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

E. A. da Costa ; A. N. Krasilnikov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 3, pp. 69-76.

Résumé

Let $K$ be a field and let $\mathbb N=\{1,2,\dots\}$ be the set of all positive integers. Let $R_n=K[x_{ij}\mid1\le i\le n,\ j\in\mathbb N]$ be the ring of polynomials in $x_{ij}$ ($1\le i\le n$, $j\in\mathbb N$) over $K$. Let $\mathrm S_n=\mathrm{Sym}(\{1,2,\dots,n\})$ and $\mathrm{Sym}(\mathbb N)$ be the groups of permutations of the sets $\{1,2,\dots,n\}$ and $\mathbb N$, respectively. Then $\mathrm S_n$ and $\mathrm{Sym}(\mathbb N)$ 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 $i\in\{1,2,\dots,n\}$ and $j\in\mathbb N$, $\tau\in\mathrm S_n$ and $\sigma\in\mathrm{Sym}(\mathbb N)$. Let $\bar R_n$ be the subalgebra of ($\mathrm S_n$-)symmetric polynomials in $R_n$, i.e., $$ \bar R_n=\{f\in R_n\mid\tau(f)=f\ \text{for each}\ \tau\in\mathrm S_n\}. $$ An ideal $I$ in $\bar R_n$ is called $\mathrm{Sym}(\mathbb N)$-invariant if $\sigma(I)=I$ for each $\sigma\in\mathrm{Sym}(\mathbb N)$. In 1992, the second author proved that if $\mathrm{char}(K)=0$ or $\mathrm{char}(K)=p>n$, then every $\mathrm{Sym}(\mathbb N)$-invariant ideal in $\bar R_n$ is finitely generated (as such). In this note, we prove that this is not the case if $\mathrm{char}(K)=p\le n$. We also survey some results about $\mathrm{Sym}(\mathbb N)$-invariant ideals in polynomial algebras and some related topics.

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