  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 $\frak{S}_n$-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 $\ell$.
