closure of $\k$. Let us denote by $\rep^\alpha_{\rm a.i.}(Q,\R)$ the subset of $\rep^\alpha(Q,\R)$ of absolutely indecomposable representations.

If $\k$ is a finite field we put

$$a_\alpha(\R Q):=\card(\rep^\alpha_{\rm a.i.}(Q,\R)/\G_\alpha(\R)).$$Otherwise we define $a_\alpha(\R Q)$ as the pair $(c,d)$ where $c$ is the number of irreducible components of $\{(x,g)\in\rep^\alpha_{\rm a.i.}(Q,\R)\times\G_\alpha(\R)\,|\,g\cdot x=x\}$ of maximal dimension and $d$ is

$${\rm dim}\,\{(x,g)\in\rep^\alpha_{\rm a.i.}(Q,\R)\times\G_\alpha(\R)\,|\,g\cdot x=x\}-{\rm dim}\, \G_\alpha(\R).$$ Denote by $|\alpha|$ the integer $\sum_{i\in I}\alpha(i)$.

\[main1\] Assume that $\R$ is a Frobenius algebra over a finite or algebraically closed field $\k$. Let $Q$ be a quiver and let $Q'$ be another quiver obtained from $Q$ by changing the orientation of some of the arrows. Then $$m_\alpha(\R Q)=m_\alpha(\R Q').$$If moreover $\R$ is local, split and if $\k$ contains a primitive $|\alpha|$-th root of unity then $$a_\alpha(\R Q)=a_\alpha(\R Q').$$ \[indeptheo\]

The version of this theorem when $\R=\k$ was called the [ *fundamental lemma*]{} in [@kraft-riedtmann] in Kac’s characterization [@kac] of roots in terms of $a_\alpha(\k Q).$

To formulate our second main result consider the double quiver $\overline{Q}=(I,E\cup E^*)$ obtained from $Q$ by adjoining a new arrow $a^*:j\leftarrow i$ for each arrow $a:i\rightarrow j\in E$, and denote by $Q^*$ the *opposite* quiver $(I,E^*)$ of $Q$. Then define \_=\_[,]{}:

[ccc]{}\^(,) && \_()\
