A combinatorial model for the decomposition of multivariate polynomial rings as $S_n$-modules

We consider the symmetric group $S_n$-module of the polynomial ring with $m$ sets of $n$ commuting variables and $m'$ sets of $n$ anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition $λ$ (a partition of $n$) is the number of multiset tableaux of shape $λ$ satisfying certain column and row strict conditions. We also present a finite generating set for the ring of $S_n$ invariant polynomials of this ring.

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