TY - JOUR

T1 - On non-conjugate Coxeter elements in well-generated reflection groups

AU - Reiner, Victor

AU - Ripoll, Vivien

AU - Stump, Christian

N1 - Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2017/4/1

Y1 - 2017/4/1

N2 - 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.

AB - 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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U2 - 10.1007/s00209-016-1736-4

DO - 10.1007/s00209-016-1736-4

M3 - Article

AN - SCOPUS:84984780846

VL - 285

SP - 1041

EP - 1062

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 3-4

ER -