The Enumeration of Coxeter Elements

Let (W, S, 0) be a Coxeter system: a Coxeter group W with S its distinguished generator set and 0 its Coxeter graph. In the present paper, we always assume that the cardinality l = |S| of S is finite. A Coxeter element of W is by definition a product of all generators s ∈ S in any fixed order. We use the notation C(W ) to denote the set of all the Coxeter elements in W . These elements play an important role in the theory of Coxeter groups, e.g., the determination of polynomial invariants, the Poincaré polynomial, the Coxeter number and the group order of W (see [1–5] for example). They are also important in representation theory (see [6]). In the present paper, we show that the set C(W ) is in one-to-one correspondence with the set C(0) of all acyclic orientations of 0. Then we use some graph-theoretic tricks to compute the cardinality c(W ) of the set C(W ) for any Coxeter group W . We deduce a recurrence formula for this number. Furthermore, we obtain some direct formulae of c(W ) for a large family of Coxeter groups, which include all the finite, affine and hyperbolic Coxeter groups. The content of the paper is organized as below. In Section 1, we discuss some properties of Coxeter elements for simplifying the computation of the value c(W ). In particular, we establish a bijection between the sets C(W ) and C(0). Then among the other results, we give a recurrence formula of c(W ) in Section 2. Subsequently we deduce some closed formulae of c(W ) for certain families of Coxeter groups in Section 3.