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

Abstract

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 $S_n=Sym(\{1,2,\dots ,n\})$ 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: $\tau (x_{ij})=x_{\tau (i)j}$ and $\sigma (x_{ij})=x_{i\sigma (j)}$ for all $\tau \in S_n$ and $\sigma \in Sym(\mathbb{N})$. 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.

8 pages

References in corpus (3)

• Bounded-rank tensors are defined in bounded degree
• Noetherianity up to symmetry
• Noetherianity for infinite-dimensional toric varieties