Decomposition into irreducible representations unique in characteristic p? Let $G$ be a finite group and $K$ algebraically closed with $\operatorname{char}(K)=p \in prim$ and $p \nmid |G|$, then with Maschke's theorem we get a decomposition of irreducible representations.
Is this decomposition unique?
In the case $\operatorname{char}(K)=0$ one uses character theory to show the uniqueness using that the multiplicity of the irred. repr. equals the "inner product" of the characters... This just shows that the number of occurrence of the irred. repr. is independent of the decomposition. But this only works because $\operatorname{char}(K)=0$, i.e. the equations are in $\mathbb{Z}$. If $\operatorname{char}(K)=p \in prim$  the equations only are true in $\mathbb{Z}/p\mathbb{Z}$. So the number of occurrence may differ by multiples of p.
Does anyone of you know if there is proof or counterexample?
 A: You don't need character theory to do this even in characteristic zero. If $V$ is a completely reducible representation then the number of times an irreducible representation $V_i$ occurs in a direct sum decomposition is $\dim \text{Hom}(V_i, V)$, which is just an ordinary integer. This follows from Schur's lemma. If $K$ isn't algebraically closed then we need to modify the expression to $\frac{\dim \text{Hom}(V_i, V)}{\dim \text{End}(V_i)}$.
In general, any completely reducible module (over any ring, and more generally in any suitable abelian category) has a canonical "multiplicity decomposition"
$$V \cong \bigoplus_i \text{Hom}(V_i, V) \otimes V_i$$
where $\text{Hom}(V_i, V)$ is the "multiplicity space" associated to $V_i$. This is a completely canonical and in particular uniquely determined version of the irreducible decomposition, which by contrast is not unique if the multiplicities are greater than $1$ (a choice of such a decomposition amounts to a choice of splitting of the multiplicity spaces into lines).
