Structure of conjugacy classes in Coxeter groups

This paper gives a definitive solution to the problem of describing conjugacy classes in arbitrary Coxeter groups in terms of cyclic shifts. Let (W, S) be a Coxeter system. A cyclic shift of an element w W is a conjugate of w of the form sws for some simple reflection s S such that S (sws) S (w). The cyclic shift class of w is then the set of elements of W that can be obtained from w by a sequence of cyclic shifts. Given a subset K S such that WK: = K W is finite, we also call two elements w, w' W K-conjugate if w, w' normalise WK and w'=w₀ (K) ww₀ (K), where w₀ (K) is the longest element of WK. Let O be a conjugacy class in W, and let O^ be the set of elements of minimal length in O. Then O^ is the disjoint union of finitely many cyclic shift classes C₁, , Cₖ. We define the structural conjugation graph associated to O to be the graph with vertices C₁, , Cₖ, and with an edge between distinct vertices Cᵢ, Cⱼ if they contain representatives u Cᵢ and v Cⱼ such that u, v are K-conjugate for some K S. In this paper, we compute explicitely the structural conjugation graph associated to any conjugacy class in W, and show in particular that it is connected (that is, any two conjugate elements of W differ only by a sequence of cyclic shifts and K-conjugations). Along the way, we obtain several results of independent interest, such as a description of the centraliser of an infinite order element w W, as well as the existence of natural decompositions of w as a product of a torsion part and of a straight part, with useful properties.