Fokko Du Cloux
This paper is a report on a computer check of some positivity properties of the Hecke algebra in type H4, including the non-negativity of the coefficients of the structure constants in the Kazhdan-Lusztig basis. This answers a long-standing question of Lusztig's. The same algorithm, carried out by hand, also allows us to deal with the case of dihedral Coxeter groups.
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Let $W$ be a Coxeter group and let $M$ be the free $Z[v,v^{-1}]$-module with basis indexed by the involutions of $W$. We show how recent results of Elias and Williamson on Soergel bimodules can be used to give an alternative definition of an action of the Hecke algebra of $W$ on $M$.
Standard Representations of Crystallographic Coxeter Groups
该论文研究将特定Coxeter群表示为Kac-Moody代数Weyl群的Cartan表示，证明了相关同构条件、有限特征域上的部分结果，明确了晶体学Coxeter群的含义并给出对应Weyl群的条件，还提供了示例。
We prove some injectivity results: that a Coxeter monoid $\mathbb{Z}$-algebra (or $0$-Hecke algebra) injects in the incidence $\mathbb{Z}$-algebra of the corresponding Bruhat poset, for any Coxeter group; that the Hecke algebra of a right-angled Coxeter group injects in the Coxeter monoid $\mathbb{Z}[q,q^{-1}]$-algebra (and then in the incidence $\mathbb{Z}[q,q^{-1}]$-algebra of the corresponding Bruhat poset); that a right-angled Artin group injects in the group of invertible elements of the Hecke algebra of the corresponding Coxeter group (and then in the group of invertible elements of a Co…
The character values of Iwahori--Hecke algebras on Coxeter basis elements
该1992-1993年未发表笔记作为Lusztig近期Coxeter元素迹论文的补充，含迹的明确表格，其证明依赖于与Lusztig关于Coxeter轨道和Frobenius特征空间工作的联系，具独立意义。
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…
Pursuing Coxeter theory for Kac-Moody affine Hecke algebras
本文针对Kac-Moody仿射Hecke代数的类Coxeter结构展开研究，证明了长度函数构造、经典仿射Hecke代数乘积支撑等三个结果，提出相关猜想并给出仿射Demazure乘积的猜想定义，为其Coxeter理论构建迈出第一步。
Let $(W,S)$ be a Coxeter system with $I\subseteq S$ such that the parabolic subgroup $W_I$ is finite. Associated to this data there is a \textit{Hecke algebra} $\scH$ and a \textit{parabolic Hecke algebra} $\scH^I=\mathbf{1}_I\scH\mathbf{1}_I$ (over a ring $\ZZ[q_s]_{s\in S}$). We give a complete classification of the commutative parabolic Hecke algebras across all Coxeter types.
We refine the infinitesimal Hecke algebra associated to a 2-reflection group into a $\Z/2\Z$-graded Lie algebra, as a first step towards a global understanding of a natural $\mathbbm{N}$-graded object. We provide an interpretation of this Lie algebra in terms of the image of the braid group into the Hecke algebra, in connection with unitary structures. This connection is particularly strong when $W$ is a Coxeter group, and when in addition we can use generalizations of Drinfeld's rational even associators. Finally, in the Coxeter ADE case, we provide a generating set of the even part of this L…
In [LV] the authors defined a Hecke algebra action and a bar involution on a vector space spanned by the involutions in a Weyl group. In this paper we give a new definition of the Hecke algebra action and the bar operator which, unlike the one in [LV], is completely elementary (does not use geometry) and in particular it makes sense for an arbitrary Coxeter group.
Centralizers in Hecke Algebras of Any Coxeter Group
针对任意Coxeter群Hecke代数中抛物子代数中心化子在不定情形下研究不足的问题，结合代数技术与Davis复形几何工具，给出中心化子的基、分类无限Coxeter群有限部分共轭类并定义适配的类多项式变体，推广了中心相关已知结果。
Koszul duality for Coxeter groups
本文构造了Coxeter系统在给定实现与对偶实现下Hecke范畴之间的Koszul对偶等价，推广了几何语境下的相关工作，并证明倾斜反常层的组合学可由对偶实现下Hecke代数的典范基编码。
Quasiparabolic Gelfand models for finite irreducible Coxeter groups and their canonical Hecke structures
本文分类了有限不可约Coxeter群中由抛物子群诱导特征构造的拟抛物Gelfand模型，并推导了Gelfand对及典范Hecke结构。
This paper constructs a representation of a Hecke algebra on a vector space spanned by the involutions in a Coxeter group.
In this paper, we give explicit descriptions of the centers and cocenters of $0$-Hecke algebras associated to finite Coxeter groups.
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.
By Theorem~1.12 of the paper "A Class of Representations of Hecke Algebras", if $W$ is a Coxeter group whose proper parabolic subgroups are finite, and if the module of a finite $W$-digraph $Γ$ is isomorphic to the module of a $W$-graph, then $Γ$ must be acyclic. Here we extend this result to Coxeter groups with finite dihedral parabolic subgroups and $W$-graphs with arbitrary scalar edge labels. Also, errata for the paper are listed in the last section.
Hecke algebras and edge contractions
本文建立了一个从边收缩后的Coxeter系统所关联的Hecke代数到原始Coxeter系统所关联的Hecke代数的嵌入映射。
Hecke-Hopf algebras were defined by A. Berenstein and D. Kazhdan. We give an explicit presentation of an Hecke-Hopf algebra when the parameter $m_{ij},$ associated to any two distinct vertices $i$ and $j$ in the presentation of a Coxeter group, equals $4,$ $5$ or $6$. As an application, we give a proof of a conjecture of Berenstein and Kazhdan when the Coxeter group is crystallographic and non-simply-laced. As another application, we show that another conjecture of Berenstein and Kazhdan holds when $m_{ij},$ associated to any two distinct vertices $i$ and $j,$ equals $4$ and that the conjectur…
Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving operator algebra, as poset algebra, as commutant algebra, ...). In type A, the Hecke-group algebra can be described as the algebra generated simultaneously by the elementary transpositions and the elementa…
On overview of Coxeter groups $W$, their Iwahori-Hecke algebras $H$, Lusztig's asymptotic algebras $J$, cellular algebras in general and cellularity of $H$ in particular, as well as an introduction to Gyoja's $W$-graph algebra and the $W$-graph decomposition conjecture.
Let $(W, I)$ be a finite Coxeter group. In the case where $W$ is a Weyl group, Berenstein and Kazhdan in \cite{BK} constructed a monoid structure on the set of all subsets of $I$ using unipotent $χ$-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.
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.
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.
Let $W$ be a Coxeter group whose proper parabolic subgroups are finite. According to Theorem~1.12 of [1], if the module of a finite $W$-digraph $Γ$ is isomorphic to the module of a $W$-graph over $Q$, then $Γ$ is acyclic. We extend this result to Coxeter groups with finite dihedral parabolic subgroups and $W$-graphs over arbitrary fields $F$ of $C$. Also, an example is provided showing the converse of this theorem is false. That is, there is an example of a finite, acyclic $W$-digraph whose module does not afford a $W$-graph.
It is shown that the Euler characteristic $χ_{(\mathcal{H},\mathcal{B},ε_q)}$ of a $\mathbb{Z}[[q]]$-Hecke algebra $\mathcal{H}$ associated with a finitely generated Coxeter group $(W,S)$ coincides with $p_{(W,S)}(q)^{-1}$, where $p_{(W,S)}(t)$ is the Poincaré series of $(W,S)$.
We study the Hecke algebra $\H(\bq)$ over an arbitrary field $\FF$ of a Coxeter system $(W,S)$ with independent parameters $\bq=(q_s\in\FF:s\in S)$ for all generators. This algebra is always linearly spanned by elements indexed by the Coxeter group $W$. This spanning set is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receive the same parameter. In general, the dimension of $\H(\bq)$ could be as small as $1$. We construct a basis for $\H(\bq)$ when $(W,S)$ is simply laced. We also characterize when $\H(\bq)$ is commutative, which happens o…
By a result of Mathas, the basis element T_{w_0} of the Hecke algebra of a finite Coxeter group acts in the canonical basis of a cell module as a permutation matrix times plus or minus a power of v. We generalize this result to the unequal parameter case,. We also show that the image of T_{w_0} in the corresponding asymptotic Hecke algebra is given by a simple formula.
On the trace of Coxeter elements
针对Weyl群极小长度Coxeter元在不可约表示上的迹问题，本文证明了Macdonald提出的迹为0、1或-1的结论，并给出其Iwahori-Hecke代数版本的证明。
In this paper we give a version of Bergman's diamond lemma which applies to certain monoidal categories presented by generators and relations. In particular, it applies to: the Coxeter presentation of the symmetric groups, the quiver Hecke algebras of Khovanov-Lauda-Rouquier, the Webster tensor product algebras, and various generalizations of these. We also give an extension of Manin-Schechtmann theory to non-reduced expressions.
Relative Serre duality for Coxeter groups
本文证明了Coxeter群的parabolic诱导 functor的左右伴随通过相对全扭联系，利用Abe-Bott-Samelson双模和Achar-Riche-Vay混合导出范畴理论。
We study Kazhdan-Lusztig cells and the corresponding representations of right-angled Coxeter groups and Hecke algebras associated to them. In case of the infinite groups generated by reflections in the hyperbolic plane about the sides of right-angled polygons we obtain an explicit description of the left and two-sided cells. In particular, we prove that there are infinitely many left cells but they all form only three two-sided cells.
Let $(W,S)$ be a Coxeter system. A $W$-graph encodes a representation of the Hecke algebra $\mathcal{H}$ of $W$. We construct universal representations of multi-parameter Hecke algebras on certain quotients of path algebras, and study their relationships with $W$-graph representations. We also study the quotients of path algebras on their own, motivated by one example where the quotient path algebra is isomorphic to an ideal of Lusztig asymptotic Hecke algebra. Finally, we describe a method to obtain a generating set for the ideals by which we quotient the path algebras.
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 usual combinatorial model for the 0-Hecke algebra of the symmetric group is to consider the algebra (or monoid) generated by the bubble sort operators. This construction generalizes to any finite Coxeter group W. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it has |W| simple and projective modules. In order to construct a combinatorial model for the simp…
This is an extended and corrected version of the author's Diplomarbeit. A class of algebras called generic pro-$p$ Hecke algebras is introduced, enlarging the class of generic Hecke algebras by considering certain extensions of (extended) Coxeter groups. Examples of generic pro-$p$ Hecke algebras are given by pro-$p$-Iwahori Hecke algebras and Yokonuma-Hecke algebras. The notion of an orientation of a Coxeter group is introduced and used to define `Bernstein maps' intimately related to Bernstein's presentation and to Cherednik's cocycle. It is shown that certain relations in the Hecke algebra …
Spin Representations of Finite Coxeter Groups and Generalisations of Saxl's Conjecture
本文从李理论视角推广了Saxl猜想，利用旋量表示在经典型取得进展并给出cusp族的等价描述，同时将猜想推广至有限Coxeter群并证明了非晶体学情形。
Intersections of blocks of cyclotomic Hecke algebras
本文证明了关于循环Hecke代数块交集的猜想，并推广到更广泛的反射群类别。
For W a finite (2-)reflection group and B its (generalized) braid group, we determine the Zariski closure of the image of B inside the corresponding Iwahori-Hecke algebra. The Lie algebra of this closure is reductive and generated in the group algebra of W by the reflections of W. We determine its decomposition in simple factors. In case W is a Coxeter group, we prove that the representations involved are unitarizable when the parameters of the representations have modulus 1 and are close to 1. We consequently determine the topological closure in this case.
In this paper we construct an abelian category of "mixed perverse sheaves" attached to any realization of a Coxeter group, in terms of the associated Elias-Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias-Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.
We study the (complex) Hecke algebra $\mathcal{H}_S(\mathbf{q})$ of a finite simply-laced Coxeter system $(W,S)$ with independent parameters $\mathbf{q} \in \left( \mathbb{C} \setminus\{\text{roots of unity}\} \right)^S$. We construct its irreducible representations and projective indecomposable representations. We obtain the quiver of this algebra and determine when it is of finite representation type. We provide decomposition formulas for induced and restricted representations between the algebra $\mathcal{H}_S(\mathbf{q})$ and the algebra $\mathcal{H}_R(\mathbf{q}|_R)$ with $R\subseteq S$. …
We compute the trace decategorification of the Hecke category for an arbitrary Coxeter group. More generally, we introduce the notion of a strictly object-adapted cellular category and calculate the trace for such categories.
Let $W$ be a Coxeter group. The goal of the paper is to construct new Hopf algebras that contain Hecke algebras $H_{\bf q}(W)$ as (left) coideal subalgebras. Our Hecke-Hopf algebras ${\bf H}(W)$ have a number of applications. In particular they provide new solutions of quantum Yang-Baxter equation and lead to a construction of a new family of endo-functors of the category of $H_{\bf q}(W)$-modules. Hecke-Hopf algebras for the symmetric group are related to Fomin-Kirillov algebras, for an arbitrary Coxeter group $W$ the "Demazure" part of ${\bf H}(W)$ is being acted upon by generalized braided …
The irreducible representations of full support in the rational Cherednik category $\mathcal{O}_c(W)$ attached to a Coxeter group $W$ are in bijection with the irreducible representations of an associated Iwahori-Hecke algebra. Recent work has shown that the irreducible representations in $\mathcal{O}_c(W)$ of arbitrary given support are similarly governed by certain generalized Hecke algebras. In this paper we compute the parameters for these generalized Hecke algebras in the remaining previously unknown cases, corresponding to the parabolic subgroup $B_n \times S_k$ in $B_{n+k}$ for $k \geq …
Let $W$ be an extended affine Weyl group. We prove that minimal length elements $w_{\co}$ of any conjugacy class $\co$ of $W$ satisfy some special properties, generalizing results of Geck and Pfeiffer \cite{GP} on finite Weyl groups. We then introduce the "class polynomials" for affine Hecke algebra $H$ and prove that $T_{w_\co}$, where $\co$ runs over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H, H]$. We also classify the conjugacy classes satisfying a generalization of Lusztig's conjecture \cite{L4}.
We construct two-dimensional non-commutative topological quantum field theories (TQFTs), one for each Hecke algebra corresponding to a finite Coxeter system. These TQFTs associate an invariant to each ciliated surface, which is a Laurent polynomial for punctured surfaces. There is a graphical way to compute the invariant using minimal colored graphs. We give explicit formulas in terms of the Schur elements of the Hecke algebra and prove positivity properties for the invariants when the Coxeter group is of classical type, or one of the exceptional types $H_3$, $E_6$ and $E_7$.
Some remarks on realizations defining diagrammatic Hecke categories
本文针对定义图示Hecke范畴的实现，证明了确保范畴性态良好的常见假设非必要，还给出非平衡实现旋转标量的非归纳描述与平衡性检验简化准则，补充了图示Hecke范畴相关研究的关键结论。
A basis of the center of the 0-Hecke algebra of an arbitrary finite Coxeter group was described by He in 2015. This basis is indexed by certain equivalence classes of the Coxeter group whose explicit description is rather complicated. Even their number is not obvious. We consider case of the symmetric group $\mathfrak{S}_n$. Building on work of Geck, Kim and Pfeiffer, we obtain a complete set of representatives of the equivalence classes. This set is naturally parametrized by certain compositions of $n$ called maximal. It follows that the maximal compositions of $n$ index the basis of the cent…
We investigate the extensions of the Hecke algebras of finite (complex) reflection groups by lattices of reflection subgroups that we introduced, for some of them, in our previous work on the Yokonuma-Hecke algebras and their connections with Artin groups. When the Hecke algebra is attached to the symmetric group, and the lattice contains all reflection subgroups, then these algebras are the diagram algebras of braids and ties of Aicardi and Juyumaya. We prove a stucture theorem for these algebras, generalizing a result of Espinoza and Ryom-Hansen from the case of the symmetric group to the ge…
An axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras was presented in [1]. Combinatorial aspects of this construction are studied in this paper. In particular, the symmetric group case is investigated in detail. The resulting representations are completely classified and include the irreducible ones.
The structure of subring $J^{Γ\cap Γ^{-1}}$ of the asymptotic Hecke algebra is described for $Γ$ a left cell of the Coxeter group of type $H_4$. A small set of generators is produced. The subalgebras spanned by a subset of the basis ${t_x}_{x\in Γ\capΓ^{-1}}$ are determined.
For a complex reflection group $W$ with reflection representation $\mathfrak{h}$, we define and study a natural filtration by Serre subcategories of the category $\mathcal{O}_c(W, \mathfrak{h})$ of representations of the rational Cherednik algebra $H_c(W, \mathfrak{h})$. This filtration refines the filtration by supports and is analogous to the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. Using the monodromy of the Bezrukavnikov-Etingof parabolic restriction functors, we show that the subquotients of this filtration are equivalent to categories of …
Quiver data of representations of Hecke algebras and Coxeter groups
研究Hecke代数和Coxeter群表示的箭图数据，提出[Γ;q]型H-表示概念，通过其与Hq(Γ)-表示的关系构造非相交表示，还引入H²-表示，开发构造程序并证明特定条件下H²-表示的可实现性。
In this paper, we will fully describe the representations of the crystallographic rank two affine Hecke algebras using elementary methods, for all possible values of q. The focus is on the case when q is a root of unity of small order.
We obtain a complete characterisation of factorial multiparameter Hecke von Neumann algebras associated with right-angled Coxeter groups. Considering their $\ell^p$-convolution algebra analogues, we exhibit an interesting parameter dependence, contrasting phenomena observed earlier for group Banach algebras. Translated to Iwahori-Hecke von Neumann algebras, these results allow us to draw conclusions on spherical representation theory of groups acting on right-angled buildings, which are in strong contrast to behaviour of spherical representations in the affine case. We also investigate certain…
A type of directed multigraph called a W-digraph is introduced to model the structure of certain representations of Hecke algebras, including those constructed by Lusztig and Vogan from involutions in a Weyl group. Building on results of Lusztig, a complete characterization of W-digraphs is given in terms of subdigraphs for dihedral parabolic subgroups. Graph-theoretic properties of W-digraphs are established including, under certain assumptions, acyclicity. In case the Coxeter system is finite, a bound on the number of vertices of a connected W-digraph is obtained, and a graph-theoretic versi…
We give a new proof of the Macdonald-Mehta-Opdam integral identity for finite Coxeter groups. This identity was conjectured by Macdonald and proved by Opdam in 1993 using the theory of multivariable Bessel functions, but in non-crystallographic cases the proof relied on a computer calculation by F. Garvan. Our proof is somewhat more elementary (in particular, it does not use multivariable Bessel functions), and uniform (does not refer to the classification of finite Coxeter groups and does not use computers).
We describe a positive characteristic analogue of the Kazhdan-Lusztig basis of the Hecke algebra of a crystallographic Coxeter system and investigate some of its properties. Using Soergel calculus we describe an algorithm to calculate this basis. We outline some known or expected applications in modular representation theory. We conclude by giving several examples.
The center of the asymptotic Hecke category and unipotent character sheaves
本文证明了渐近Hecke范畴的中心为模ularity张量范畴，并研究了其在二面体群及H3、H4部分细胞中的应用。
We describe an algorithm for computing the $p$-canonical basis of the Hecke algebra, or one of its antispherical modules. The algorithm does not operate in the Hecke category directly, but rather uses a faithful embedding of the Hecke category inside a semisimple category to build a "model" for indecomposable objects and bases of their morphism spaces. Inside this semisimple category, objects are sequences of Coxeter group elements, and morphisms are (sparse) matrices over a fraction field, making it quite amenable to computations. This strategy works for the full Hecke category over any base …
In this paper we classify all Markov traces on Iwahori-Hecke algebras associated with the finite Coxeter groups of type B. We then use these traces for constructing Jones-type invariants for oriented knots inside a solid torus, and finally we give skein-theoretical interpretations.
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