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

Abstract

We consider the symmetric group Sn-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 ? (a partition of n) is the number of multiset tableaux of shape ? satisfying certain column and row strict conditions. We also present a finite generating set for the ring of Sn invariant polynomials of this ring.