Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem

TL;DR

This paper introduces multiplicative preprojective algebras, linking them to middle convolution and applying these concepts to solve the Deligne-Simpson problem by providing conditions for the existence of certain irreducible solutions.

Contribution

It develops a new algebraic framework for multiplicative preprojective algebras, connecting them to middle convolution and advancing the understanding of the Deligne-Simpson problem.

Findings

Established a homological formula for Hom and Ext spaces.

Studied varieties of representations of multiplicative preprojective algebras.

Provided a sufficient condition for the existence of irreducible solutions to the Deligne-Simpson problem.

Abstract

We introduce a family of algebras which are multiplicative analogues of preprojective algebras, and their deformations, as introduced by M. P. Holland and the first author. We show that these algebras provide a natural setting for the 'middle convolution' operation introduced by N. M. Katz in his book 'Rigid local systems', and put in an algebraic setting by M. Dettweiler and S. Reiter, and by H. Volklein. We prove a homological formula relating the dimensions of Hom and Ext spaces, study varieties of representations of multiplicative preprojective algebras, and use these results to study simple representations. We apply this work to the Deligne-Simpson problem, obtaining a sufficient (and conjecturally necessary) condition for the existence of an irreducible solution to the equation $A_1 A_2 ... A_k = 1$ with the $A_i$ in prescribed conjugacy classes in $GL_n(C)$.