Commuting matrices and volumes of linear stacks

TL;DR

This paper explores counting problems related to quiver representations and unipotent groups over finite fields, developing a framework of linear stacks to connect these invariants and address Higman's conjecture.

Contribution

It introduces a general framework of linear stacks over small etale sites and derives explicit formulas linking various counting invariants to Higman's conjecture.

Findings

Derived explicit formulas for invariants related to quiver representations

Connected counting problems to Higman's polynomial conjecture

Developed a new theoretical framework for volumes of linear stacks

Abstract

A conjecture by Higman asserts that the number of conjugacy classes in the unipotent group of upper triangular matrices over a finite field depends polynomially on the number of elements of the field. We will study several alternative counting problems arising from quiver representations and prove explicit formulas relating the corresponding invariants to the invariants of Higman's conjecture. To do this, we develop a general framework of linear stacks over small etale sites and study volumes of these stacks and of their substacks of absolutely indecomposable objects.