Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups

Eulerian quasisymmetric functions

We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprisi…

Quasisymmetric functions and Kazhdan-Lusztig polynomials

We associate a quasisymmetric function to any Bruhat interval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, leading to an extension of the cd-index of convex polytopes. We show how the Kazhdan-Lusztig polynomial of the Bruhat interval can be expressed in terms of this complete cd-index and otherwise explicit combinatorially defined polynomials. In particular, we obtain the simplest closed formula for the Kazhdan-Lusztig polynomials that holds in complete generality.

Generalized coinvariant algebras for wreath products

Let $r$ be a positive integer and let $G_n$ be the reflection group of $n \times n$ monomial matrices whose entries are $r^{th}$ complex roots of unity and let $k \leq n$. We define and study two new graded quotients $R_{n,k}$ and $S_{n,k}$ of the polynomial ring $\mathbb{C}[x_1, \dots, x_n]$ in $n$ variables. When $k = n$, both of these quotients coincide with the classical coinvariant algebra attached to $G_n$. The algebraic properties of our quotients are governed by the combinatorial properties of $k$-dimensional faces in the Coxeter complex attached to $G_n$ (in the case of $R_{n,k}$) and…

A domino tableau-based view on type B Schur-positivity

Over the past years, major attention has been drawn to the question of identifying Schur-positive sets, i.e. sets of permutations whose associated quasisymmetric function is symmetric and can be written as a non-negative sum of Schur symmetric functions. The set of arc permutations, i.e. the set of permutations $π$ in $S_n$ such that for any $1\leq j \leq n$, $\{π(1),π(2),\dots,π(j)\}$ is an interval in $\mathbb{Z}_n$ is one of the most noticeable examples. This paper introduces a new type B extension of Schur-positivity to signed permutations based on Chow's quasisymmetric functions and gener…

0-Hecke 模、多米诺骨牌表与 B 型拟对称函数

0-Hecke Modules, Domino Tableaux, and Type-$B$ Quasisymmetric Functions

本文将升序相容推广至所有 Coxeter 群以构造 0-Hecke 模，并在 B 型中解释拟对称、峰函数及移位多米诺函数，证明正展开与限制特征公式。

The Flag Major Index and Group Actions on Polynomial Rings

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.

A tableau approach to the representation theory of 0-Hecke algebras

A 0-Hecke algebra is a deformation of the group algebra of a Coxeter group. Based on work of Norton and Krob--Thibon, we introduce a tableau approach to the representation theory of 0-Hecke algebras of type A, which resembles the classic approach to the representation theory of symmetric groups by Young tableaux and tabloids. We extend this approach to type B and D, and obtain a correspondence between the representation theory of 0-Hecke algebras of type B and D and quasisymmetric functions and noncommutative symmetric functions of type B and D. Other applications are also provided.

On the group of alternating colored permutations

The group of alternating colored permutations is the natural analogue of the classical alternating group, inside the wreath product $\mathbb{Z}_r \wr S_n$. We present a 'Coxeter-like' presentation for this group and compute the length function with respect to that presentation. Then, we present this group as a covering of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ and use this point of view to give another expression for the length function. We also use this covering to lift several known parameters of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ to the group of alternating colored permutations.

$0$-Hecke 代数的弱 Bruhat 区间模的分类

Classification of weak Bruhat interval modules of $0$-Hecke algebras

本文以类型无关的方式给出了 Coxeter 群中两个左弱 Bruhat 区间保降序同构的等价描述，并分类了 $0$-Hecke 代数的所有弱 Bruhat 区间模，解决了 Jung-Kim-Lee-Oh 的公开问题并证实了 Kim-Lee-Oh 的猜想。

模素数维数的组合

Dimensions of compositions modulo a prime

本文研究0-Hecke代数模p的投射不可分解模的维数（带状数），利用Dickson定理计算其模p同余类分布，并推广到有限Coxeter群。