Timeline for Isotypic components of the action of the symmetric group on polynomials
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 15, 2014 at 21:35 | vote | accept | Nicholas Proudfoot | ||
| Oct 15, 2014 at 21:35 | |||||
| Oct 15, 2014 at 19:13 | comment | added | Vladimir Dotsenko | @NicholasProudfoot: yes, you are right, you say it at some point. (I genuinely think that asking about $\mathbb{C}[x_1,\ldots,x_n]^{S_n}\otimes\mathbb{C}S_n$-modules would be a much shorter and cleaner way to express what you want). Then the second answer I give certainly provides a very explicit recipe. | |
| Oct 15, 2014 at 18:58 | comment | added | Nicholas Proudfoot | @VladimirDotsenko: $M$ is by definition the sum of the non-trivial isotypic components for the action of $S_n$. No matter what you consider $M$ to be a module over, it doesn't contain any monomials. | |
| Oct 15, 2014 at 18:39 | comment | added | Vladimir Dotsenko | @NicholasProudfoot: you probably mean "modules over $\mathbb{C}[x_1,\ldots,x_n]^{S_n}\otimes\mathbb{C}S_n$"? As $\mathbb{C}[x_1,\ldots,x_n]^{S_n}$-modules, you of course can use monomials as generators, right?.. | |
| Oct 15, 2014 at 18:34 | comment | added | Nicholas Proudfoot | @VladimirDotsenko: We have a decomposition $\mathbb{C}[x_1,\ldots,x_n] = \mathbb{C}[x_1,\ldots,x_n]^{S_n} \oplus M$ of modules over $\mathbb{C}[x_1,\ldots,x_n]^{S_n}$, and I'm looking for a set of module generators of $M$. Your first answer didn't make sense because $M$ doesn't contain any monomials. Your second answer may be correct, and I thank you for it. Steven Sam's answer is also correct, but yours may be more useful. | |
| Oct 15, 2014 at 18:25 | comment | added | Vladimir Dotsenko | @NicholasProudfoot: more to your question, if you want the space your module generators to be closed under the action of the symmetric group, then the second set (partial derivatives of the Vandermonde) gives you that. The first set gives a module decomposition which indeed does not agree with symmetries. | |
| Oct 15, 2014 at 18:22 | comment | added | David E Speyer | continued later, student just arrived. | |
| Oct 15, 2014 at 18:21 | comment | added | Vladimir Dotsenko | @NicholasProudfoot: OK, so you are taking the complement of the isotypic component of the trivial module, and you want a set of generators of this as what? (As for your question: it is a free module over the ring of symmetric polynomials, so I assume that regardless of the answer to my clarifying question it is enough to find generators over that ring, since then you can just multiply everything by arbitrary symmetric polynomials, n'est-ce pas?) | |
| Oct 15, 2014 at 18:20 | comment | added | David E Speyer | @NicholasProudfoot Because $R:=\mathbb{C}[x_1, \ldots, x_n]^{S_n}$ is a polynomial ring, and $S:=\mathbb{C}[x_1,\ldots, x_n]$ is Cohen-Macualay, $S$ is a free $R$-module. Moreover, each isotypic summand of $S$ is likewise a free $R$-module. (A graded summand of a graded free module over a polynomial ring is graded free.) continued.. | |
| Oct 15, 2014 at 18:08 | comment | added | Nicholas Proudfoot | Vladimir: I'm sorry, but I don't understand your answer. In the (very easy) case n=2, I am asking for a set of generators for the isotypic component of the sign representation. This isotypic component does not contain ANY monomials at all; in particular, it cannot be spanned by monomials. Also, the statement that you meant to make about the regular representation is that the ring of all polynomials modulo positive degree symmetric polynomials is isomorphic to the regular representation of $S_n$. The module that I'm asking about is infinite dimensional as a vector space. | |
| Oct 15, 2014 at 17:05 | comment | added | Vladimir Dotsenko | @darijgrinberg Well, roughly because geometrically the fiber over a generic point consists of a single orbit of $S_n$ acting via permutations of coordinates. Probably you can make it more algebraic passing to fields of quotients, looking at the field extension $\mathbb{C}(x_1,\ldots,x_n):\mathbb{C}(x_1,\ldots,x_n)^{S_n}$, and doing some basic Galois theory. | |
| Oct 15, 2014 at 16:58 | comment | added | darij grinberg | "it is a regular representation": why? | |
| Oct 15, 2014 at 16:56 | comment | added | Vladimir Dotsenko | @darijgrinberg: this is not clear to me from the way this question is stated: the way it is written, he asks for generators of the complement of the isotypic component of the trivial representation viewed as a module over the ring of symmetric functions. In any case, as an $S_n$-module it is a regular representation, so one presumably can act on elements I mention by Young symmetrizers (the second description actually gives a basis in the space of all partial derivatives of the Vandermonde, so it is a space of generators which is closed under the $S_n$-action). | |
| Oct 15, 2014 at 16:48 | comment | added | darij grinberg | I think he wants a decomposition into $S_n$-submodules. | |
| Oct 15, 2014 at 16:47 | history | answered | Vladimir Dotsenko | CC BY-SA 3.0 |