Involutions in Coxeter groups

Abstract

We combinatorially characterize the number $\mathrm{cc}_2$ of conjugacy classes of involutions in any Coxeter group in terms of higher rank odd graphs. This notion naturally generalizes the concept of odd graphs, used previously to count the number of conjugacy classes of reflections. We provide uniform bounds and discuss some extremal cases, where the number $\mathrm{cc}_2$ is smallest or largest possible. Moreover, we provide formulae for $\mathrm{cc}_2$ in free and direct products as well as for some finite and affine types, besides computing $\mathrm{cc}_2$ for all triangle groups, and all affine irreducible Coxeter groups of rank up to eleven.