A combinatorial model for the decomposition of multivariate polynomials rings as an $S_n$-module

Rosa Orellana, Mike Zabrocki

Published 2019-06-03Version 1

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 calculate that the multiplicity of an irreducible indexed by the partition $\lambda$ (a partition of $n$) is the number of multiset tableaux of shape $\lambda$ 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.