In characteristic zero one can use the Clebsch-Gordan rule to decompose tensor products of SL(2)-modules. In characteristic $p$ things are more complicated.

I am interested in the special case $S^dV\otimes V$ (where $V$ is the 2-dimesional standard representation) for fields $k$ of characteristic $p>0$. In fact, I mainly want to know about $d=3$.

If one computes the Clebsch-Gordan isomorphism explicitly, one can see that the denominator is $(d+1)$. So there will be a problem for $p|(d+1)$.

What is known in this case? I'd be happy just to know the case $d=3$, especially an explicit decomposition. I'd also like to know about the invariant theory.