Lengths of involutions in finite Coxeter groups

Abstract

Let W be a finite Coxeter group and X a subset of W . The length polynomial L W , X ( t ) is defined by L W , X ( t ) = ∑ x ∈ X t ℓ ( x ) , where ℓ is the length function on W . If X = { x ∈ W : x 2 = 1 } then we call L W , X ( t ) the involution length polynomial of W . In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, and the involution length polynomial, in any finite Coxeter group W . In particular, these results correct errors in [11] for the involution length polynomials of Coxeter groups of type B n and D n . Moreover, we give a counterexample to a unimodality conjecture stated in [11] .