On non-conjugate Coxeter elements in well-generated reflection groups

TL;DR

This paper explores the properties and symmetries of Coxeter elements in well-generated complex reflection groups, revealing their orbit structure, isomorphisms of noncrossing partitions, and Galois group actions.

Contribution

It extends the concept of Coxeter elements to regular elements of order h and demonstrates their unified orbit and related structural properties.

Findings

Coxeter elements form a single orbit under reflection automorphisms.

Noncrossing partition lattices are isomorphic for different Coxeter elements.

Galois group acts simply transitively on conjugacy classes of Coxeter elements.

Abstract

Given an irreducible well-generated complex reflection group W with Coxeter number h, we call a Coxeter element any regular element (in the sense of Springer) of order h in W; this is a slight extension of the most common notion of Coxeter element. We show that the class of these Coxeter elements forms a single orbit in W under the action of reflection automorphisms. For Coxeter and Shephard groups, this implies that an element c is a Coxeter element if and only if there exists a simple system S of reflections such that c is the product of the generators in S. We moreover deduce multiple further implications of this property. In particular, we obtain that all noncrossing partition lattices of W associated to different Coxeter elements are isomorphic. We also prove that there is a simply transitive action of the Galois group of the field of definition of W on the set of conjugacy classes of Coxeter elements. Finally, we extend several of these properties to Springer's regular elements of arbitrary order.