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Start with the really well-known fact that $R[x_1, \ldots, x_n]^{S_n}$, where $R$ is any commutative ring, is polynomial on elementary symmetric polynomials. Now consider the slight generalization of multiple collections of variables, namely $R[x(i)_1, \ldots, x(i)_n]^{S_n}$, where $i$ runs over some finite indexing set and $S_n$ still acts by permuting subscripts. These rings are generally not polynomial algebras, in particular when $R$ is ${\mathbb F}_p$.

Ten years ago, in the context of computing the cohomology of symmetric groups, Mark Feshbach gave generators and inductively-defined relations for these rings when $R$ is ${\mathbb F}_2$. My questions are:

(1) Does anyone know of calculations over ${\mathbb F}_p$ or other approaches over ${\mathbb F}_2$?

(2) Restricting to $R = {\mathbb F}_p$ and replacing $S_n$ by $GL_n({\mathbb F}_p)$ we get the Dickson algebras in the case of one collection of variables. Has anyone studied the analogues of Dickson algebras where there are multiple collections of variables?

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As the answer below mentions, this has beeen studied a lot by combinatorialists when $k=0$ and $R$ is a field of characteristic 0, and it is interesting, but hard. Search for "diagonal coinvariants" or "Cherednik algebras" (which are a deformation of the smash product). –  Ben Webster Apr 1 '10 at 22:01

3 Answers 3

Four quick references that contain substantial info on your questions (for more, it'd be good to know what exactly you would like to know...):

de Concini, C.; Procesi, C. A characteristic free approach to invariant theory. Advances in Math. 21 (1976), no. 3, 330--354.

Grosshans, F. D. Vector invariants in arbitrary characteristic. Transform. Groups 12 (2007), no. 3, 499--514.

Stepanov, S. A. Vector invariants of symmetric groups in the case of a field of prime characteristic. Discrete Math. Appl. 10 (2000), no. 5, 455--468

Stepanov, Serguei A. Orbit sums and modular vector invariants. Diophantine approximation, 381--412, Dev. Math., 16, Vienna, 2008.

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The paper [P. FLEISCHMANN, A NEW DEGREE BOUND FOR VECTOR INVARIANTS OF SYMMETRIC GROUPS, TRANS. AMS Volume 350, Number 4, April 1998, Pages 1703-1712] shows that this ring is generated by homogeneous invariants whose degree does not exceed max{n, k(n − 1)} (where i runs over an index set of size k). Also this bound is sharp if $n=p^s$ for some prime $p$ and either $R=\mathbb Z$ or $R$ has characteristic $p$.

Some work has been done on the Dickson invariants version as well. I think that is considered in the article [Steinberg, Robert, On Dickson's theorem on invariants. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 3, 699–707.]

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Thank you for these references. –  Dev Sinha Aug 2 '11 at 5:28
@Dev: You are welcome. Looking back at my answer I realize I misread question 2. Let i run over an index set of size k. Some things may be known for k=2 but I don't think anything is known for larger values of k except there is a complete answer for general k for $GL(2,F)$. See the paper [Vector invariants for the two dimensional Modular representation of a cyclic group of prime order, Campbell, Shank, Wehlau, Adv. in Math., 225(2) 1069-1094]. –  David Wehlau Dec 3 '11 at 15:38

Do you mean the ring of diagonal invariants with $k > 1$? This appears in the combinatorics literature (Garsia-Haiman and developments thereof) for $k=2$ and $R$ a field of characteristic 0, but the definition can be given for all $k$ and $R$. It is just the $S_n$ invariants of the following object:

The polynomial ring (with $R$ as coefficients) generated by $nk$ variables, with the variables partitioned into $k$ disjoint sets of size $n$, and $S_n$ simultaneously (that is, "diagonally"), permuting all of the $n$-sets.

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Yes, I am asking about this ring of diagonal invariants, but in positive characteristic, as well as diagonal invariants (so to speak) of GL_n acting on $k$ disjoint sets of size $n$, in positive characteristic. –  Dev Sinha Apr 2 '10 at 4:30

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