A Combinatorial Model for the Decomposition of Multivariate Polynomial Rings as $S_n$-Modules

  • Rosa Orellana
  • Michael Zabrocki


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 $\lambda$ (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. 

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