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Let $V$ be the space of $n$ by $n$ complex matrices with the conjugate action of the symmetric group $G=S_n$. Is any explicit set of generators for the invariant ring $C[V]^G$ known?

flag	asked Apr 22 2012 at 18:29 Ketan Mulmuley 141●3

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Dima Pasechnik · Answer 1 · 2012-04-23 07:09:14Z UTC

For a related question, invariants of the action of $G$ on the space of pairs of {1,...,n}, (this is a quotient ring of $C[V]^{G}$) see Sect. 2 of Algebraic invariants of graphs; a study based on computer exploration, by Nicolas M. Thiéry. However, the generating set given there is certainly very far from a minimal, and degrees are high. Sect. 10 of this paper also discusses the ring you are asking about.

Ketan Mulmuley · Answer 2 · 2012-04-23 13:17:15Z UTC

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thanks for all the answers. I found a paper by Garcia and Stanton, "Group actions on Stanley Reisner rings and .." (Advances in Maths, 1984), which provides a reasonable answer to this question.

Ketan Mulmuley

answered Apr 23 2012 at 13:17

Ketan Mulmuley
141●3

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Are you your own evil twin? There seem to be two KMs with different reps... – Igor Rivin Apr 23 2012 at 13:28

Explicit generators for matrix invariants of the symmetric group

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