On Coxeter elements whose powers afford the longest word (Combinatorics of Lie Type)

Abstract

The purpose of this note is to present a condition for the power of a Coxeter element of mathfrak{S}_{n+1} to become the longest word. To be precise, given a product C of n distinct adjacent transpositions of 6_{n+1} in any order, we describe a condition for C such that the (n+1)/2-th power C^{(n+1)/2} of C becomes the longest element, in terms of the Amida diagrams.