Regular theory in complex braid groups

TL;DR

This paper extends the theory of regular elements and Garside structures from well-generated irreducible complex reflection groups to all reflection groups, establishing new algebraic properties and connections.

Contribution

It generalizes the connection between regular elements and roots of the full twist to all reflection groups, broadening the scope of previous results.

Findings

Regular elements' centralizers are well-structured in all reflection groups.

The Garside structure approach applies broadly beyond initial classes.

New algebraic properties of complex braid groups are established.

Abstract

In his seminal paper on complex reflection arrangements, Bessis introduces a Garside structure for the braid group of a well-generated irreducible complex reflection group. Using this Garside structure, he establishes a strong connection between regular elements in the reflection group, and roots of the "full twist" element of the pure braid group. He then suggests that it would be possible to extend the conclusion of this theorem to centralizers of regular elements in well-generated groups. In this paper we give a positive answer to this question and we show moreover that these results hold for an arbitrary reflection group.