Let $W=S_n$ (the symmetric group) acting on $V=k^n$ via permutation of the indices, where $k$ is an algebraically closed field. Closely related to this are the reflection (sub-)representation $V_0=\{ (v_1,\ldots ,v_n)\in k^n : \sum_{i=1}^n v_i=0\}$ and its dual $V/{\mathfrak z}$, where ${\mathfrak z}$ is the subspace spanned by $(1,1,\ldots ,1)$. Of course, if ${\rm char}\, k=0$ or is a prime not dividing $n$, then these representations are isomorphic. But this is no longer true if ${\rm char}\, k$ divides $p$.

It's well known that the $W$-invariant polynomials in $k[V]=S(V^*)$ are freely generated by the elementary symmetric polynomials in the coefficients. Since $V_0$ is the zero set of the first elementary symmetric polynomial $(v_1,\ldots ,v_n)\mapsto v_1+\ldots +v_n$, it follows that $k[V_0]^W$ is freely generated by homogeneous polynomials of degree $2,3,\ldots ,n$.

What (if anything) is known about the invariants $k[V/{\mathfrak z}]^W\cong S(V_0)^W$ when $p|n$?

The motivation for the question is: by the Chevalley restriction theorem (proved in maximal generality by Chaput and Romagny https://arxiv.org/abs/0805.2140), $k[V/{\mathfrak z}]^W$ is isomorphic to $k[\mathfrak{pgl}_n]^{{\rm GL}_n}$, and the generators of the ideal $\{ f\in k[V/{\mathfrak z}]^W: f(0)=0\}$ are mapped to generators of the ideal determining the nilpotent cone ${\mathcal N}(\mathfrak{pgl}_n)$. The geometry here may be somewhat different to the geometry of ${\mathcal N}(\mathfrak{sl}_n)$ (e.g. ${\mathcal N}(\mathfrak{pgl}_2)$ is smooth in characteristic 2).

There is always a linear generator, namely $(1,1,\ldots ,1)$. To see how different this can be from $k[V_0]$: for $n=p=3$ I think $S(V_0)^W$ is freely generated by elements of degree $1$ and $6$. For $n=p=5$ there are generators of the invariant ring of degree $1, 4, 6, 8$, but I don't know whether they generate all of $S(V_0)^W$. (I suspect not.)