For the standard representation of $S_5$, the ring of invariants is generated by the elementary symmetric polynomials and hence it is a polynomial ring. Now if we take the tensor product of standard and sign representation, it is again irreducible of degree $4$ but the ring of invariants is not a polynomial ring anymore. Now my question is what are the generators and relations for this ring? Is it known for general $n$ ?
2026-03-26 19:25:05.1774553105
Invariant ring for $S_5$
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