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 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 has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite Coxeter groups. Moreover, to each element of 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.