Counting Quiver Representations over Finite Fields Via 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 $\alpha \in {\mathbb{N}}^n$. 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 $A_{\Gamma }(\mathbf{\alpha },q)$ 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.

16 pages