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I am looking for a reference on certain modules of invariants. I think that the question is quite natural so that I believe there should be some results already, but I am not able to find anything.

The situation is the following: let $\mathfrak{S}_n$ be the symmetric group that acts on $\mathbb{C}[x_1,y_1,x_2,y_2,\dots,x_n,y_n]$ by permuting simultaneously the $x_i,y_i$. Then the ring of invariants $S=\mathbb{C}[x_1,y_1,x_2,y_2,\dots,x_n,y_n]^{\mathfrak{S}_n}$ is sometimes called the ring of multisymmetric functions.

The generators of $S$ have been computed by Vaccarino in https://arxiv.org/pdf/math/0205233.pdf, but I am interested in the module of alternating covariants $Alt = \mathbb{C}[x_1,y_1,x_2,y_2,\dots,x_n,y_n]^{alt_n}$, where $alt_n$ denotes the alternating representation of $\mathfrak{S}_n$. More precisely,

$$ Alt = \{ f \in \mathbb{C}[x_1,y_1,\dots,x_n,y_n] \,|\, \sigma\cdot f = \operatorname{sgn}(\sigma)f \quad \text{ for all } \sigma\in \mathfrak{S}_n \} $$ where $\operatorname{sgn}$ denotes the sign of a permutation.

So my question is

  • Is there any reference about the generators of $Alt$ as an $S$-module? I would already be interested in the cases $n\leq 4$.

Any help would be greatly appreciated.

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    $\begingroup$ In the highly cited paper "Conjectures on the Quotient Ring by Diagonal Invariants", ams.org/mathscinet-getitem?mr=1256101, your $S$ is called diagonal invariants. I have seen it in many places unter this name. $\endgroup$
    – Christian Stump
    May 16, 2017 at 16:51
  • $\begingroup$ I think that she wants the polynomials $f(x,y)$ satisfying $w\cdot f(x,y) = \mathrm{sgn}(w)f(x,y)$ for all $w\in\mathfrak{S}_n$. If it were just $f(x)$ then the module is a free module of rank one over the symmetric functions, generated by $\prod_{i<j}(x_i-x_j)$. $\endgroup$
    – Richard Stanley
    May 16, 2017 at 17:57
  • $\begingroup$ Yes, thank you, that's precisely what I am looking for. I am sorry if I was not clear before, I added more details now. $\endgroup$
    – Daniele A
    May 17, 2017 at 9:09
  • $\begingroup$ @ChristianStump: thank you for the pointers to the diagonal invariants and the paper of Haiman. I'll have a look at it and see whether it says something on the alternating case. $\endgroup$
    – Daniele A
    May 17, 2017 at 9:12

1 Answer 1

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If I am not mistaken, the space $Alt$ has a basis given by $\{ \Delta_M(\mathbf{x},\mathbf{y}) \mid M \subset \mathbb{N}\times\mathbb{N} \text{ finite}\}$. Here, $\Delta_M(\mathbf{x},\mathbf{y}) = \det[x_i^{p_j}y_i^{q_j}]_{1\leq i,j\leq n}$ for $M = \{ (p_1,q_1),\ldots,(p_n,q_n) \}$.

You find this for example in Section 1.3 of Notes on Macdonald polynomials and the geometry of Hilbert schemes by Mark Haiman, see also the references therin.

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