Fox-Neuwirth cells, quantum shuffle algebras, and the homology of type-B Artin groups

TL;DR

This paper develops new braid representations for type-B Artin groups, linking their homology to bimodule cohomology over quantum shuffle algebras, and provides a complete characterization of their homology with specific coefficients.

Contribution

It introduces a novel family of braid representations for type-B Artin groups and connects their homology to quantum shuffle algebra bimodules, offering new computational methods.

Findings

Homology of type-B Artin groups characterized with one-dimensional braid coefficients.

Two approaches for homology computation: induced representations and cellular stratification.

Complete description of homology over characteristic zero fields.

Abstract

In this paper, we will develop a family of braid representations of Artin groups of type B from braided vector spaces, and identify the homology of these groups with these coefficients with the cohomology of a specific bimodule over a quantum shuffle algebra. As an application, we give a complete characterization of the homology of type-B Artin groups with coefficients in one-dimensional braid representations over a field of characteristic 0. We will also discuss two different approaches to this computation: the first method extends a computation of the homology of braid groups due to Ellenberg-Tran-Westerland by means of induced representation, while the second method involves constructing a cellular stratification for configuration spaces of the punctured complex plane.