Counting Quiver Representations over Finite Fields Via Graph Enumeration

2008/10/12 by Helleloid, Geir T., Villegas, Fernando Rodriguez · 1 citation
#16G20 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.0810.2127

Abstract

Let Γ be a quiver on n vertices v₁, v₂, ..., v_n with g_ij edges between v_i and v_j, and let α∈ \Nⁿ. 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.

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