A Quiver Analogue of Higman's Conjecture

TL;DR

This paper introduces a quiver-based generalization of Higman's conjecture on conjugacy class counts in unitriangular groups, proving the conjecture for specific quivers with limited path lengths.

Contribution

It generalizes Higman's conjecture to quivers and proves it for quivers with no paths longer than two, providing explicit formulas.

Findings

Proved invariance of conjugacy class counts under certain quiver operations.

Solved the conjecture for quivers with maximum path length of two.

Provided explicit formulas for these cases.

Abstract

An unresolved conjecture by Graham Higman states that for all $n\geq 1$ the number of conjugacy classes of the group of $n \times n$ unitriangular matrices with entries in the finite field $\mathbb{F}_q$ is a polynomial in $q$. In this paper we introduce a new quiver generalization of the conjecture. Motivated by this generalization, we prove that certain operations on quivers leave the relevant counts unchanged. Based on these invariance properties, we solve the introduced conjecture for quivers containing no path of length exceeding two, providing explicit formulas.