Cells in Coxeter groups I

Abstract

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 determine the partition of $W$ into left (right) cells. We are able to prove these conjectures only in a special case, but even from these partial results we can deduce some interesting corollaries.

14 pages, v4: fixed a small gap in the proof of Theorem 3.1 in the published version of the paper, we thank Jian-yi Shi for pointing the gap and for his help with the correction

References in corpus (2)

• Lusztig's $a$-function for Coxeter groups with complete graphs
• Lusztig's $a$ function for Coxeter groups of rank 3