Symmetric polynomials and non-finitely generated $Sym (\mathbb N)$-invariant ideals

Let $K$ be a field and let $\mathbb N = \{1,2, \dots \}$. Let $R_n=K[x_{ij} \mid 1\le i\le n, j\in \mathbb N]$ be the ring of polynomials in $x_{ij}$ $(1 \le i \le n, j \in \mathbb N)$ over $K$. Let $S_n = Sym (\{1,2, \ldots, n \})$ and $Sym (\mathbb N)$ be the groups of the permutations of the sets $\{1,2,\dots, n \}$ and $\mathbb N$, respectively. Then $S_n$ and $Sym (\mathbb N)$ act on $R_n$ in a natural way: $τ(x_{ij})=x_{τ(i)j}$ and $σ(x_{ij})=x_{iσ(j)}$ for all $τ\in S_n$ and $σ\in Sym(\mathbb N)$. Let $\overline{R}_n$ be the subalgebra of the symmetric polynomials in $R_n$, \[ \overline{R}_n = \{f \in R_n \mid τ(f) = f \mbox{for each} τ\in S_n \} . \] In 1992 the second author proved that if $char (K)= 0$ or $char(K)=p > n$ then every $Sym (\mathbb N)$-invariant ideal in $\overline{R}_n$ is finitely generated (as such). In this note we prove that this is not the case if $char (K)=p\le n$. We also survey some results about $Sym (\mathbb N)$-invariant ideals in polynomial algebras and some related results.

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