Cells in Coxeter groups I

Details Cells in Coxeter groups I Cells in Coxeter groups I

2010-12-02

arxiv.org/abs/1012.0489v2

Mathematics

Representation Theory

13 pages, added a reference to Shi's counterexample to Lusztig's conjecture

Scientific paper

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.

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