field the first part of Theorem \[main2\] is [@jambor-plesken Theorem 1] of Jambor–Plesken, which was the starting point of our research. In this case Theorem 1.2 amounts to the agreement of the number of similarity classes of $n\times n$ matrices over $\k_2$ with the number of isomorphism classes of commuting pairs of $n\times n$ matrices over $\k$. We will discuss the case of the Jordan quiver in more detail in a forthcoming publication.

We start in Section 2 by introducing Frobenius algebras. In Section 3 we discuss the Fourier transform for finitely generated modules over a finite Frobenius algebra. In Section 4 we apply such Fourier transforms for representations of algebraic group schemes, culminating in Proposition \[mainprop\] and its many corollaries among them formula . In Section 4.3 we give counterexamples to , when $\R$ is not a Frobenius algebra. In Section 5 we develop the technical machinery to deduce our main Theorems \[main1\] and \[main2\] from the finite field versions. In Section 6 we classify all quivers and Frobenius algebras with finitely many locally free indecomposable representations. Finally in Section 7 we compute the number of indecomposable representations of a quiver of rank vector $\bf 1$ over $\k_d$, prove 