Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups

Proof of Stasinski and Voll's Hyperoctahedral Group Conjecture

In a recent paper, Stasinski and Voll introduced a length-like statistic on hyperoctahedral groups and conjectured a product formula for this statistic's signed distribution over arbitrary quotients. Stasinski and Voll proved this conjecture for a few special types of quotients. We prove this conjecture in full, showing it holds for all quotients. In the case of signed permutations with at most one descent, this formula gives the Poincare polynomials for the varieties of symmetric matrices of a fixed rank.

On some analogues of Carlitz's identity for the hyperoctahedral group

We give a new description of the flag major index, introduced by Adin and Roichman, by using a major index defined by Reiner. This allows us to establish a connection between an identity of Reiner and some more recent results due to Chow and Gessel. Furthermore we generalize the main identity of Chow and Gessel by computing the four-variate generating series of descents, major index, length, and number of negative entries over Coxeter groups of type $B$ and $D$.

Tensorial square of the Hyperoctahedral group Coinvariant Space

The purpose of this paper is to give an explicit description of the trivial and alternating components of the irreducible representation decomposition of the bigraded module obtained as the tensor square of the coinvariant space for hyperoctahedral groups.

On the Descent Algebra of Type $D$

Here we give a combinatorial interpretation of Solomon's rule for multiplication in the descent algebra of Weyl groups of type $D$, $ΣD_n$. From here we show that $ΣD_n$ is a homomorphic image of the descent algebra of the hyperoctahedral group, $ΣB_{n-2}$.

Branching rules for the Weyl Groups of the Unitary and Orthogonal Lie Groups

This note presents a procedure to determine the reduction of the irreducible and the induced characters of the symmetric group in terms of the irreducible and induced characters of the hyperoctahedral group Key Words: Symmetric Group, Hyperoctahedral group, Representations, Characters, Re- duction.

具有超菱形群作用的Thagomizer矩阵的等变Kazhdan-Lusztig多项式

Equivariant Kazhdan--Lusztig Polynomials of Thagomizer Matroids with a Hyperoctahedral Group Action

本文研究了具有超菱形群作用的Thagomizer矩阵的等变Kazhdan-Lusztig多项式，利用回文性与对称群理论计算了多项式及其逆多项式，并通过Frobenius特征表达系数。

The weak order on the hyperoctahedral group and the monomial basis for the Hopf algebra of signed permutations

We give a combinatorial description for the weak order on the hyperoctahedral group. This characterization is then used to analyze the order-theoretic properties of the shifted products of hyperoctahedral groups. It is shown that each shifted product is a disjoint union of some intervals, which can be convex embedded into a hyperoctahedral group. As an application, we investigate the monomial basis for the Hopf algebra $\mathfrak{H}Sym$ of signed permutations, related to the fundamental basis via Möbius inversion on the weak order on hyperoctahedral groups. It turns out that the image of a mon…

超八面体群的Specht理想偏序集

The poset of Specht ideals for hyperoctahedral groups

本文针对超八面体群$B_n$的Specht理想，引入二划分上的双支配序刻画其包含偏序，并研究其对一般$B_n$-不变理想与簇的代数影响，从而简化相关计算。

Generating series formulas for the structure constants of Solomon's descent algebra

Introduced by Solomon in his 1976 paper, the descent algebra of a finite Coxeter group received significant attention over the past decades. As proved by Gessel, in the case of the symmetric group its structure constants give the comultiplication table for the fundamental basis of quasisymmetric functions. We show that this property actually implies several well known relations linked to the Robinson-Schensted-Knuth correspondence and some of its generalisations. We further use the theory of type B quasisymmetric functions introduced by Chow to provide analogue results when the Coxeter group i…

The Colored Eulerian Descent Algebra

Using a new colored analogue of P-partitions, we prove the existence of a colored Eulerian descent algebra which is a subalgebra of the Mantaci-Reutenauer algebra. This algebra has a basis consisting of formal sums of colored permutations with the same number of descents (using Steingrímsson's definition of the descent set of a colored permutation). The colored Eulerian descent algebra extends familiar Eulerian descent algebras from the symmetric group algebra and the hyperoctahedral group algebra to colored permutation group algebras. We also describe a set of orthogonal idempotents that span…

Cyclic descents and P-partitions

Louis Solomon showed that the group algebra of the symmetric group $\mathfrak{S}_{n}$ has a subalgebra called the descent algebra, generated by sums of permutations with a given descent set. In fact, he showed that every Coxeter group has something that can be called a descent algebra. There is also a commutative, semisimple subalgebra of Solomon's descent algebra generated by sums of permutations with the same number of descents: an "Eulerian" descent algebra. For any Coxeter group that is also a Weyl group, Paola Cellini proved the existence of a different Eulerian subalgebra based on a modi…

The Flag Descent Algebra and the Colored Eulerian Descent Algebra

We prove that the group algebra of the hyperoctahedral group contains a subalgebra corresponding to the flag descent number of Adin, Brenti, and Roichman. This algebra is in fact the span of the basis elements of the type A and type B Eulerian descent algebras. We describe a set of orthogonal idempotents which spans the flag descent algebra and prove that it contains the type A Eulerian descent algebra as a two-sided ideal. Using a new colored analogue of Stanley's $P$-partitions, we prove the existence of a colored Eulerian descent algebra which is a subalgebra of the Mantaci-Reutenauer algeb…

Macdonald trees and determinants of representations for finite Coxeter groups

Every irreducible odd dimensional representation of the $n$'th symmetric or hyperoctahedral group, when restricted to the $(n-1)$'th, has a unique irreducible odd-dimensional constituent. Furthermore, the subgraph induced by odd-dimensional representations in the Bratteli diagram of symmetric and hyperoctahedral groups is a binary tree with a simple recursive description. We survey the description of this tree, known as the Macdonald tree, for symmetric groups, from our earlier work. We describe analogous results for hyperoctahedral groups. A partition $λ$ of $n$ is said to be chiral if the co…

Two descent statistics over 321-avoiding centrosymmetric involutions

Centrosymmetric involutions in the symmetric group S_{2n} are permutations πsuch that π=π^{-1} and π(i)+π(2n+1-i)=2n+1 for all i, and they are in bijection with involutions of the hyperoctahedral group. We describe the distribution of some natural descent statistics on 321-avoiding centrosymmetric involutions, including the number of descents in the first half of the involution, and the sum of the positions of these descents. Our results are based on two new bijections, one between centrosymmetric involutions in S_{2n} and subsets of {1,...,n}, and another one showing that certain statistics o…

On the joint distribution of descents and signs of permutations

We study the joint distribution of descents and sign for elements of the symmetric group and the hyperoctahedral group (Coxeter groups of types $A$ and $B$). For both groups, this has an application to riffle shuffling: for large decks of cards the sign is close to random after a single shuffle. In both groups, we derive generating functions for the Eulerian distribution refined according to sign, and use them to give two proofs of central limit theorems for positive and negative Eulerian numbers.

Properties of the descent algebras of type $D$

We establish simple combinatorial descriptions of the radical and irreducible representations specifically for the descent algebra of a Coxeter group of type $D$ over any field.

On characters of wreath products

A character identity which relates irreducible character values of the hyperoctahedral group $B_n$ to those of the symmetric group $S_{2n}$ was recently proved by Lübeck and Prasad. Their proof is algebraic and involves Lie theory. We present a short combinatorial proof of this identity, as well as a generalization to other wreath products.

Enriched $P$-partitions and peak algebras (extended abstract)

We generalize Stembridge's enriched $P$-partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group. Whereas Stembridge's enriched $P$-partitions are related to quasisymmetric functions (the coalgebra dual to Solomon's type A descent algebra), our generalized enriched $P$-partitions are related to type B quasisymmetric functions (the coalgebra dual to Solomon's type B descent algebra). Using these functions, we explore three different peak algebras: the "interior" and "left" peak algebras of type A, and a new type B peak algebr…

The p-modular Descent Algebra of the Symmetric Group

The descent algebra of the symmetric group, over a field of non-zero characteristic p, is studied. A homomorphism into the algebra of generalised p-modular characters of the symmetric group is defined. This is then used to determine the radical, and its nilpotency index. It also allows the irreducible representations of the descent algebra to be described.

关于下降数、翻转数和弱序的一个猜想

A conjecture on descents, inversions and the weak order

本文提出并证明了关于对称群和超立方群中元素划分与右下降数关系的猜想。

超立方体图的 Vietoris-Rips 复形的同调：通过群作用

Homology of Vietoris-Rips complexes of hypercube graphs via group actions

利用超八面体群作用，完全描述了超立方体图在 Hamming 距离下 Vietoris-Rips 复形在尺度 r≤3 和 r=n-1 时的同调群分解为不可约表示。

The $p$-modular descent algebras

The concept of descent algebras over a field of characteristic zero is extended to define descent algebras over a field of prime characteristic. Some basic algebraic structure of the latter, including its radical and irreducible modules, is then determined. The decomposition matrix of the descent algebras of Coxeter group types $A$, $B$, and $D$ are calculated, and used to derive a description of the decomposition matrix of an arbitrary descent algebra. The Cartan matrix of a variety of descent algebras over a finite field is then obtained.

配置空间与峰表示

Configuration spaces and peak representations

本文为峰代数中幂等元生成的峰表示提供了类似的上同调解释，将其描述为Thrall高阶Lie特征标的和，并给出Hilbert级数、分支规则递归及与Jordan括号的联系。

圈积特征值与对称群特征值之间的一种关系

A Relationship Between Character Values Of Wreath Products And The Symmetric Group

本文基于Adin-Roichman的组合推广结果，建立了有限阿贝尔群圈积$G\wr S_n$与对称群$S_{rn}$之间又一不可约特征值关系。

$A$型和$B$型中可分元素的组合等价性

Combinatorial equivalence of separable elements in types $A$ and $B$

研究$A$型和$B$型可分元素的组合等价性，构造对称群与超八面体群间的双射，证明降序多项式$\gamma$-正性并推导秩生成函数显式公式。

Colored-Descent Representations of Complex Reflection Groups G(r,p,n)

We study the complex reflection groups G(r,p,n). By considering these groups as subgroups of the wreath products Z_r wr S_n, and by using Clifford theory, we define combinatorial parameters and descent representations of G(r,p,n), previously known for classical Weyl groups. One of these parameters is the flag major index, which also has an important role in the decomposition of these representations into irreducibles. A Carlitz type identity relating the combinatorial parameters with the degrees of the group, is presented.

Affine descents and the Steinberg torus

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…

The Steinberg torus of a Weyl group as a module over the Coxeter complex

Associated to each irreducible crystallographic root system $Φ$, there is a certain cell complex structure on the torus obtained as the quotient of the ambient space by the coroot lattice of $Φ$. This is the Steinberg torus. A main goal of this paper is to exhibit a module structure on (the set of faces of) this complex over the (set of faces of the) Coxeter complex of $Φ$. The latter is a monoid under the Tits product of faces. The module structure is obtained from geometric considerations involving affine hyperplane arrangements. As a consequence, a module structure is obtained on the space …

Diagonal invariants and the refined multimahonian distribution

Combinatorial aspects of multivariate diagonal invariants of the symmetric group are studied. As a consequence it is proved the existence of a multivariate extension of the classical Robinson-Schensted correspondence. Further byproduct are a pure combinatorial algorithm to describe the irreducible decomposition of the tensor product of two irreducible representations of the symmetric group, and new symmetry results on permutation enumeration with respect to descent sets.

有限 Coxeter 群中 Mallows 分布元素的双侧下降数的中心极限定理

A Central Limit Theorem on Two-Sided Descents of Mallows Distributed Elements of Finite Coxeter Groups

本文研究有限不可约 Coxeter 群中 Mallows 分布元素的双侧下降统计量，并通过大小偏倚耦合与 Stein 方法证明其满足高斯中心极限定理。

Cohomology of Coxeter arrangements and Solomon's descent algebra

We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of $W$. We give a uniform proof of the claim for symmetric groups. In addition, we prove that a relative version of the conjecture holds for every pair $(W, W_L)$, where $W$ is arbitrary and $W_…

有限群的特征标次数和与特征标表和相比有多大？

How large is the character degree sum compared to the character table sum for a finite group?

本文研究有限群特征标表和与特征标次数和的比值，提出其至多为2且至少为1（等号当且仅当群为阿贝尔群）的猜想，并对所有有限不可约Coxeter群及广义对称群证明了该猜想，同时给出了相关特征标表和的渐近公式与生成函数。

Cyclotomic Solomon Algebras

This paper introduces an analogue of the Solomon descent algebra for the complex reflection groups of type $G(r,1,n)$. As with the Solomon descent algebra, our algebra has a basis given by sums of `distinguished' coset representatives for certain `reflection subgroups'. We explicitly describe the structure constants with respect to this basis and show that they are polynomials in $r$. This allows us to define a deformation, or $q$-analogue, of these algebras which depends on a parameter $q$. We determine the irreducible representations of all of these algebras and give a basis for their radica…

The peak algebra of the symmetric group revisited

The linear span P_n of the sums of all permutations in the symmetric group S_n with a given set of peaks is a sub-algebra of the symmetric group algebra, due to Nyman. This peak algebra is a left ideal of the descent algebra D_n; and the direct sum P of all P_n is a Hopf sub-algebra of the direct sum D of all D_n, dual to the Stembridge algebra of peak functions. In our self-contained approach, peak counterparts of several results on the descent algebra are established, including a simple combinatorial characterization of the algebra P_n; an algebraic characterization of P_n based on the actio…

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。

A diagrammatic representation of an affine $C$ Temperley--Lieb algebra

In this thesis, I present an associative diagram algebra that is a faithful representation of a particular Temperley--Lieb algebra of type affine $C$, which has a basis indexed by the fully commutative elements of the Coxeter group of the same type. The Coxeter group of type affine $C$ contains an infinite number of fully commutative elements, and so the corresponding Temperley--Lieb algebra is of infinite rank. With the exception of type affine $A$, all other generalized Temperley--Lieb algebras with known diagrammatic representations are of finite rank. In the finite rank case, counting argu…

Mutual Interlacing and Eulerian-like Polynomials for Weyl Groups

We use the method of mutual interlacing to prove two conjectures on the real-rootedness of Eulerian-like polynomials: Brenti's conjecture on $q$-Eulerian polynomials for Weyl groups of type $D$, and Dilks, Petersen, and Stembridge's conjecture on affine Eulerian polynomials for irreducible finite Weyl groups. For the former, we obtain a refinement of Brenti's $q$-Eulerian polynomials of type $D$, and then show that these refined Eulerian polynomials satisfy certain recurrence relation. By using the Routh--Hurwitz theory and the recurrence relation, we prove that these polynomials form a mutual…

Block characters of the symmetric groups

Block character of a finite symmetric group S(n) is a positive definite function which depends only on the number of cycles in permutation. We describe the cone of block characters by identifying its extreme rays, and find relations of the characters to descent representations and the coinvariant algebra of S(n). The decomposition of extreme block characters into the sum of characters of irreducible representations gives rise to certain limit shape theorems for random Young diagrams. We also study counterparts of the block characters for the infinite symmetric group S(\infty) along with their …

Tableau posets and the fake degrees of coinvariant algebras

We introduce two new partial orders on the standard Young tableaux of a given partition shape, in analogy with the strong and weak Bruhat orders on permutations. Both posets are ranked by the major index statistic offset by a fixed shift. The existence of such ranked poset structures allows us to classify the realizable major index statistics on standard tableaux of arbitrary straight shape and certain skew shapes. By a theorem of Lusztig--Stanley, this classification can be interpreted as determining which irreducible representations of the symmetric group exist in which homogeneous component…

Signatures of representations of Hecke algebras and rational Cherednik algebras

Determining whether an irreducible representation of a group (or $*$-algebra) admits a non-degenerate invariant, positive-definite Hermitian form is an important problem in representation theory. In this paper, we study a related notion: that of signatures. We study representations $S^λ(q)$ of $\mathcal{H}_{n}(q)$, the Hecke algebra of type $A$ ($|q| = 1$), and representations $M_{c}(λ)$ of $\mathbb{H}_{c}$, the rational Cherednik algebra of type $A$ ($c \in \mathbb{R}$), which have unique (up to scaling) invariant Hermitian forms (here $λ$ is a partition of $n$). The signature is the number o…