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arXiv · 1404.5522

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

Victor Reiner·Vivien Ripoll·Christian Stump
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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.

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Victor Reiner, Vivien Ripoll, Christian Stump. 2014-12-14. On non-conjugate Coxeter elements in well-generated reflection groups. https://arxiv.org/abs/1404.5522

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