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

Victor Reiner, Vivien Ripoll, Christian Stump

Research output: Contribution to journalConference articlepeer-review

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 regular elements of arbitrary order.

Original languageEnglish (US)
Pages (from-to)109-120
Number of pages12
JournalDiscrete Mathematics and Theoretical Computer Science
StatePublished - 2015
Externally publishedYes
Event27th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2015 - Daejeon, Korea, Republic of
Duration: Jul 6 2015Jul 10 2015

Bibliographical note

Funding Information:
†Email: reiner@math.umn.edu. Supported by NSF grant DMS-1001933. ‡Email: vivien.ripoll@univie.ac.at. 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”. §Email: christian.stump@fu-berlin.de. Supported by the German Research Foundation DFG, grant STU 563/2-1 “Coxeter-Catalan combinatorics”.

Publisher Copyright:
© 2015 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.

Keywords

  • Coxeter elements
  • Coxeter groups
  • Noncrossing partitions
  • Reflection groups
  • Shephard groups

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