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Dev Sinha
Dev Sinha
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Start with the remarkablyreally 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?

Start with the remarkably 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?

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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Dev Sinha
Dev Sinha
  • 5k
  • 27
  • 42

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

Start with the remarkably 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?