Generalized Polarization Modules

TL;DR

This paper generalizes the Operator Theorem to polynomials in multiple variable sets, defining polarization modules and computing their representation decompositions to understand their structure.

Contribution

It introduces polarization modules for polynomials in multiple variable sets and provides formulas for their graded Frobenius characteristics, extending previous work on the Operator Theorem.

Findings

Derived general formulas for the graded Frobenius characteristic of polarization modules

Extended the Operator Theorem to polynomials in multiple variable sets

Analyzed the irreducible decomposition of these modules

Abstract

This work enrols the research line of M. Haiman on the Operator Theorem (the old operator conjecture). This theorem states that the smallest $\mathfrak{S}_n$-module closed under taking partial derivatives and closed under the action of polarization operators that contains the Vandermonde determinant is the space of diagonal harmonics polynomials. We start generalizing the context of this theorem to the context of polynomials in $\ell$ sets of $n$ variables $x_{ij}$ with $1\leq i\leq \ell$ et $1\leq j\leq n$. Given a $\mathfrak{S}_n$-stable family of homogeneous polynomials in the variables $x_{ij}$ the smallest vector space closed under taking partial derivatives and closed under the action of polarization operators that contains $F$ is the polarization module generated by the family $F$. These polarization modules are all representation of the direct product $\mathfrak{S}_n\times{GL}_{\ell}(\mathbb{C})$. In order to study the decomposition into irreducible submodules, we compute the graded Frobenius characteristic of these modules. For several cases of $\mathfrak{S}_n$-stable families of homogeneous polynomials in $n$ variables, for every $n\geq 1$, we show general formulas for this graded characteristic in a global manner, independent of the value of $\ell$.