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.



Alternating multisymmetric functions
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](https://math.berkeley.edu/~mhaiman/ftp/newt-sf-2001/newt.pdf) by Mark Haiman, see also the references therin.


