Counting Quiver Representations over Finite Fields Via Graph Enumeration

Let Γ be a quiver on n vertices v1, v2, . . . , vn with gij edges between vi and vj , and let α ∈ Nn. Hua gave a formula for AΓ(α, q), the number of isomorphism classes of absolutely indecomposable representations of Γ over the finite field Fq with dimension vector α. Kac showed that AΓ(α, q) is a polynomial in q with integer coefficients. Using Hua’s formula, we show that for each integer s ≥ 0, the s-th derivative of AΓ(α, q) with respect to q, when evaluated at q = 1, is a polynomial in the variables gij , 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.