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?
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2 Answers
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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.
answered Apr 23, 2012 at 0:41
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1$\begingroup$ @Dima: excellent reference. It seems to me that the ring in Ketan's question is exactly the Grigoriev digraph invariant ring mentioned in Section 10 of Thiery's article. $\endgroup$ Apr 23, 2012 at 16:19
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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
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3$\begingroup$ Are you your own evil twin? There seem to be two KMs with different reps... $\endgroup$ Apr 23, 2012 at 13:28