Given an algebra $A$ over $k$ with characteristic zero and a positive integer $n$, the subspace of $A^{\otimes n}$ consisting of all tensors invariant under the action of all permutations $\sigma\in\mathfrak{S}_n$ permuting its factors forms a subalgebra of $A^{\otimes n}$ often denoted by $S^n(A)$, the $n$-th symmetric power of $A$. Is there a simple relationship between the irreducible representations of $A$ and the irreducible representations of $S^n(A)$? The only reference I have found to the symmetric power of an algebra (as opposed to a vector space) is in Etingof's lecture notes on page 65.
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