The Braid Group and Linear Rigidity

Abstract

We introduce the BC-operation (short for braid-companion) on tuples of matrices. It corresponds to Katz's middle convolution operation on local systems, and generalizes it to fields K of arbitrary characteristic. It is based on the Burau–Gassner representation of the braid group. We expect many applications, in Galois theory (for finite K) as well as in Katz's original set-up of local systems and linear differential equations (where K = ℂ).