$\DeclareMathOperator\SL{SL}$Let $K$ be an algebraically closed field. Let $V$ be a 2-dimensional vector space over $K$. The group $G=\SL_2(K)$ acts naturally on $V$ by left-multiplication on column vectors.
If $K=\mathbb{C}$ then we have an isomorphism of $KG$-modules
\begin{equation} S^d(S^m(V)^{\star}) \cong \Lambda^d(S^{m+d-1}(V)^{\star}) \tag{1}\label{469536_1} \end{equation}
for all $d>0$ and $m>0$. This is a consequence of basic character theory. The stars aren't actually needed because the modules $S^m(V)$ are self-dual for all $m$. Since symmetric and exterior powers commute with restriction, this also implies the same isomorphism holds when $G$ is replaced by a subgroup of $\SL_2(K)$, such as the additive group of $\mathbb{C}$ which is embedded in $G$ as the subgroup of upper unitriangular matrices, or its discrete subgroup $H$ isomorphic to $\mathbb{Z}$ generated by the matrix
$$\begin{pmatrix} 1&1 \\ 0 &1 \end{pmatrix}.$$
Now let $K$ be an algebraically closed field of characteristic $p$ and let $H$ be the subgroup of $G$ generated by the above matrix. So $H$ is finite group of order $p$. There is an isomorphism of $KH$-modules given by the formula \eqref{469536_1}. This is usually stated somewhat differently — there are exactly $p$ indecomposable modules for $H$, namely $V_1,V=V_2, \dotsc, V_p$, where the generator of $H$ acts on $V_n$ as multiplication by a $n \times n$ Jordan block. Each of these is self-dual, and $S^m(V_2) \cong V_{m+1}$ for $m<p$, so we usually write \eqref{469536_1} as $S^d(V_m) \cong \Lambda^d(V_{m+d})$ where $m+d \leq p$. You can't prove it by reducing the proof in the characteristic 0 case modulo $p$, it's considerably more involved than that.
I'd like to know whether an isomorphism of $KG$-modules \eqref{469536_1} exists when $K$ is an algebraically closed field of characteristic $p$. I've trawled around and found some work on tensor products for representations of $\SL_2(K)$ but nothing on symmetric and exterior powers.
For context: I'm really interested in proving \eqref{469536_1} for finite subgroups of the additive group of $K$, which are elementary abelian $p$-groups of order $q=p^n$, when $d+m<q$. I've found a plausible looking formula for an explicit isomorphism, which doesn't seems to have any dependence on $q$, so actually my proof should apply to the additive group of $K$. I thought this question for the additive group of $K$ was unlikely to have attracted much attention in itself, but that the analogue for $\SL_2(K)$ might already be known to the experts in representation theory of reductive linear algebraic groups, which would imply everything I want.
