$\DeclareMathOperator\SL{SL}$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.



Representations of $\mathrm{SL}(2)$ in characteristic 2
$\newcommand{\Hom}{\operatorname{Hom}}$Here is an elaboration of my comment with what happens in characteristic $2$ when $d=3$:


The usual decomposition rule gives us a filtration of $S^dV\otimes V$ with two factors: $H^0(4)$ and $H^0(2)$ (I use the notation from Jantzen's Representations of Algebraic Groups and write all weights in terms of the fundamental weight).


Applying the Jantzen sum formula, we see that $H^0(4)$ has a composition series consisting of $L(4)$, $L(2)$ and $L(0)$.


We also see that $H^0(2)$ has a composition series consisting of $L(2)$ and $L(0)$.


All this gives us a composition series $$0 \subseteq M\_1 \subseteq M\_2 \subseteq M\_3 \subseteq M\_4 \subseteq M\_5 = S^3V\otimes V$$ where $M\_1\cong L(2)$, $M\_2/M\_1\cong L(0)$, $M\_3/M\_2\cong L(4)$ and $\{M\_4/M\_3,M\_5/M\_4\}\cong \{L(2),L(0)\}$. Which order the two top factors come in is less obvious (I will need to think a bit about it), and whether we actually have $S^3V\otimes V\cong H^0(4)\oplus H^0(2)$ I will also need to think a bit more to figure out.


Added: So, after some further thought, we can actually say a bit more.


First note that as mentioned by Jim Humphreys, we have $S^3V\otimes V\cong L(1)\otimes L(1)\otimes L(1)^{(1)}$ which means that it is self-dual. In particular, we see that our composition series can be chosen to be "symmetric", so we get $M\_4/M\_3\cong L(0)$ and $M\_5/M\_4\cong L(2)$ (it is also good to notice that we actually have $M\_2\cong H^0(2)$ and $M\_5/M\_2\cong H^0(4)$ as these are sometimes easier to work with).


We can also show that $S^3V\otimes V$ is indecomposable. In fact, we have $\operatorname{soc}\_{SL\_2}(S^3V\otimes V) = L(2)$.


To see this, we need a bit more machinery (it might be possible to do this in a more elementary way). Let $G = SL\_2$ and let $G\_1$ be the first Frobenius kernel of $G$. We let $\lambda = \lambda\_0 + p\lambda\_1$ be a dominant weight with $\lambda\_0 < p$ and use that $L(\lambda) \cong L(\lambda\_0)\otimes L(\lambda\_1)^{(1)}$. Now we note that $$\Hom\_G(L(\lambda),L(1)\otimes L(1)\otimes L(1)^{(1)})$$ $$\cong \Hom\_{G/G\_1}(L(\lambda\_1)^{(1)},\Hom\_{G\_1}(L(\lambda\_0),L(1)\otimes L(1))\otimes L(1)^{(1)})$$ so it is sufficient to show that $\operatorname{soc}\_{G\_1}(L(1)\otimes L(1)) = L(0)$.


To see this we further note that it will suffice to show that $\operatorname{soc}\_G(L(1)\otimes L(1)) = L(0)$ since the $G\_1$-socle is a $G$-submodule. But this final part is a simple calculation, as we clearly just need to check that neither $L(1)$ nor $L(2)$ are submodules. That $L(1)$ is not a submodule is clear by parity (all highest weights of composition factors in $L(1)\otimes L(1)$ must be even), and that $L(2)$ is not a submodule is seen by noting that $$\Hom\_G(L(2),L(1)\otimes L(1))\cong \Hom\_G(L(1),L(1)\otimes L(2))\cong \Hom\_G(L(1),L(3))$$ and $L(3)$ is simple (it is the 2'nd Steinberg module as also mentioned by Jim Humphreys).


A few final notes: The above actually shows that as a $G\_1$-module, $L(1)\otimes L(1)$ is the injective hull of the trivial module. This is a general fact about $SL\_2$ in characteristic $2$, ie, that for all $r$, $St\_r\otimes St\_r$ is the injective hull of the trivial module as a $G\_r$-module (this does not generalize to other groups, nor to other primes).


Also, the conclusion about the module $S^3V\otimes V$ is in fact that it is indecomposable tilting (in the notation from Jantzen, it is denoted $T(4)$).


