c © 2004 Heldermann Verlag Lengths of Involutions in Coxeter Groups

Abstract. Let t be an involution in a Coxeter group W. We determine the minimal and maximal (in the case of finite W) length of an involution in the conjugacy class of t. Mathematics Subject Classification 20F55. Let W be a finitely generated Coxeter group whose distinguished set – the set of fundamental reflections – is R. The length l(w) of a non-trivial element w in W is defined to be l(w) = min{l ∈ N: w = r1r2 · · · rl some ri ∈ R} and l(1) = 0. Suppose t is an involution in W, and let C = tW be the conjugacy class of t in W. The aim of this short paper is to determine the minimal and maximal (in which case W is assumed finite) length of an involution in C. Associated to any Coxeter group W is the root system Φ, which is the disjoint union of its positive and negative roots (denoted Φ+ and Φ − respectively). The fundamental reflections r ∈ R are in one-to-one correspondence with the fundamental roots αr, r ∈ R and W acts faithfully on Φ (see [1]). For w ∈ W