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 $1\le i \le \ell$ and $1 \le j \le n$. 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 $n \ge 1$, we show general formulas for this graded characteristic in a global manner, independent of the value of $\ell$. Ce travail s'inscrit dans la lignée de recherche des travaux de M. Haiman sur le théorème de l'opérateur (ex-conjecture de l'opérateur). Ce théorème affirme que le plus petit $\mathfrak{S}_n$-module clos par dérivation partielle et clos par l'action des opérateurs de polarisation qui contient le déterminant de Vandermonde est l'espace des polynômes harmoniques diagonaux. On commence par généraliser le contexte du théorème de l'opérateur au contexte de polynômes à ensembles de $n$ variables $x_{ij}$ avec $1\le i \le \ell$ et $1 \le j \le n$. Étant donnée une famille $\mathfrak{S}_n$-stable $F$ des polynômes homogènes en les variables $x_{ij}$, le plus petit espace vectoriel $\mathcal{M}_F$ clos par dérivation partielle et clos par léaction des opérateurs de polarisation contenant $F$ est le module de polarisation engendré par la famille $F$. Les modules $\mathcal{M}_F$ sont tous des représentations du produit direct $\mathfrak{S}_n \times GL_\ell(\mathbb{C})$. Dans le but d'étudier la décomposition en sous-modules irréductibles on calcule la caractéristique de Frobenius graduée de ces modules. Pour plusieurs cas de familles homogènes $\mathfrak{S}_n$-stables constituées des polynômes homogènes à $n$ variables, pour tout $n \ge 1$, on démontre des formules générales pour cette caractéristique graduée de façon globale, indépendante de la valeur de $\ell$.

Highlights

  • This work is inspired by the Operator Theorem of M

  • The goal of this paper is to study the decomposition into irreducible submodules under the action of Sn × GL (C) of polarization modules MF

  • In this paper we completely describe the decomposition into irreducible submodules of the polarization modules generated by each of the polynomials pd[1], pd, and ed for any d ≥ 1

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Summary

Introduction

This work is inspired by the Operator Theorem (ex-operator conjecture) of M. Haiman conjectured that this space coincides with the polarization module generated by the Vandermonde determinant when = 3 (see (7)) Bergeron extend Haiman’s conjecture concerning to the identification of the space Dn( ) of multivariate diagonal harmonics polynomials with polarization module generated by the Vandermonde determinant for > 3. We completely determine the classification of polarization modules generated by a single homogeneous symmetric polynomial when the degree is 2 or 3. We have open problems about the constraints for the multiplicities of irreducible submodules of polarization modules generated by any Sn-stable family consisting of homogeneous polynomials of degree at most 2. We compute the graded Frobenius characteristic of polarization modules generated by the family Td of all monomials of degree d, when d = 2, 3 in any number of variables n. This implies that the multiplicities of irreducible Sn × GL (C)-modules of MF are least or equal to the corresponding multiplicities in MTd

Preliminaries
Definitions and discussions
Generalized Polarization Modules
Properties of Polarization Modules
Frobenius characteristics of some polarization modules
Exceptions
Open problems
Tables for the graded Frobenius characteristic
Full Text
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