Generalized symmetric algebras and Dickson algebras over \${\mathbb F}_p\$. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:15:08Z http://mathoverflow.net/feeds/question/20106 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20106/generalized-symmetric-algebras-and-dickson-algebras-over-mathbb-f-p Generalized symmetric algebras and Dickson algebras over \${\mathbb F}_p\$. Dev Sinha 2010-04-01T20:30:50Z 2011-07-28T03:27:37Z <p>Start with the really well-known fact that <code>\$R[x_1, \ldots, x_n]^{S_n}\$</code>, where \$R\$ is any commutative ring, is polynomial on elementary symmetric polynomials. Now consider the slight generalization of multiple collections of variables, namely <code>\$R[x(i)_1, \ldots, x(i)_n]^{S_n}\$</code>, where \$i\$ runs over some finite indexing set and <code>\$S_n\$</code> still acts by permuting subscripts. These rings are generally not polynomial algebras, in particular when \$R\$ is <code>\${\mathbb F}_p\$</code>.</p> <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:</p> <p>(1) Does anyone know of calculations over <code>\${\mathbb F}_p\$</code> or other approaches over <code>\${\mathbb F}_2\$</code>?</p> <p>(2) Restricting to <code>\$R = {\mathbb F}_p\$</code> and replacing <code>\$S_n\$</code> by <code>\$GL_n({\mathbb F}_p)\$</code> 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?</p> http://mathoverflow.net/questions/20106/generalized-symmetric-algebras-and-dickson-algebras-over-mathbb-f-p/20110#20110 Answer by clarifythequestion for Generalized symmetric algebras and Dickson algebras over \${\mathbb F}_p\$. clarifythequestion 2010-04-01T21:48:36Z 2010-04-01T21:48:36Z <p>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:</p> <p>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.</p> http://mathoverflow.net/questions/20106/generalized-symmetric-algebras-and-dickson-algebras-over-mathbb-f-p/20158#20158 Answer by Vladimir Dotsenko for Generalized symmetric algebras and Dickson algebras over \${\mathbb F}_p\$. Vladimir Dotsenko 2010-04-02T11:26:06Z 2010-04-02T11:26:06Z <p>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...):</p> <p>de Concini, C.; Procesi, C. A characteristic free approach to invariant theory. Advances in Math. 21 (1976), no. 3, 330--354. </p> <p>Grosshans, F. D. Vector invariants in arbitrary characteristic. Transform. Groups 12 (2007), no. 3, 499--514. </p> <p>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 </p> <p>Stepanov, Serguei A. Orbit sums and modular vector invariants. Diophantine approximation, 381--412, Dev. Math., 16, Vienna, 2008.</p> http://mathoverflow.net/questions/20106/generalized-symmetric-algebras-and-dickson-algebras-over-mathbb-f-p/71460#71460 Answer by David Wehlau for Generalized symmetric algebras and Dickson algebras over \${\mathbb F}_p\$. David Wehlau 2011-07-28T03:27:37Z 2011-07-28T03:27:37Z <p>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\$. </p> <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.]</p>