Invertible tensor powers span the Schur algebra

{title2=$\operatorname{span}\{a^{\otimes n}\}=\bigl(U^{\otimes n}\bigr)^{S_n}$}

Over a field of <characteristic zero>, degree-$n$ <symmetric tensors> are spanned by pure powers $a^{\otimes n}$. Inclusion-exclusion expands $\sum_{J\subseteq[n]}(-1)^{n-|J|}(\sum_{j\in J}a_j)^{\otimes n}$ into the sum of all <permutation> orderings of $a_1\otimes\cdots\otimes a_n$. Every surviving term uses each of the $n$ labels once. This converts an invariant-tensor spanning problem into pure <tensor powers>.


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 Incoming links: Polarization spanning of symmetric tensors