W-graph determining elements in type A

Van Minh Nguyen

Abstract

Let $(W,S)$ be a Coxeter system of type $A$, so that $W$ can be identified with the symmetric group $\mathrm{Sym}(n)$ for some positive integer $n$ and $S$ with the set of simple transpositions $\{(i,i+1)\mid 1⩽i⩽n-1\}$. Let ${⩽}_{\mathsf{L}}$ denote the left weak order on $W$, and for each $J\subseteq S$ let $w_J$ be the longest element of the subgroup $W_J$ generated by $J$. We show that the basic skew diagrams with $n$ boxes are in bijective correspondence with the pairs $(w,J)$ such that the set $\{x\in W\mid w_J{⩽}_{\mathsf{L}}x{⩽}_{\mathsf{L}}w{w}_J\}$ is a nonempty union of Kazhdan–Lusztig left cells. These are also the pairs $(w,J)$ such that $\mathcal{I}(w)=\{v\in W\mid v{⩽}_{\mathsf{L}}w\}$ is a $W$-graph ideal with respect to $J$. Moreover, for each such pair the elements of $\mathcal{I}(w)$ are in bijective correspondence with the standard tableaux associated with the corresponding skew diagram.

Keywords: Coxeter group, W-graph, Kazhdan–Lusztig cell, skew diagram, standard tableau.

AMS Subject Classification: Primary 20C08; secondary 20.85.