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Reduced words for reflections in Weyl groups

Elizabeth Milićević

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.

Reduced words for reflections in Weyl 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.
Paper Structure (3 sections, 1 theorem, 4 equations)

This paper contains 3 sections, 1 theorem, 4 equations.

Table of Contents

  1. Introduction
  2. Reduced words for reflections
  3. Discussion of the results

Key Result

Theorem 1.1

The following are palindromic reduced expressions, one for each distinct reflection in the Weyl group of the type specified.

Theorems & Definitions (2)

  • Theorem 1.1
  • Example 1.2