Locally free representations of quivers over commutative Frobenius algebras

In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra \(\textrm{R}\) $\text{R}$ by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over \(\textrm{R}\) $\text{R}$ equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over \(\textrm{R}[t]/(t^2)\) $\text{R}[t]/(t^2)$ . Using these results together with results of Geiss, Leclerc and Schröer we give, when \(\textrm{k}\) $\text{k}$ is algebraically closed, a classification of pairs \((Q,\textrm{R})\) $(Q,\text{R})$ such that the set of isomorphism classes of indecomposable locally free representations of Q over \(\textrm{R}\) $\text{R}$ is finite. Finally when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra \(\mathbb {F}_q[t]/(t^r)\) ${\mathbb{F}}_q[t]/(t^r)$ . We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.