Cells in Coxeter groups I

TL;DR

This paper explores the combinatorial structure of Kazhdan-Lusztig cells in infinite Coxeter groups, focusing on distinguished involutions and their recursive enumeration, with partial proofs supporting the conjectured cell partitioning.

Contribution

It introduces a conjectural recursive structure for distinguished involutions in Coxeter groups and proposes explicit equivalence relations to determine cell partitions, with partial proofs provided.

Findings

Conjecture on recursive enumeration of distinguished involutions.

Explicit equivalence relations for cell partitioning.

Partial proof of conjectures in special cases.

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.