在 Weyl 群上提出一类新的 Markov 链。
$B_{n-1}$-orbits on the flag variety and the Bruhat graph of the symmetric group
本文研究复一般线性群旗簇上 $B_{n-1}$-轨道的结构，通过将其表示为两个 Borel 子群轨道的交集，用 Weyl 群元素对给出组合描述，简化闭包关系与幺半群作用分析并得到轨道维数公式。
A finite Coxeter group $W$ has a natural metric $d$ and if $\mathcal{M}$ is a subset of $W$, then for each $u\in W$, there is $q\in \mathcal{M}$ such that $d(u,q)=d(u,\mathcal{M})$. Such $q$ is not unique in general but if $\mathcal{M}$ is a Coxeter matroid, then it is unique, and we define a retraction $\mathcal{R}^m_{\mathcal{M}}\colon W\to \mathcal{M}\subset W$ so that $\mathcal{R}^m_{\mathcal{M}}(u)=q$. The $T$-fixed point set $Y^T$ of a $T$-orbit closure $Y$ in a flag variety $G/B$ is a Coxeter matroid, where $G$ is a semisimple algebraic group, $B$ is a Borel subgroup, and $T$ is a maxim…
We examine the relationship between the (double) Schubert polynomials of Billey-Haiman and Ikeda-Mihalcea-Naruse and the (double) theta and eta polynomials of Buch-Kresch-Tamvakis and Wilson from the perspective of Weyl group invariants. We obtain generators for the kernel of the natural map from the corresponding ring of Schubert polynomials to the (equivariant) cohomology ring of symplectic and orthogonal flag manifolds.
Flag domains are open orbits of real semisimple Lie groups in flag manifolds of their complexifications. Certain group theoretically defined compact complex submanifolds, which are regarded as cycles, are of basic importance for their complex geometric and representation theoretic properties. It is known that there are optimal Schubert varieties which intersect the cycles transversally in finitely many points and in particular determine them in homology. Here we give a precise description of these Schubert varieties in terms of certain subsets of the Weyl group and compute their total number f…
Integer Representations of the Generalized Symmetric Groups
本研究针对广义对称群$G(m,1,n)$，构建混合基数系统与次上溢函数实现其元素与正整数的一一对应，定义逆序统计量给出新枚举系统，证明标志主指标与逆序统计量等分布，确立其Mahonian性质。
Degeneracy locus formulas for amenable Weyl group elements
本文提出了一类可容Weyl群元素，并推导了其在辛和正交退化领域中的上同调类公式，推广了双theta和eta多项式。
Combinatorial Curve Neighborhood of the Affine Flag Manifold of Type $A_{n-1}^1$
本文针对$A_{n-1}^{(1)}$型仿射旗流形的组合曲线邻域问题，给出了计算其矩图中对应极大元的组合公式与算法，为相关代数组合研究提供了工具。
Schubert varieties in the full flag variety of Kac-Moody type are indexed by elements of the corresponding Weyl group. We give a practical criterion for when two such Schubert varieties (from potentially different flag varieties) are isomorphic, in terms of the Cartan matrix and reduced words for the indexing Weyl group elements. As a corollary, we show that two such Schubert varieties are isomorphic if and only if there is an isomorphism between their integral cohomology rings that preserves the Schubert basis.
Schubert varieties in finite dimensional flag manifolds G/P are a well-studied family of projective varieties indexed by elements of the corresponding Weyl group W. In particular, there are many tests for smoothness and rational smoothness of these varieties. One key result due to Lakshmibai-Sandhya is that in type A the smooth Schubert varieties are precisely those that are indexed by permutations that avoid the patterns 4231 and 3412. Recently, there has been a flurry of research related to the infinite dimensional analogs of flag manifolds corresponding with G being a Kac-Moody group and W …
We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P which is a direct generalization of the classical Chevalley formula. Our formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, for the Demazure characters. This model can be viewed as a discrete counterpart of the Littelmann path model, and has several advantages. Our construction is given in terms of a certain R-matrix, that is, a collection of operators satisfying the Yang-Baxter equation. It reduces to …
Affine Deligne-Lusztig varieties can be thought of as affine analogs of classical Deligne-Lusztig varieties, or Frobenius-twisted analogs of Schubert varieties. We provide a method for proving a non-emptiness statement for affine Deligne-Lusztig varieties inside the affine flag variety associated to affine Weyl group elements satisfying a certain length additivity hypothesis. In particular, we prove that non-emptiness holds whenever it is conjectured to do so for alcoves in the shrunken dominant Weyl chamber, providing a partial converse to the emptiness results of Goertz, Haines, Kottwitz, an…
For any Lie algebra of classical type or type $G_2$ we define a $K$-theoretic analog of Dunkl's elements, the so-called truncated {\it Ruijsenaars-Schneider-Macdonald elements}, $RSM$-elements for short, in the corresponding {\it Yang-Baxter group}, which form a commuting family of elements in the latter. For the root systems of type $A$ we prove that the subalgebra of the {\it bracket algebra} generated by the RSM-elements is isomorphic to the Grothendieck ring of the flag variety. In general, we prove that the subalgebra generated by the {\it images} of the RSM-elements in the corresponding …
Let G be a semi-simple simply connected group over complex numbers. In this paper we give a geometric definition of the (dual) Weyl modules over the group G[t] and show that their characters form an eigen-function of the lattice version of the q-Toda integrable integrable system (defined by means of the quantum group version of Kostant-Whittaker reduction due to Etingof and Sevostyanov). All the proofs are algebro-geometric and rely on our previous work which interprets the universal eigen-function of the q-Toda system in terms of rings of functions on the spaces of based quasi-maps from P^1 t…
A presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part II: quantum double Grothendieck polynomials
本文证明了量子双重Grothendieck多项式在A型旗流形的环面等变量子K-理论环的呈现中代表对偶Schubert类，其关键工具是半无限Schubert类的显式公式。
The purpose of this note is to give a refinement of the product formula proved in [1] for the Poincare polynomial of a smooth Schubert variety in the flag variety of an algebraic group G over C. This yields a factorization of the number of elements in a Bruhat interval [e,w] in the Weyl group W of G provided the Schubert variety associated to w is smooth. This gives an elementary necessary condition for a Schubert variety in the flag variety to be smooth.
Let G be a classical complex Lie group, P any parabolic subgroup of G, and X = G/P the corresponding homogeneous space, which parametrizes (isotropic) partial flags of subspaces of a fixed vector space. In the mid 1990s, Fulton, Pragacz, and Ratajski asked for global formulas which express the cohomology classes of the universal Schubert varieties in flag bundles -- when the space X varies in an algebraic family -- in terms of the Chern classes of the vector bundles involved in their definition. This has applications to the theory of degeneracy loci of vector bundles and is closely related to …
In this paper we develop machinery for studying sequences of representations of any of the three families of classical Weyl groups, extending work of Church, Ellenberg, Farb, and Nagpal on the symmetric groups S_n to the signed permutation groups B_n and the even-signed permutation groups D_n. For each family W_n, we present an algebraic framework where a sequence V_n of W_n-representations is encoded into a single object we call an FI_W-module. We prove that if an FI_W-module V satisfies a simple finite generation condition then the structure of the sequence is highly constrained. One consequ…
Let G be a simply-connected simple compact Lie group over the complex numbers. The affine Grassmannian is a projective ind-variety, homotopy-equivalent to the loop space of G and closely analogous to a maximal flag variety of the classical Grassmannian manifold. It has a Schubert cell decomposition indexed by the coroot lattice or equivalently by the minimal length coset representatives for the affine Weyl group modulo the Weyl group for G. The closure of an affine Schubert cell is a finite dimensional projective variety that we call an affine Schubert variety. In this paper we completely dete…
Let $G=PSO(2n+1, \mathbb{C}) (n \ge 3)$ and $B$ be the Borel subgroup of $G$ containing maximal torus $T$ of $G.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline{i}$ of $w.$ In this article, we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ is the longest element of the Weyl group $W.$ We describe all the reduced express…
Let $W\ltimes L$ be an irreducible affine Weyl group with Coxeter complex $Σ$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $Σ$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $γ$-vector, and hence, symmetric and unimodal coefficients. In the class…
For each infinite series of the classical Lie groups of type B,C or D, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the appropriate flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. They are also positive in a certain sense, and when indexed by maximal Grassmannian elements, or by the longest element in a fini…
Restrictions of the dot action representation
本文证明外尔群点作用表示限制到A型抛物子群时分解为A型点作用表示的直和，并为B/C和D型经典外尔群提供组合公式，进而证明Lesnevich猜想及加泰罗尼亚数计数。
Let $G$ be a simple algebraic group of adjoint type over $\mathbb{C},$ whose root system is of type $F_{4}.$ Let $T$ be a maximal torus of $G$ and $B$ be a Borel subgroup of $G$ containing $T.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline{i}$ of $w.$ In this article, we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ i…
The aim of this article is to present a smoothness criterion for Schubert varieties in generalized flag manifolds $G/B$ in terms of patterns in root systems. We generalize Lakshmibai-Sandhya's well-known result that says that a Schubert variety in $SL(n)/B$ is smooth if and only if the corresponding permutation avoids the patterns 3412 and 4231. Our criterion is formulated uniformly in general Lie theoretic terms. We define a notion of pattern in Weyl group elements and show that a Schubert variety is smooth (or rationally smooth) if and only if the corresponding element of the Weyl group avoi…
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