A note on Bruhat order and double coset representatives

Generic Newton points and the Newton poset in Iwahori double cosets

We consider the Newton stratification on Iwahori double cosets in the loop group of a reductive group. We describe a group-theoretic condition on the generic Newton point, called cordiality, under which the Newton poset (i.e. the index set for non-empty Newton strata) is saturated and Grothendieck's conjecture on closures of the Newton strata holds. Finally, we give several large classes of Iwahori double cosets for which this condition is satisfied by studying certain paths in the associated quantum Bruhat graph.

The Möbius function on Affine Grassmannian elements

To any saturated chain in the affine Weyl group whose translation parts are sufficiently regular, we associate a near path and a far path in the quantum Bruhat graph. Using this, working in the Bruhat order on the minimal-length representatives of the cosets in the affine Weyl group with respect to the finite Weyl group, we characterize the pairs of elements for which the Möbius function is nonzero. This is applied to obtain explicit expansions in the $K$-theory of affine Grassmannians, of the basis of ideal sheaves into the basis of structure sheaves of Schubert varieties.

On the shape of Bruhat intervals

Let (W,S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and J a subset of S. Let $W^J$ denote the set of minimal coset representatives modulo the parabolic subgroup $W_J$. For w in $W^J$, let $f^{w,J}_{i}$ denote the number of elements of length i below w in Bruhat order on $W^J$ (notation simplified to $f^{w}_{i}$ in the case when J=S). We show that $f^{w,J}_{i}$ is less than or equal to $f^{w,J}_{j}$ when i < j and j is less than or equal to the length of w minus i. Furthermore, we express when an initial and final interval of the f's is symmetric aro…

Cyclic sieving of finite Grassmannians and flag varieties

In this paper we prove instances of the cyclic sieving phenomenon for finite Grassmannians and partial flag varieties, which carry the action of various tori in the finite general linear group GL_n(F_q). The polynomials involved are sums of certain weights of the minimal length parabolic coset representatives of the symmetric group S_n, where the weight of a coset representative can be written as a product over its inversions.

一个原子 Coxeter 呈现

An atomic Coxeter presentation

本文通过将抛物双陪集分解为原子并分类原子复合间的辫关系，证明了 Matsumoto 定理，进而给出 nilCoxeter 代数胚的呈现，并由此构造出带长度函数和 Bruhat 序的新组合结构。

Carries, group theory, and additive combinatorics

Given a group G and a normal subgroup H we study the problem of choosing coset representatives with few carries.

基于Brunn--Minkowski不等式的支配下Bruhat区间的渐近对数凹性

Asymptotic Log-concavity of Dominant Lower Bruhat Intervals via Brunn--Minkowski Inequality

本文针对仿射Weyl群中抛物型下Bruhat区间的长度计数序列，通过构造伸缩序列的离散测度与阶梯函数，证明其渐近收敛于多胞体对应的连续测度密度，并借助Brunn--Minkowski不等式得出该密度的对数凹性，从而确立序列的渐近对数凹性。

Deformations of permutation representations of Coxeter groups

The permutation representation afforded by a Coxeter group W acting on the cosets of a standard parabolic subgroup inherits many nice properties from W such as a shellable Bruhat order and a flat deformation over Z[q] to a representation of the corresponding Hecke algebra. In this paper we define a larger class of ``quasiparabolic" subgroups (more generally, quasiparabolic W-sets), and show that they also inherit these properties. Our motivating example is the action of the symmetric group on fixed-point-free involutions by conjugation.

Probability distributions of multi-species q-TAZRP and ASEP as double cosets of parabolic subgroups

We write explicit contour integral formulas for probability distributions of the multi-species q-TAZRP and the multi-species ASEP starting with q-exchangeable initial conditions. The formulas are equal to the corresponding explicit contour integral formulas for the single-species q-TAZRP ([Korhonen-Lee 2014, Wang-Waugh 2016]) and ASEP [Tracy-Widom 2007], with a factor in front of the integral. For the multi-species q-TAZRP, we use a decomposition theorem for elements of double cosets of parabolic subgroups in a Coxeter group. The set of distinguished double coset representatives with minimal l…

Special matchings and parabolic Kazhdan-Lusztig polynomials

We prove that the combinatorial concept of a special matching can be used to compute the parabolic Kazhdan-Lusztig polynomials of doubly laced Coxeter groups and of dihedral Coxeter groups. In particular, for this class of groups which includes all Weyl groups, our results generalize to the parabolic setting the main results in [Advances in Math. {202} (2006), 555-601]. As a consequence, the parabolic Kazhdan-Lusztig polynomial indexed by $u$ and $v$ depends only on the poset structure of the Bruhat interval from the identity element to $v$ and on which elements of that interval are minimal co…

Poincare series of subsets of affine Weyl groups

In this note, we identify a natural class of subsets of affine Weyl groups whose Poincare series are rational functions. This class includes the sets of minimal coset representatives of reflection subgroups. As an application, we construct a generalization of the classical length-descent generating function, and prove its rationality.

A Note on Element Centralizers in Finite Coxeter Groups

The normalizer $N_W(W_J)$ of a standard parabolic subgroup $W_J$ of a finite Coxeter group $W$ splits over the parabolic subgroup with complement $N_J$ consisting of certain minimal length coset representatives of $W_J$ in $W$. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type $D_n$) the centralizer $C_W(w)$ of an element $w \in W$ is in a similar way a semidirect product of the centralizer of $w$ in a suitable small parabolic subgroup $W_J$ with complement isomorphic to the normalizer complement $N_J$.

Commuting elements in conjugacy classes: An application of Hall's Marriage Theorem

Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c \in C and d \in D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ~. One of the proofs involves an interesting application of the famous Marriage Theorem of Philip Hall. The paper concludes by discussing some aspects of these theorems and of the relation ~ in the particular cases of symmetric and general linear grou…

Equidistribution of negative statistics and quotients of Coxeter groups of type B and D

We generalize some identities and q-identities previously known for the symmetric group to Coxeter groups of type B and D. The extended results include theorems of Foata and Schützenberger, Gessel, and Roselle on various distributions of inversion number, major index, and descent number. In order to show our results we provide caracterizations of the systems of minimal coset representatives of Coxeter groups of type B and D.

Group Coding with Complex Isometries

We investigate group coding for arbitrary finite groups acting linearly on a vector space. These yield robust codes based on real or complex matrix groups. We give necessary and sufficient conditions for correct subgroup decoding using geometric notions of minimal length coset representatives. The infinite family of complex reflection groups G(r,1,n) produces effective codes of arbitrarily large size that can be decoded in relatively few steps.

A two-sided analogue of the Coxeter complex

For any Coxeter system $(W,S)$ of rank $n$, we introduce an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples $(I,w,J)$, where $I$ and $J$ are subsets of the set $S$ of simple generators, and $w$ is a minimal length representative for the parabolic double coset $W_I w W_J$. There is exactly one maximal face for each element of the group $W$. The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the $h$-pol…

有限阿贝尔群中瓦片（tile）的周期性

Periodicity of tiles in finite Abelian groups

本文为有限阿贝尔群引入周期平铺（PT）性质，完整分类了具有该性质的循环群，并构造了具有 PT 性质但不满足 Hajós 性质的非循环群，同时得到新的 Tile⟹Spectral 群族，并刻画了初等 p-群中瓦片作为陪集代表系的结构。

On parabolic Kazhdan-Lusztig R-polynomials for the symmetric group

Parabolic $R$-polynomials were introduced by Deodhar as parabolic analogues of ordinary $R$-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic $R$-polynomials for the symmetric group. Let $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$, and let $S=\{s_i\,|\, 1\leq i\leq n-1\}$ be the generating set of $S_n$, where for $1\leq i\leq n-1$, $s_i$ is the adjacent transposition. For a subset $J\subseteq S$, let $(S_n)_J$ be the parabolic subgroup generated by $J$, and let $(S_n)^{J}$ be the set of minimal coset representatives for $S_n/(S…

Variations of the Poincaré series for affine Weyl groups and q-analogues of Chebyshev polynomials

Let $(W,S)$ be a Coxeter system and write $P_W(q)$ for its Poincaré series. Lusztig has shown that the quotient $P_W(q^2)/P_W(q)$ is equal to a certain power series $L_{W}(q)$, defined by specializing one variable in the generating function recording the lengths and absolute lengths of the involutions in $W$. The simplest inductive method of proving this result for finite Coxeter groups suggests a natural bivariate generalization $L^J_W(s,q) \in \mathbb{Z}[[s,q]]$ depending on a subset $J\subset S$. This new power series specializes to $L_W(q)$ when $s=-1$ and is given explicitly by a sum of r…

$G_{r,qr+1}$ 中 Richardson 簇的环面商

Torus quotients of Richardson varieties in $G_{r,qr+1}$

本文研究 $G_{r,qr+1}$ 中 Richardson 簇的环面商，给出条件使得其 GIT 商为射影空间乘积。