Abstract
Let (W, S, Γ) be an irreducible finitely presented Coxeter system. The present paper is mainly concerned with a conjugacy relation on Coxeter elements in the case where Γ contains 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.
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