Bar operators for quasiparabolic conjugacy classes in a Coxeter group

A finiteness theorem for W-graphs

On involutions in symmetric groups and a conjecture of Lusztig

The purpose of this paper is to describe a general procedure for computing analogues of Young’s seminormal representations of the symmetric groups . The method is to generalize the Jucys – Murphy elements in the group algebras of the symmetric groups to arbitrary Weyl groups and Iwahori – Hecke algebras . The combinatorics of these elements allow one to compute irreducible representations explicitly and often very easily . In this paper we do these computations for Weyl groups and Iwahori – Hecke algebras of types A n , B n , D n , G 2 . Although these computations are within reach for types F 4 , E 6 , and E 7 , we shall , in view of the length of the current paper , postpone this to another work . In reading this paper , I would suggest that the reader begin with § 3 , the symmetric group case , and go back and pick up the generalities from §§ 1 and 2 as they are needed . This will make the motivation for the material in the earlier sections much more clear and the further examples in the later sections very easy . The realization that the Jucys – Murphy elements for Weyl groups and Iwahori – Hecke algebras come from the very natural central elements in (2 . 1) and Proposition 2 . 4 is one of the main points of this paper . There is a simple concrete connection (Proposition 2 . 8) between Jucys – Murphy type elements in Iwahori – Hecke algebras and Jucys – Murphy elements in group algebras of Weyl groups . I know that the analogues of the Jucys – Murphy elements in Weyl groups of types B and D will be new to some of the experts and known to others . These Jucys – Murphy elements for types B and D are not new ; similar elements appear in the paper of Cherednik [ 7 ] , but I was not able to recognize them there until they were pointed out to me by M . Nazarov . I extend my thanks to him for this . Some people were asking me for Jucys – Murphy elements in type G 2 as late as June 1995 . In July 1995 I was told that it was not known how to quantize the elements of Cherednik , that is , to find analogues of them in the Iwahori – Hecke algebras of types B and D . Of course , this had been done already in 1974 , by Hoefsmit . I have chosen to state my results in terms of the general mechanism of path algebras which I have defined in § 1 . This is a technique which I learned from H . Wenzl during our work on the paper [ 30 ] . It is a well-known method in several fields (with many dif ferent terminologies) . I shall mention here only a few of the

The character table of the Iwahori-Hecke algebra of the symmetric group: Starkey's rule

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 satisfying a certain compatibility condition, which admits its own canonical basis which can be viewed as a generalization of the Kazhdan-Lusztig basis. One can modify the parameters defining Lusztig and Vogan's module to obtain other pre-canonical structures, each of which admits a unique canonical basis indexed by twisted involutions. We classify all of the pre-canonical structures which arise in this fashion, and explain the relationships between their resulting canonical bases. While some of these canonical bases are related in a trivial fashion to Lusztig and Vogan's construction, others appear to have no simple relation to what has been previously studied. Along the way, we also clarify the differences between Webster's notion of a canonical basis and the related concepts of an IC basis and a $P$-kernel.

Some Generic Representations, W-Graphs, and Duality

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 $\mathcal{B}_*(x,y) \subset W$ with a similar definition, whose elements we refer to as Hecke atoms. Our main results concern some interesting properties of the sets $\hat{\mathcal{R}}_*(x,y)$ and $\mathcal{A}_*(x,y) \subset \mathcal{B}_*(x,y)$. For finite Coxeter groups we prove that $\mathcal{A}_*(1,y)$ consists of exactly the minimal-length elements $w \in W$ such that $w^* y \leq w$ in Bruhat order, and conjecture a more general property for arbitrary Coxeter groups. In type $A$, we describe a simple set of conditions characterizing the sets $\mathcal{A}_*(x,y)$ for all involutions $x,y \in S_n$, giving a common generalization of three recent theorems of Can, Joyce, and Wyser. We show that the atoms of a fixed involution in the symmetric group (relative to $x=1$) naturally form a graded poset, while the Hecke atoms surprisingly form an equivalence class under the "Chinese relation" studied by Cassaigne, Espie, et al. These facts allow us to recover a recent theorem of Hu and Zhang describing a set of "braid relations" spanning the involution words of any self-inverse permutation. We prove a generalization of this result giving an analogue of Matsumoto's theorem for involution words in arbitrary Coxeter groups.

Let w ↦ ( P ( w ) , Q ( w ) ) be the Robinson-Schensted correspondence between the symmetric group S n and the set of pairs of standard tableaux with the same shapes. We show that each Kazhdan-Lusztig basis (KL basis for short) element C ′ w can be expressed as a linear combination of some f s t which satisfies that s ¥ P ( w ) ∗ , t ¥ Q ( w ) ∗ , where “ ¥ ” is the dominance (partial) order between standard tableaux, u ∗ denotes the conjugate of u for each standard tableau u , { f s t | s , t ∈ Std ( λ ) , λ⊢n } is the seminormal basis of the Iwahori-Hecke algebra associated to S n . As a result, we generalize an earlier result of Geck on the relation between the KL basis and the Murphy basis. Similar relations between the twisted KL basis, the dual seminormal basis and the dual Murphy basis are obtained.

The article proves a relative version of one of the results from the influential article [4] of Kazhdan and Lusztig which introduced the Kazhdan–Lusztig polynomials. Given a Coxeter group W and a set S of simple reflections, let ℋ denote the corresponding Hecke algebra, it has a “standard” basis T w and another basis C w with many remarkable properties. The Kazhdan–Lusztig polynomials p x, w give the transition matrix between these bases. One of their results proved by Kazhdan and Lusztig is an inversion formula, which states that if W is finite with longest element w 0, then for all x ≤ w in W. The main result of this article generalizes this result to the following setting: for any subset J of S, we define elements η J , and , and consider the two “dual” ideals and , where their standard basis and a Kazhdan–Lusztig basis are, respectively, indexed by subsets E J and of W.

Let $H$ be the Iwahori–Hecke algebra associated with $S_n$, the symmetric group on $n$ symbols. This algebra has two important bases: the Kazhdan–Lusztig basis and the Murphy basis. We establish a precise connection between the two bases, allowing us to give, for the first time, purely algebraic proofs for a number of fundamental properties of the Kazhdan–Lusztig basis and Lusztig's results on the $a$-function.

In this first chapter, we introduce the main objects of our study: Finite Coxeter groups, generic Iwahori-Hecke algebras, and their representations. An Iwahori-Hecke algebra H is here seen as a deformation of the group algebra of a finite Coxeter group W, where the deformation depends on a choice of a certain “weight function” L. Following Lusztig, to each simple module E of a Coxeter group over ℂ, we canonically associate a numerical invariant a E . The study of the a-function, and its subtle relation with Kazhdan–Lusztig basis of H, will be one of the main themes of this book. As a first step we shall introduce an “asymptotic” version of H and use this to define partitions of W into left, right and two sided cells.

We introduce the computer algebra package PyCox, written entirely in the Python language. It implements a set of algorithms, in a spirit similar to the older CHEVIE system, for working with Coxeter groups and Hecke algebras. This includes a new variation of the traditional algorithm for computing Kazhdan–Lusztig cells andW-graphs, which works efficiently for all finite groups of rank ≤8 (exceptE8). We also discuss the computation of Lusztig’s leading coefficients of character values and distinguished involutions (which works forE8as well). Our experiments suggest a re-definition of Lusztig’s ‘special’ representations which, conjecturally, should also apply to the unequal parameter case. Supplementary materials are available with this article.

We introduce and study certain topological spaces associated with connected rooted graphs. These spaces reflect combinatorial and order theoretic properties of the underlying graph and relate in the case of hyperbolic graphs to Gromov's hyperbolic compactification. They are particularly tractable in the case of Cayley graphs of finite rank Coxeter systems and are intimately related to the corresponding Iwahori-Hecke algebras. We study this connection by considering dynamical properties of the induced action of the Coxeter group.

For an element 𝑤 of a Coxeter group 𝑊, there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on Φ, the root system of 𝑊. This paper investigates the interaction between these two features of 𝑤, introducing the notion of the crossing number of 𝑤, κ ⁢ ( w ) \kappa(w) . Writing w = c 1 ⁢ ⋯ ⁢ c r w=c_{1}\cdots c_{r} as a product of disjoint cycles, we associate to each cycle c i c_{i} a “crossing number” κ ⁢ ( c i ) \kappa(c_{i}) , which is the number of positive roots 𝛼 in c i c_{i} for which w ⋅ α w\cdot\alpha is negative. Let Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) be the sequence of κ ⁢ ( c i ) \kappa(c_{i}) written in increasing order, and let κ ⁢ ( w ) = max ⁡ Seq κ ⁢ ( w ) \kappa(w)=\max{\mathrm{Seq}}_{\kappa}({w}) . The length of 𝑤 can be retrieved from this sequence, but Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) provides much more information. For a conjugacy class 𝑋 of 𝑊, let κ min ⁢ ( X ) = min ⁡ { κ ⁢ ( w ) ∣ w ∈ X } \kappa_{\min}(X)=\min\{\kappa(w)\mid w\in X\} and let κ ⁢ ( W ) \kappa(W) be the maximum value of κ min \kappa_{\min} across all conjugacy classes of 𝑊. We call κ ⁢ ( w ) \kappa(w) and κ ⁢ ( W ) \kappa(W) , respectively, the crossing numbers of 𝑤 and 𝑊. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups, if 𝑢 and 𝑣 are two elements of minimal length in the same conjugacy class 𝑋, then Seq κ ⁢ ( u ) = Seq κ ⁢ ( v ) {\mathrm{Seq}}_{\kappa}({u})={\mathrm{Seq}}_{\kappa}({v}) and κ min ⁢ ( X ) = κ ⁢ ( u ) = κ ⁢ ( v ) \kappa_{\min}(X)=\kappa(u)=\kappa(v) .

We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written \(x^{-1} y\) where x and y are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type \(A_n\) and \(B_n\) and then show that in spherical types different from \(D_n\) the simple elements of the dual braid monoid (for arbitrary choice of Coxeter element) embedded in the braid group are rational permutation braids (we conjecture this to hold also in type \(D_n\)). This property implies positivity properties of the polynomials arising in the linear expansion of their images in the Iwahori-Hecke algebra when expressed in the Kazhdan-Lusztig basis. In type \(A_n\), it implies positivity properties of their images in the Temperley-Lieb algebra when expressed in the diagram basis.