On the trace of Coxeter elements

TL;DR

This paper proves that the trace of a minimal-length Coxeter element on any irreducible representation of a Weyl group is always 0, 1, or -1, confirming a statement by Macdonald from the 1970s.

Contribution

The paper provides a proof of Macdonald's conjecture regarding the trace values of minimal-length Coxeter elements on irreducible representations of Weyl groups, including an extension to Iwahori-Hecke algebras.

Findings

Trace of Coxeter element is always 0, 1, or -1

Proof extends to Iwahori-Hecke algebra representations

Confirms Macdonald's 1970s conjecture

Abstract

Let W be a Weyl group and let w be a Coxeter elememt of minimal length of W. In the early 1970's I.G.Macdonald stated that the trace of w on an irreducible representation of W is 0,1 or -1. In this paper we give a proof of this statement and of an Iwahori-Hecke algebra version of it.