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I know it fails but is there an answer?

More precisely, let $V$ be the standard complex $n$-dimensional representation of the alternating group $A_n$, $kV$ the direct sum of its $k$ copies, $S(kV)$ its symmetric algebra. What are the generators of the invariants $S(kV)^{A_n}$?

Comment: the generators of $S(kV)^{S_n}$ are polarizations of the generators of $S(V)^{S_n}$, the fact known as the first fundamental theorem. It is no longer true in this case: polarizations of elements of $S(V)^{A_n}$ generate a smaller subalgebra than $S(kV)^{A_n}$ except some small cases.

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  • $\begingroup$ Do you care about general k, or is characteristic zero good enough? $\endgroup$
    – Noah Snyder
    Dec 5, 2017 at 14:26

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