Proof of Shvartsman's conjecture on braid groups of projective complex reflection groups

TL;DR

This paper proves Shvartsman's conjecture connecting complex projective reflection groups with braid group quotients, extending previous real cases to all complex groups and correcting related earlier results.

Contribution

It extends Shvartsman's conjecture proof from real to all complex projective reflection groups and corrects a prior result by Broué, Malle, Rouquier.

Findings

Confirmed the conjecture for all complex projective reflection groups

Extended the proof from real to complex cases

Corrected a previous result in the literature

Abstract

The purpose of this note is to prove a conjecture of Shvartsman relating a complex projective reflection group with the quotient of a suitable complex braid group by its center. Shvartsman originally proved this result in the case of real projective reflection groups, and we extend it to all complex projective reflection groups. Our study also allows us to correct a result of Brou\'e, Malle, Rouquier on projective reflection groups.