Involution Statistics in Finite Coxeter Groups

2014/03/28 by Hart, Sarah B., Rowley, Peter J.
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1403.7506

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. In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, or the set of all involutions, in any finite Coxeter group W. In particular, these results correct errors in the paper "Permutation statistics on involutions", W.M.B. Dukes., European J. Combin. 28 (2007), 186--198. for the involution length polynomials of Coxeter groups of type B_n and D_n. Moreover, we give a counterexample to a unimodality conjecture of Dukes.

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