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.