A Clebsch-Gordan decomposition in positive characteristic

Abstract: Let $G$ be the special linear group of degree $2$ over an algebraically closed field $K$. Let $E$ be the natural module and $S^rE$ the $r$th symmetric power. We consider here, for $r,s\geq 0$, the tensor product of $S^rE$ and the dual of $S^sE$. In characteristic zero this tensor product decomposes according to the Clebsch-Gordan formula. We consider here the situation when $K$ is a field of positive characteristic. We show that each indecomposable component occurs with multiplicity one and identify which modules occur for given $r$ and $s$.