Proof of Shvartsman's conjecture on braid groups of projective complex reflection groups
Owen Garnier
TL;DR
This work extends Shvartsman’s conjecture to all complex projective reflection groups by linking the projective braid group to the quotient of a complex braid group by its center via an enlarged braid-group framework. It introduces X_S and the enlarged braid groups B_S and P_S, clarifying their centers and showing that pi1(X̂_S/W) computes as B_S/⟨beta_S⟩ and, in favorable cases, yields pi1(X̂/G) ≃ B/Z(B). The authors provide a corrected account of Broué–Malle–Rouquier results, describing precisely when a natural diagram of exact sequences can be completed, and they characterize the instances in which all regular elements are central, listing the finite types for which this holds. In remaining cases, they establish a surjective relation from B to pi1(X̂/widehat{W}) with a kernel generated by central-power elements, thus delineating the limits of the diagrammatic approach and clarifying the structure of projective braid groups in the complex setting.
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 Broué, Malle, Rouquier on projective reflection groups.