case when $\R$ is a Frobenius algebra.

These results have been achieved by use of reflection functors. In this paper we will concentrate on the subcategory of locally free representations. It is not stable under reflection functors, which use kernels and cokernels. Thus usual techniques will not be available. Instead we will use arithmetic harmonic analysis and prove results showing that locally free representations behave better when changing orientations of the quiver, and have interesting connections to representations of preprojective algebras.

Let $\alpha\in \N^I$ be a rank vector and put $$\rep^\alpha(Q,\R):=\bigoplus_{i\rightarrow j\in E} 
\operatorname{Hom}_\R(\R^{\alpha(i)},\R^{\alpha(j)})$$ the $\R$-module of locally free $\R$-representations of the quiver $Q$ of rank $\alpha$. The group $$\G_\alpha(\R):=\varprod_{i\in I} \GL(\R^{\alpha(i)})$$ acts on $\rep^\alpha(Q,\R)$ and so gives a homomorphism :\_()\_(\^(Q,)).Finally let $\g_\alpha(\R)=\oplus_{i\in} \gl(\R^{\alpha(i)})$ be the Lie algebra of $\G_\alpha(\R)$ and $\varrho:\g_\alpha(\R)\to \End(\rep^\alpha(Q,\R))$ the derivative of , the infinitesimal action. The set of isomorphism classes of locally free $\R$-representations of $Q$ of dimension $\alpha$, can be identified with the orbit space $\rep^\alpha(Q,\R)/\G_\alpha(\R)$.

If $\k$ is a finite field we put

m\_(Q):=(\^(Q,)/\_())the number of isomorphism classes of locally free $\R$-representations of $Q$ of dimension $\alpha$.

If $\k$ is algebraically closed, the number of isomorphism classes could be infinite and so instead 