is only one $\R^\times$-orbit in this case.

\(ii) If $a_0=b_0$ and $A,B$ both non-zero, then there are exactly $(q^n-1)^2/(q-1)$ orbits.

\(iii) If $a=0$, $b_0=0$ and $B\neq 0$ (resp. $b=0$, $a_0=0$ and $A\neq 0$) then there are $(q^n-1)/(q-1)$ orbits.

\(iv) If $a=b=0$, there is one orbit.

We then find

$$B_n(q)=3+(q-1)\left(\sum_{i=0}^{n-1}q^i\right)^2+2\left(\sum_{i=0}^{n-1}q^i\right).$$ A simple calculation shows that $B_n-A_n =
 (q^n-1)(q^{n-1}-1)$. Therefore the equality $A_n=B_n$ holds only for $n=1$. In fact only in these cases $\R$ a finite Frobenius algebra and the equality $A_1=B_1$ is hence an instance of our main theorem.

Representations of quivers
==========================

We use the notation of the introduction.

The finite field case
---------------------

Here we prove Theorems \[main1\] and\[maintheoquiv\] in the finite field case. We thus assume that $\k$ is a finite field and that $\R$ is a Frobenius $\k$-algebra.

\(i) Assume that $Q'$ is a quiver obtained from $Q$ by changing the orientation of some arrows. Then there exists an isomorphism $\C[\rep^\alpha(Q,\R)]\rightarrow\C[\rep^\alpha(Q',\R)]$ of $\G_\alpha(\R)$-modules.

\(ii) Let $\mu_\R$ be the moment map (\[moment\]). There exists an isomorphism $\C[\rep^\alpha(Q,\R[\epsilon])]^{\g_\alpha(\R)}\rightarrow\C[\mu_\R^{-1}(0)]$ of $\G_\alpha(\R)$-modules. \[proprep\]

The assertion (ii) is a particular case of the first assertion of Corollary \[maincoro\] with $\V:=\rep^\alpha(Q,\k)$, $\G=\G_\alpha$ and $\xi=0$. The implication of Assertion (i) on the dimension of 