Generalized Polarization Modules

This work enrols the research line of M. Haiman on the Operator Theorem (the former Operator Conjecture). Given a Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{S}_n}$$\end{document}-stable family F of homogeneous polynomials in the variables xij\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x_i j}$$\end{document} with 1≤i≤ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${1 \leq i \leq \ell}$$\end{document} and 1≤j≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${1 \leq j \leq n}$$\end{document}. We define the polarization module generated by the family F, as the smallest vector space closed under taking partial derivatives and closed under the action of polarization operators that contains F. These spaces are representations of the direct product Sn×GLℓ(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{S}_n \times GL_\ell(\mathbb{C})}$$\end{document}. We compute the graded Frobenius characteristic of these modules. We use some basic tools to study these spaces and give some in-depth calculations of low degree examples of a family or a single symmetric polynomial.