Victor Reiner ; Vivien Ripoll ; Christian Stump - On non-conjugate Coxeter elements in well-generated reflection groups

dmtcs:2466 - Discrete Mathematics & Theoretical Computer Science, January 1, 2015, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) - https://doi.org/10.46298/dmtcs.2466
On non-conjugate Coxeter elements in well-generated reflection groupsArticle

Authors: Victor Reiner ORCID1; Vivien Ripoll 2; Christian Stump 3

  • 1 School of Mathematics
  • 2 Fakultät für Mathematik [Wien]
  • 3 Department of Mathematics and Computer Science

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.


Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Section: Proceedings
Published on: January 1, 2015
Imported on: November 21, 2016
Keywords: Shephard groups,reflection groups,Coxeter groups,Coxeter elements,noncrossing partitions,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Reflection Group Combinatorics; Funder: National Science Foundation; Code: 1001933
  • Klassische Kombinatorik und Anwendungen; Funder: Austrian Science Fund (FWF); Code: Z 130

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