Counting Representations of Valued Quivers over Finite Fields

Abstract

For a symmetrizable Borcherds–Cartan matrix A with integer entries and even diagonal entries, we show that there exists a k-species 𝓢 over the finite field k such that 𝓢 and the Borcherds–Cartan matrix provide the same bilinear form. We also show that the number of isomorphism classes of indecomposable representations of any valued graph with fixed dimension vector is a polynomial, and is independent of the orientation of the valued graph. This extends to the situation of valued graphs with loops.