Let $\Sigma_m$ be the permutation group on $m$ and $R=\mathbb{C}[x_1,x_2,\dots,x_m]$. Let $\Sigma_m$ act on $R$ in the usual way. Let $R^{\Sigma_m}$ denote the ring of $\Sigma_m$ invariant polynomials and $R^{\Sigma_r \times \Sigma_{m-r}}$ denote the ring of polynomials invariant under permutation of the first $r$ and the last $m-r$ variables. I found a claim in a reference that says $R^{\Sigma_r \times \Sigma_{m-r}}$ is a free $R^{\Sigma_m}$-module of free rank $\binom{m}{r}$. The reference claims that this is "well-known" but doesn't give a specific reference or cite a theorem.

1. Can anyone explain why this is true?

2. Is there any knowledge of the specific form of the generators?

3. Can we say anything similar over more general fields?

Let $\Sigma_m$ be the permutation group on $m$ letters and $R=\mathbb{C}[x_1,x_2,\dots,x_m]$. Let $\Sigma_m$ act on $R$ in the usual way. Let $R^{\Sigma_m}$ denote the ring of $\Sigma_m$ invariant polynomials and $R^{\Sigma_r \times \Sigma_{m-r}}$ denote the ring of polynomials invariant under permutation of the first $r$ and the last $m-r$ variables. I found a claim in a reference that says $R^{\Sigma_r \times \Sigma_{m-r}}$ is a free $R^{\Sigma_m}$-module of free rank $\binom{m}{r}$. The reference claims that this is "well-known" but doesn't give a specific reference or cite a theorem.

1. Can anyone explain why this is true?

2. Is there any knowledge of the specific form of the generators?

3. Can we say anything similar over more general fields?