Maximal length elements of excess zero in finite Coxeter groups

Abstract

Abstract In this paper we prove that for W a finite Coxeter group and C a conjugacy class of W, there is always an element of C of maximal length in C which has excess zero. An element w ∈ W {w\in W} has excess zero if there exist elements σ , τ ∈ W {\sigma,\tau\in W} such that σ 2 = τ 2 = 1 , w = σ ⁢ τ {\sigma^{2}=\tau^{2}=1,w=\sigma\tau} and ℓ ⁢ ( w ) = ℓ ⁢ ( σ ) + ℓ ⁢ ( τ ) {\ell(w)=\ell(\sigma)+\ell(\tau)} , ℓ {\ell} being the length function on W.