Multiplicative middle convolution for KZ equations

Abstract

We extend the middle convolution for monodromy of Fuchsian ordinary differential equations due to Katz to an operation for monodromy of KZ equations. This operation, also called the middle convolution, gives a monodromy of KZ equation obtained by the additive middle convolution. Since the fundamental group of the domain of definition of KZ equation is the pure braid group, our middle convolution also gives a recursive way of constructing irreducible representations of the pure braid group. Several applications are considered.