Generalized symmetric algebras and Dickson algebras over ${\mathbb F}_p$. - MathOverflow

Generalized symmetric algebras and Dickson algebras over ${\mathbb F}_p$.

2010-04-01T20:30:50Z

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?

Answer by clarifythequestion for Generalized symmetric algebras and Dickson algebras over ${\mathbb F}_p$.

2010-04-01T21:48:36Z

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.

Answer by Vladimir Dotsenko for Generalized symmetric algebras and Dickson algebras over ${\mathbb F}_p$.

2010-04-02T11:26:06Z

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.

Answer by David Wehlau for Generalized symmetric algebras and Dickson algebras over ${\mathbb F}_p$.

2011-07-28T03:27:37Z

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.]