Given a $S_n$-stable family $F$ of homogeneous polynomials in the variables $x_{ij}$ with $1≤i≤\ell$ and $1≤j≤n$, the polarization module generated by the family $F$ is the smallest vector space closed under taking partial derivatives and closed under the action of polarization operators that contains $F$.
Given a $S_n$-stable family $F$ of homogeneous polynomials in the variables $x_{ij}$ with $1≤i≤\ell$ and $1≤j≤n$, the polarization module generated by the family $F$ is 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 $S_n×GL_\ell(\mathbb C)$.