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

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.

Set partitions, fermions, and skein relations

“…Returning to the symmetric group, various authors [2,4,10,12] have considered a 'multidiagonal' version of the fermionic coinvariants defined as follows. Consider an k × n matrix Θ of fermionic variables θ i,j where 1 ≤ i ≤ k and 1 ≤ j ≤ n. Let ∧{Θ} be the exterior algebra over these variables, a C-vector space of dimension 2 nk .…”

“…, y n ) of commuting variables. Various authors [2,3,4,8,10,12,17,18,21,22,24,25] have considered versions of DR n involving mixtures of commuting and anticommuting variables.…”

“…In general F DR(n; k) is a k-fold graded S n -module. Orellana and Zabrocki[10] found generators for the ideal I, as well as a combinatorial formula for the multigraded Frobenius image of the numerator ∧{Θ}. Give a combinatorial interpretation for the multigraded pieces of F DR(n; k).…”

Kronecker powers of harmonics, polynomial rings, and generalized principal evaluations

“…Orellana and Zabrocki [14] have recently done this (with the inclusion of anti-commuting variables) by viewing…”

“…This theorem gives a new way of writing the multigraded multiplicity of the irreducible representation corresponding to λ in the polynomial ring with k sets of variables. Orellana and Zabrocki [14] have a combinatorial interpretation for this expression in terms of monomial symmetric functions in the q i . It can also be viewed as a character on GL k resulting from the duality.…”