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

## TL;DR

This paper introduces a combinatorial model for decomposing multivariate polynomial rings as modules over the symmetric group, linking irreducible representations to multiset tableaux and providing generators for invariant polynomials.

## Contribution

It presents a novel combinatorial framework for understanding the $S_n$-module structure of multivariate polynomial rings with commuting and anti-commuting variables, including explicit multiplicity formulas and generators.

## Key findings

- Multiplicity of irreducible modules equals the count of specific multiset tableaux.
- Provides a finite generating set for the ring of $S_n$-invariant polynomials.
- Establishes a combinatorial correspondence between tableaux and module decomposition.

## Abstract

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.