Conjugacy Relation on Coxeter Elements

Let (W, S, Γ) be an irreducible finitely presented Coxeter system. The present paper is mainly concerned with conjugacy relation on Coxeter elements in the case where Γ containing just one circle, in particular when Γ is itself a circle. In the cases where Γ is either a three multiple circle or a circle with three nodes, we show that the ss-equivalence relation on Coxeter elements of W is the same as the W -conjugacy relation. An explicit formula is given for the characteristic polynomial of a Coxeter element in the natural reflection representation of W when Γ is a circle. We also give the answers to some questions raised by Coleman and extend some results of Geck and Pfeiffer concerning conjugacy relation in Coxeter groups. Let (W,S,Γ) be a Coxeter system, where W is a Coxeter group, S a distinguished generator set, and Γ the corresponding Coxeter graph. In the present paper, we always assume S finite (write S = {s1, s2, · · · , sr}) and Γ connected unless otherwise specified. By a Coxeter element w ∈ W , we mean a product si1si2 · · · sir with i1, i2, · · · , ir a permutation of 1, 2, · · · , r. Let C(W ) be the set of all the Coxeter elements in W . The spectral classes in C(W ) have been studied extensively by a number of people (see [2, 3, 8, 9, 12, 13] and §5.). But the conjugacy relation in C(W ) is relatively less known except for the case where Γ is a tree, in the latter case, C(W ) is wholly contained in a single W -conjugacy class (see, for example, [7, 3.16]). In the present paper, we mainly consider the case where Γ contains just one circle. We first introduce the concept of ss-equivalence