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Let $G$ be a finite group and $V$ be an integral representation of $G$, i.e. a free abelian group of finite rank with $G$-action. Now consider the symmetric power $Sym(V)$ of $V$ over $\mathbb{Z}$, which has again a $G$-action by the functoriality of $Sym$. While for the tensor algebra instead, one could compute its group cohomology via a Künneth theorem [edit: as Thorsten Ekedahl pointed out, this computes only the cohomology of $G\times G$], I have not found a similar theorem for symmetric powers in the literature. So my question is:

Is there a general procedure for computing the cohomology groups $H^i(G; Sym(V))$ (with $H^i(G; V)$ as input)?

It is not clear for me how to do this in a conceptual way even for simple examples like the two-dimensional indecomposable representation of the cyclic group $C_3$. With other words, the example I am especially interested in is to compute the $C_3 = \langle t|t^3 =1\rangle$-cohomology of $\mathbb{Z}[x,y]$ with $tx = y$ and $ty = -x - y$. Of course, it would be enough to have the decomposition in indecomposables. This computation is related to an alternative derivation of the cohomology of the sheaves $\omega^k$ on the moduli stack of elliptic curves (at the prime $3$).

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# Group Cohomology of Symmetric Powers

Let $G$ be a finite group and $V$ be an integral representation of $G$, i.e. a free abelian group of finite rank with $G$-action. Now consider the symmetric power $Sym(V)$ of $V$ over $\mathbb{Z}$, which has again a $G$-action by the functoriality of $Sym$. While for the tensor algebra instead, one could compute its group cohomology via a Künneth theorem, I have not found a similar theorem for symmetric powers in the literature. So my question is:

Is there a general procedure for computing the cohomology groups $H^i(G; Sym(V))$ (with $H^i(G; V)$ as input)?

It is not clear for me how to do this in a conceptual way even for simple examples like the two-dimensional indecomposable representation of the cyclic group $C_3$.