Abstract:

(joint work with Y. Roichman)

There is a natural graph structure on the set of all reduced words for the longest element in a finite Coxeter group W. Recently Autour and Dehornoy showed that when W = S_n, the symmetric group on n letters, the diameter of this graph grows asymptotically as O(n^4).

This talk will explain why this diameter for W = S_n is exactly n(n-1)(n-2)(3n-5)/24. The idea is to rephrase the problem as a natural general question about hyperplane arrangements, and minimal galleries from a chamber to its opposite chamber. We then show how to answer this question for hyperplane arrangements which are _supersolvable_, covering the case of W = S_n (type A_{n-1}) and also the case where W = B_n, the hyperoctahedral group. The general question is wide open.