Coxeter elements of the symmetric groups whose powers afford the longest

TL;DR

This paper investigates Coxeter elements in symmetric groups, identifying which elements' powers produce the longest elements, with specific results for even and odd n.

Contribution

It characterizes Coxeter elements that generate the longest elements through their powers in symmetric groups, providing new insights into their structure.

Findings

For even n, the n/2-th power of certain Coxeter elements yields the longest element.

For odd n, specific Coxeter elements directly produce the longest element.

The paper clarifies the relationship between Coxeter elements and longest elements in symmetric groups.

Abstract

In this article, we first show that in case $n$ is even which Coxeter element in $\mathfrak{S}_{n}$ affords the longest by taking its power to $n/2$. We also show that in case $n$ is odd which Coxeter element affords the longest in $\mathfrak{S}_{n}$.