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.
Bibliographical noteFunding Information:
Victor Reiner was supported by NSF Grant DMS-1001933. Vivien Ripoll was supported by the Austrian Science Foundation FWF, Grants Z130-N13 and F50-N15, the latter in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”. Christian Stump was supported by the German Research Foundation DFG, Grant STU 563/2-1 “Coxeter-Catalan combinatorics”.
© 2016, Springer-Verlag Berlin Heidelberg.