Counting Quiver Representations over Finite Fields Via Graph Enumeration

TL;DR

This paper explores the enumeration of quiver representations over finite fields, deriving formulas for derivatives of counting polynomials at q=1, linking algebraic representation counts to graph enumeration.

Contribution

It introduces a method to compute derivatives of representation counting polynomials at q=1 using graph enumeration, extending Hua's and Kac's formulas.

Findings

Derivatives of counting polynomials are polynomials in graph edge counts.

Highest degree terms of these derivatives are explicitly computed.

Connections established between algebraic counts and graph enumeration.

Abstract

Let $\Gamma$ be a quiver on n vertices $v_1, v_2, ..., v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\alpha \in \N^n$. Hua gave a formula for $A_{\Gamma}(\alpha, q)$, the number of isomorphism classes of absolutely indecomposable representations of $\Gamma$ over the finite field $\F_q$ with dimension vector $\alpha$. Kac showed that $A_{\Gamma}(\bm{\alpha}, q)$ is a polynomial in q with integer coefficients. Using Hua's formula, we show that for each non-negative integer s, the s-th derivative of $A_{\Gamma}(\alpha,q)$ with respect to q, when evaluated at q = 1, is a polynomial in the variables $g_{ij}$, and we compute the highest degree terms in this polynomial. Our formulas for these coefficients depend on the enumeration of certain families of connected graphs.