Conjugacy classes and straight elements in Coxeter groups

Let W be a Coxeter group. In this paper, we establish that, up to going to some finite index normal subgroup W-0 of W, any two cyclically reduced expressions of conjugate elements of W-0 only differ by a sequence of braid relations and cyclic shifts. This thus provides a very simple description of conjugacy classes in W-0. As a byproduct of our methods, we also obtain a characterisation of straight elements of W, namely of those elements w is an element of W for which l(w(n)) = n l(w) for any n is an element of Z. In particular, we generalise previous characterisations of straight elements within the class of so-called cyclically fully commutative (CFC) elements, and we give a shorter and more transparent proof that Coxeter elements are straight. (c) 2014 Elsevier Inc. All rights reserved.