On cycle decompositions in Coxeter groups

Abstract

The aim of this note is to show that the cycle decomposition of elements of the symmetric group admits a quite natural formulation in the framework of dual Coxeter theory, yielding a generalization of it to the family of so-called parabolic quasi-Coxeter elements of Coxeter groups (in the symmetric group every element is a parabolic quasi-Coxeter element). We show that such an element admits an analogue of the cycle decomposition. Elements which are not in this family still admit a generalized cycle decomposition, but it is not unique in general.

10 pages

References in corpus (1)

• Clusters, Coxeter-sortable elements and noncrossing partitions

Cited by in corpus (3)

• Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements
• Hurwitz numbers for reflection groups III: Uniform formulas
• In which it is proven that, for each parabolic quasi-Coxeter element in a finite real reflection group, the orbits of the Hurwitz action on its reflection factorizations are distinguished by the two obvious invariants