Counting Quiver Representations over Finite Fields Via Graph Enumeration

Abstract

Let $Γ$ be a quiver on n vertices $v_1, v_2, ..., v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $α\in \N^n$. Hua gave a formula for $A_Γ(α, q)$, the number of isomorphism classes of absolutely indecomposable representations of $Γ$ over the finite field $\F_q$ with dimension vector $α$. Kac showed that $A_Γ(\bmα, q)$ 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 $A_Γ(α,q)$ 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.