$\DeclareMathOperator\SL{SL}$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-dimensional 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 composition series and whether one still has some nice direct sum decompositions into representations of smaller dimension (I realize that these will no longer be simple modules as in characteristic $0$). I'd also like to know references about how the invariant theory of $\SL(2)$ works in positive characteristic.