Reduced words for reflections in Weyl groups

TL;DR

This paper provides explicit closed-form formulas for palindromic reduced expressions of reflections in any finite Weyl group, advancing the understanding of their algebraic structure.

Contribution

It introduces a general method to explicitly compute palindromic reduced expressions for all reflections in finite Weyl groups, filling a gap in existing literature.

Findings

Closed formulas for palindromic expressions in finite Weyl groups

Explicit expressions for all reflections

Enhancement of algebraic understanding of Coxeter groups

Abstract

The reflections in a Coxeter group are defined as conjugates of a single generator, and thus admit palindromic expressions as products of generators. Our main result gives closed formulas providing a palindromic reduced expression for each reflection in any finite Weyl group. There exist algorithmic methods for determining such reduced expressions, but explicit formulas have not been recorded outside of well-known special cases.