Coxeter Elements of the Symmetric Groups Whose Powers Afford the Longest Elements

Abstract

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