Counting quiver representations over finite fields via graph enumeration

JOURNAL OF ALGEBRA

卷:322

Article

Helleloid, Geir T.¹  Rodriguez-Villegas, Fernando¹ 

[1] Univ Texas Austin, Dept Math, Austin, TX 78712 USA

关键词: Quiver representation;    Absolutely indecomposable representation;    Graph enumeration;   

DOI  :  10.1016/j.jalgebra.2009.04.032

来源: Elsevier

【 摘 要 】

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 is an element of 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)(alpha, q) is a polynomial in q with integer coefficients. Using Hua's formula, we show that for each integer s >= 0, the sth 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. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free