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

Victor Reiner,Vivien Ripoll,Christian Stump

doi:10.1007/s00209-016-1736-4

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.

Highlights

Background on reflection groupsLet V = Cn, and consider a finite subgroup W of GL(V ) ∼= GLn(C)
One calls W a complex reflection group if it is generated by its subset R of reflections, that is, the elements r ∈ W for which the fixed space ker(r − 11) ⊆ V is a hyperplane
The usual definition is more restrictive than the one given here: a Coxeter element is classically taken to be regular for the specific eigenvalue e2iπ/h, and this notion is, for real reflection groups, equivalent to the definition using the product of the reflections through the walls of a chamber in the reflection arrangement

Summary

Background on reflection groups

Victor Reiner and Vivien Ripoll and Christian Stump subset of n reflections that generate W This subclass contains all (complexifications of) irreducible real reflection groups inside GLn(R), which have a well-known structure of finite Coxeter groups. As with the chamber geometry in the real case, there is a reasonably natural way to construct a set of generalized Coxeter generators consisting of reflections, using the geometry of the Shephard group [Cox67]. We call such a presentation constructed from the geometry a standard generalized Coxeter presentation. The latter are known to be all real reflection groups with unbranched Coxeter graph, together with the infinite family G(r, 1, n) with r ≥ 3, and 15 of the non-real irreducible, exceptional groups

Coxeter elements and noncrossing partition lattices

Characterizations of Coxeter elements

Characterization of Coxeter elements in real groups and Shephard groups

Regular generating sets for well-generated reflection groups

Conjugacy classes of Coxeter elements

Regular elements and reflection automorphisms