Let (W, S, 1 ) be an irreducible finitely presented Coxeter system. The present paper is mainly concerned with a conjugacy relation on Coxeter elements in the case where 1 contains just one circle, in particular when 1 is itself a circle. In the cases where 1 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 1 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.
2001 Academic Press
Let (W, S, 1 ) be a Coxeter system, where W is a Coxeter group, S a distinguished generator set, and 1 the corresponding Coxeter graph. In the present paper, we always assume S finite (write S=[s 1 , s 2 , ..., s r ]) and 1 connected unless otherwise specified.
By a Coxeter element w # W, we mean a product s i 1 s i 2 } } } s i r with i 1 , i 2 , ..., i r 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 ] and Section 5.). But the conjugacy relation in C(W ) is relatively less known except for the case where 1 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 1 contains just one circle. We first introduce the concept of ss-equivalence in C(W ) (see 1.3). Each ss-class of C(W ) is contained in some W-conjugacy class. The study of ss-classes in C(W ) with 1 containing just one circle can be reduced to the case where 1 is itself a circle (Proposition 2.3). Then we describe all the ss-classes of