Involutions in Coxeter groups

Positivity conjectures for Kazhdan-Lusztig theory on twisted involutions: the universal case

Let $(W,S)$ be a Coxeter system and let $w \mapsto w^*$ be an involution of $W$ which preserves the set of simple generators $S$. Lusztig and Vogan have recently shown that the set of twisted involutions (i.e., elements $w \in W$ with $w^{-1} = w^*$) naturally generates a module of the Hecke algebra of $(W,S)$ with two distinguished bases. The transition matrix between these bases defines a family of polynomials $P^σ_{y,w}$ which one can view as "twisted" analogues of the much-studied Kazhdan-Lusztig polynomials of $(W,S)$. The polynomials $P^σ_{y,w}$ can have negative coefficients, but displa…

Involutions in Weyl groups and nil-Hecke algebras

In a previous article we have defined an action of the Iwahori-Hecke algebra of a Coxeter group W on a free module with basis indexed by the involutions in W. In this paper we show that the specialization of this action at the parameter 0 has a simple description.

An involution based left ideal in the Hecke algebra

We define an imbedding of the Hecke algebra module carried by the involutions in a Weyl group W (defined by the author and Vogan) into a completion of the Hecke algebra. An analogous result is proved for any Coxeter group. A variant of the construction gives an imbedding of the biregular representation of a Hecke algebra of W in a completion of the Hecke algebra of W^2.

Involution words II: braid relations and atomic structures

Involution words are variations of reduced words for twisted involutions in Coxeter groups. They arise naturally in the study of the Bruhat order, of certain Iwahori-Hecke algebra modules, and of orbit closures in flag varieties. Specifically, to any twisted involutions $x$, $y$ in a Coxeter group $W$ with automorphism $*$, we associate a set of involution words $\hat{\mathcal{R}}_*(x,y)$. This set is the disjoint union of the reduced words of a set of group elements $\mathcal{A}_*(x,y)$, which we call the atoms of $y$ relative to $x$. The atoms, in turn, are contained in a larger set $\mathca…

Positivity conjectures for Kazhdan-Lusztig theory on twisted involutions: the finite case

Let $(W,S)$ be any Coxeter system and let $w \mapsto w^*$ be an involution of $W$ which preserves the set of simple generators $S$. Lusztig and Vogan have shown that the corresponding set of twisted involutions (i.e., elements $w \in W$ with $w^{-1} = w^*$) naturally generates a module of the Hecke algebra of $(W,S)$ with two distinguished bases. The transition matrix between these bases defines a family of polynomials $P^σ_{y,w}$ which one can view as a "twisted" analogue of the much-studied family of Kazhdan-Lusztig polynomials of $(W,S)$. The polynomials $P^σ_{y,w}$ can have negative coeffi…

On Kottwitz' conjecture for twisted involutions

Kottwitz' conjecture is concerned with the intersections of Kazhdan--Lusztig cells with conjugacy classes of involutions in finite Coxeter groups. In joint work with Bonnafé, we have recently found a way to prove this conjecture for groups of type $B_n$ and $D_n$. The argument for type $D_n$ relies on two ingredients which were used there without proof: (1) a strengthened version of the "branching rule" and (2) the consideration of "$\diamond$-twisted" involutions where $\diamond$ is a graph automorphism. In this paper we deal with (1), (2) and complete the argument for type $D_n$; moreover, w…

Comparing and characterizing some constructions of canonical bases from Coxeter systems

The Iwahori-Hecke algebra $\mathcal{H}$ of a Coxeter system $(W,S)$ has a "standard basis" indexed by the elements of $W$ and a "bar involution" given by a certain antilinear map. Together, these form an example of what Webster calls a pre-canonical structure, relative to which the well-known Kazhdan-Lusztig basis of $\mathcal{H}$ is a canonical basis. Lusztig and Vogan have defined a representation of a modified Iwahori-Hecke algebra on the free $\mathbb{Z}[v,v^{-1}]$-module generated by the set of twisted involutions in $W$, and shown that this module has a unique pre-canonical structure sat…

Root systems and diagram calculus. III. Semi-Coxeter orbits of linkage diagrams and the Carter theorem

A diagram obtained from the Carter diagram $Γ$ by adding one root together with its bonds such that the resulting subset of roots is linearly independent is said to be the {\it linkage diagram}. Given a linkage diagram, we associate the linkage labels vector, which is introduced like the vector of Dynkin labels. Similarly to the dual Weyl group, we introduce the group $W^{\vee}_L$ associated with $Γ$, and we call it the dual partial Weyl group. The linkage labels vectors connected under the action of $W^{\vee}_L$ constitute the linkage system $\mathscr{L}(Γ)$, which is similar to the weight sy…

Stanley symmetric functions for signed involutions

An involution in a Coxeter group has an associated set of involution words, a variation on reduced words. These words are saturated chains in a partial order first considered by Richardson and Springer in their study of symmetric varieties. In the symmetric group, involution words can be enumerated in terms of tableaux using appropriate analogues of the symmetric functions introduced by Stanley to accomplish the same task for reduced words. We adapt this approach to the group of signed permutations. We show that involution words for the longest element in the Coxeter group $C_n$ are in bijecti…

Bar operators for quasiparabolic conjugacy classes in a Coxeter group

The action of a Coxeter group $W$ on the set of left cosets of a standard parabolic subgroup deforms to define a module $\mathcal{M}^J$ of the group's Iwahori-Hecke algebra $\mathcal{H}$ with a particularly simple form. Rains and Vazirani have introduced the notion of a quasiparabolic set to characterize $W$-sets for which analogous deformations exist; a motivating example is the conjugacy class of fixed point free involutions in the symmetric group. Deodhar has shown that the module $\mathcal{M}^J$ possesses a certain antilinear involution, called the bar operator, and a certain basis invaria…

On involutions in symmetric groups and a conjecture of Lusztig

Let $(W, S)$ be a Coxeter system equipped with a fixed automorphism $\ast$ of order $\leq 2$ which preserves $S$. Lusztig (and with Vogan in some special cases) have shown that the space spanned by set of "twisted" involutions was naturally endowed with a module structure of the Hecke algebra of $(W, S)$. Lusztig has conjectured that this module is isomorphic to the right ideal of the Hecke algebra (with Hecke parameter $u^2$) associated to $(W,S)$ generated by the element $X_{\emptyset}:=\sum_{w^\ast=w}u^{-\ell(w)}T_w$. In this paper we prove this conjecture in the case when $\ast=\text{id}$ …

PyCox: Computing with (finite) Coxeter groups and Iwahori-Hecke algebras

We introduce the computer algebra package {\sf PyCox}, written entirely in the {\sf Python} language. It implements a set of algorithms - in a spirit similar to the older {\sf CHEVIE} system - for working with Coxeter groups and Hecke algebras. This includes a new variation of the traditional algorithm for computing Kazhdan--Lusztig cells and $W$-graphs, which works efficiently for all finite groups of rank $\leq 8$ (except $E_8$). We also discuss the computation of Lusztig's leading coefficients of character values and distinguished involutions (which works for $E_8$ as well). Our experiments…

On paired root systems of Coxeter groups

This paper examines a systematic method to construct a pair of (inter-related) root systems for arbitrary Coxeter groups from a class of non-standard geometric representations. This method can be employed to construct generalizations of root systems for a large family of groups generated only by involutions. We then give a characterization of Coxeter groups, among these groups, in terms of such paired root systems. Furthermore, we use this method to construct and study the paired root systems for reflection subgroups of Coxeter groups.

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.

Cells in Coxeter groups I

The purpose of this article is to shed new light on the combinatorial structure of Kazhdan-Lusztig cells in infinite Coxeter groups $W$. Our main focus is the set $\D$ of distinguished involutions in $W$, which was introduced by Lusztig in one of his first papers on cells in affine Weyl groups. We conjecture that the set $\D$ has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite Coxeter groups. Moreover, to each element of $\D$ we assign an explicitly defined set of equivalence relations on $W$ that altogether conjecturally…

Automorphisms of Coxeter groups and Lusztig's conjectures for Hecke algebras with unequal parameters

Let $(W,S)$ be a Coxeter system, let $G$ be a finite solvable group of automorphisms of $(W,S)$ and let $φ$ be a weight function which is invariant under $G$. Let $φ_G$ denote the weight function on $W^G$ obtained by restriction from $φ$. The aim of this paper is to compare the ${\mathbf{a}}$-function, the set of Duflo involutions and the Kazhdan-Lusztig cells associated to $(W,φ)$ and to $(W^G,φ_G)$, provided that Lusztig's Conjectures hold.

Automorphisms and Generalized Involution Models of Finite Complex Reflection Groups

We prove that a finite complex reflection group has a generalized involution model, as defined by Bump and Ginzburg, if and only if each of its irreducible factors is either $G(r,p,n)$ with $\gcd(p,n)=1$; $G(r,p,2)$ with $r/p$ odd; or $G_{23}$, the Coxeter group of type $H_3$. We additionally provide explicit formulas for all automorphisms of $G(r,p,n)$, and construct new Gelfand models for the groups $G(r,p,n)$ with $\gcd(p,n)=1$.

Conjectures P1-P15 for hyperbolic Coxeter groups of rank 3

We prove Lusztig's conjectures P1-P15 for hyperbolic Coxeter groups of rank 3. Our proof enables us to give a description of the a-function and Kazhdan-Lusztig cells for these Coxeter groups.

Integral $u$-deformed involution modules

Let $(W,S)$ be a Coxeter system and $\ast$ an automorphism of $W$ with order $\leq 2$ and $S^{\ast}=S$. Lusztig and Vogan ([11], [14]) have introduced a $u$-deformed version $M_u$ of Kottwitz's involution module over the Iwahori-Hecke algebra $\mathscr{H}_{u}(W)$ with Hecke parameter $u^2$, where $u$ is an indeterminate. Lusztig has proved that $M_u$ is isomorphic to the left $\mathscr{H}_{u}(W)$-submodule of ${\hat{\mathscr{H}}}_u$ generated by $X_{\emptyset}:=\sum_{w^*=w\in W}{u^{-\ell(w)}T_w}$, where ${\hat{\mathscr{H}}}_u$ is the vector space consisting of all formal (possibly infinite) su…

The boundness of weighted Coxeter groups of rank 3

We prove that a weighted Coxeter group (W,S,L) is bounded in the sense of G.Lusztig if the rank of W is 3.

Coxeter群及其忠实表示的真联合谱

Coxeter groups and the proper joint spectrums of their faithful representations

本文分析了二面体群的忠实表示，证明了Coxeter群可由其忠实表示的真联合谱唯一确定。

Minimal length elements in some double cosets of Coxeter groups

We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's $G$-stable pieces and the generalization of $G$-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in \cite{GKP}.

Perfect models for finite Coxeter groups

A model for a finite group is a set of linear characters of subgroups that can be induced to obtain every irreducible character exactly once. A perfect model for a finite Coxeter group is a model in which the relevant subgroups are the quasiparabolic centralizers of perfect involutions. In prior work, we showed that perfect models give rise to interesting examples of $W$-graphs. Here, we classify which finite Coxeter groups have perfect models. Specifically, we prove that the irreducible finite Coxeter groups with perfect models are those of types $\mathsf{A}_{n}$, $\mathsf{B}_n$, $\mathsf{D}_…

Lusztig's $a$ function for Coxeter groups of rank 3

We show that Lusztig's $a$-function of a Coxeter group is bounded if the rank of the Coxeter group is 3.

Diagrammatics for Coxeter groups and their braid groups

We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.

Representations of Coxeter groups and homology of Coxeter graphs

We classify a class of complex representations of an arbitrary Coxeter group via characters of the integral homology of certain graphs. Such representations can be viewed as a generalization of the geometric representation and correspond to the second-highest 2-sided cell in the sense of Kazhdan-Lusztig. We also give a description of the cell representation provided by this 2-sided cell, and find out all its simple quotients for simply laced Coxeter system with no more than one circuit in the Coxeter graph.

A Unified Construction of Coxeter Group Representations (I)

An elementary approach to the construction of Coxeter group representations is presented.

Coxeter群的Lusztig a函数值1的表示

Representations of Coxeter groups of Lusztig's a-function value 1

本文研究Coxeter群中Lusztig a函数值1的表示，确定了某些单理Coxeter群的不可约表示。

Preprojective roots of Coxeter groups

Certain results on representations of quivers have analogs in the structure theory of general Coxeter groups. A fixed Coxeter element turns the Coxeter graph into an acyclic quiver, allowing for the definition of a preprojective root. A positive root is an analog of an indecomposable representation of the quiver. The Coxeter group is finite if and only if every positive root is preprojective, which is analogous to the well-known result that a quiver is of finite representation type if and only if every indecomposable representation is preprojective. Combinatorics of orientation-admissible word…

Birman-Murakami-Wenzl algebras for general Coxeter groups

We introduce a BMW type algebra for every Coxeter group. These new algebras are introduced as deformations of the Brauer type algebras introduced by the author, they have the corresponding Hecke algebras as quotients.

The strong Lefschetz property of the coinvariant ring of the Coxeter group of type H4

We prove that the coinvariant ring of the irreducible Coxeter group of type H4 has the strong Lefschetz property.

有界Bruhat区间与仿射Coxeter群

Bounded Bruhat intervals and affine Coxeter groups

本文证明Dyer关于有限Coxeter群中Bruhat区间同构类型有限性的结果对仿射Coxeter群也成立，并刻画了仿射群在无限Coxeter群中的特征，同时给出Kazhdan-Lusztig多项式系数有界的充要条件。

Classification of Coxeter groups with finitely many elements of $\mathbf{a}$-value 2

We consider Lusztig's $\mathbf{a}$-function on Coxeter groups (in the equal parameter case) and classify all Coxeter groups with finitely many elements of $\mathbf{a}$-value 2 in terms of Coxeter diagrams.

Higher signs for Coxeter groups

We define and study cocycles on a Coxeter group in each degree generalizing the sign function. When the Coxeter group is a Weyl group, we explain how the degree three cocycle arises naturally from geometry representation theory.

Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures

Involution words are variations of reduced words for involutions in Coxeter groups, first studied under the name of "admissible sequences" by Richardson and Springer. They are maximal chains in Richardson and Springer's weak order on involutions. This article is the first in a series of papers on involution words, and focuses on their enumerative properties. We define involution analogues of several objects associated to permutations, including Rothe diagrams, the essential set, Schubert polynomials, and Stanley symmetric functions. These definitions have geometric interpretations for certain …

某些Coxeter群的一些无穷维表示

Some infinite-dimensional representations of certain Coxeter groups

该研究针对满足非有限且非仿射条件的特定Coxeter群，借助其对应Coxeter图的拓扑信息，构造出了该类群的部分无穷维不可约复表示。

Vanishing ranges for the mod $p$ cohomology of alternating subgroups of Coxeter groups

We obtain vanishing ranges for the mod $p$ cohomology of alternating subgroups of finite $p$-free Coxeter groups. Here a Coxeter group $W$ is $p$-free if the order of the product $st$ is prime to $p$ for every pair of Coxeter generators $s,t$ of $W$. Our result generalizes those for alternating groups formerly proved by Kleshchev-Nakano and Burichenko. As a byproduct, we obtain vanishing ranges for the twisted cohomology of finite $p$-free Coxeter groups with coefficients in the sign representations. In addition, a weak version of the main result is proved for a certain class of infinite Coxet…

Relatively compact sets in the reduced $C^{\ast}$-algebras of Coxeter groups

We characterize relatively norm compact sets in the regular $C^*$-algebra of finitely generated Coxeter groups using a geometrically defined positive semigroup acting on the algebra.

Coxeter群的反射表示与Coxeter图的同调

Reflection Representations of Coxeter Groups and Homology of Coxeter Graphs

本文利用与Coxeter图相关图的整同调群特征标，分类有限秩Coxeter群的广义几何表示，并刻画生成元以反射方式作用的表示。