Related papers: W-graph determining elements in type A

The reduced expressions for a given element $w$ of a Coxeter group $(W, S)$ can be regarded as the vertices of a directed graph $\mathcal{R}(w)$; its arcs correspond to the braid moves. Specifically, an arc goes from a reduced expression…

We attach to every Coxeter system (W,S) an extension C_W of the corresponding Iwahori-Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto C_W.…

We continue the study of separable elements in finite Weyl groups. These elements generalize the well-studied class of separable permutations. We show that the multiplication map $W/U \times U \to W$ is a length-additive bijection, or…

Let (W,S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and J a subset of S. Let $W^J$ denote the set of minimal coset representatives modulo the parabolic subgroup $W_J$. For w in $W^J$, let…

To a Coxeter system $(W,S)$ (with $S$ finite) and a weight function $L : W \to \NM$ is associated a partition of $W$ into Kazhdan-Lusztig (left, right or two-sided) $L$-cells. Let $S^\circ = \{s \in S | L(s)=0\}$, $S^+=\{s \in S | L(s) >…

Let $W$ be a finite Coxeter group and $\Omega$ be its $W$-graph algebra as defined by Gyoja. The author's previous paper \cite{hahn2016wgraphs} considered this algebra in some detail, proposed, and proved in some small cases the $W$-graph…

In a Coxeter group $W$, an element is fully commutative if any two of its reduced expressions can be linked by a series of commutation of adjacent letters. These elements have particularly nice combinatorial properties, and also index a…

We introduce the notion of 321-avoiding permutations in the affine Weyl group $W$ of type $A_{n-1}$ by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey,…

We consider the set $\Irr(W)$ of (complex) irreducible characters of a finite Coxeter group $W$. The Kazhdan--Lusztig theory of cells gives rise to a partition of $\Irr(W)$ into "families" and to a natural partial order $\leq_{\cLR}$ on…

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…

Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$, where $I$ is a finite index set. Fix a nonempty convex subset $\mathscr{L}$ of $W$. If $W$ is of type $A$, then $\mathscr{L}$ is the set of linear extensions of a poset, and…

For a Coxeter group $W$ with length function $\ell$, the excess zero graph $\mathcal{E}_0(W)$ has vertex set the non-identity involutions of $W$, with two involutions $x$ and $y$ adjacent whenever $\ell(xy)=\ell(x)+\ell(y)$. Properties of…

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…

Let $ (W,S)$ be a Coxeter system. We investigate the equation $ w(\Phi_{x}) = \Phi_{y}$ where $ w,x,y\in W$ and $ \Phi_{x}$, $\Phi_{y}$ denote the left inversion sets of $ x$ and $ y$. We then define a commutative square diagram called a…

We show that the Coxeter-sortable elements in a finite Coxeter group W are the minimal congruence-class representatives of a lattice congruence of the weak order on W. We identify this congruence as the Cambrian congruence on W, so that the…

We classify surjective lattice homomorphisms $W\to W'$ between the weak orders on finite Coxeter groups. Equivalently, we classify lattice congruences $\Theta$ on $W$ such that the quotient $W/\Theta$ is isomorphic to $W'$. Surprisingly,…

In this article, we propose to initiate the general study of involution systems. An {\em involution system}, that is, a group $W$ generated by a set of involutions $S$, is naturally endowed with a {\em weak order} arising from orienting the…

To classify the classical field theories with W-symmetry one has to classify the symplectic leaves of the corresponding W-algebra, which are the intersection of the defining constraint and the coadjoint orbit of the affine Lie algebra if…

An odd Coxeter group $W$ is one which admits a Coxeter system $(W,S)$ for which all the exponents $m_{ij}$ are either odd or infinity. The paper investigates the family of odd Coxeter groups whose associated labeled graphs…

Let $\bH$ be the generic Iwahori--Hecke algebra associated with a finite Coxeter group $W$. Recently, we have shown that $\bH$ admits a natural cellular basis in the sense of Graham--Lehrer, provided that $W$ is a Weyl group and all…