(FUNDAMENTAL AND APPLIED MATHEMATICS)

2013, VOLUME 18, NUMBER 3, PAGES 69-76

## Symmetric polynomials and nonfinitely generated $Sym\left(N\right)$-invariant ideals

E. A. da Costa
A. N. Krasilnikov

Abstract

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Let $K$ be a field and let $N = \left\{1,2,$¼} be the set of all positive integers. Let $R$n = K[xij | 1 £ i £ n, j Î N] be the ring of polynomials in $x$ij ($1$£ i £ n, $j$Î N) over $K$. Let $S$n = Sym({1,2,¼,n}) and $Sym\left(N\right)$ be the groups of permutations of the sets $\left\{1,2,$¼,n} and $N$, respectively. Then $S$n and $Sym\left(N\right)$ act on $R$n in a natural way: t(xij) = xt(i)j and s(xij) = xis(j), for all $i$Î {1,2,¼,n} and $j$Î N, t Î Sn and s Î Sym(N). Let $\bar R_n$ be the subalgebra of ($S$n-)symmetric polynomials in $R$n, i.e.,

$R$n = {f Î Rn | t(f) = f for each t Î Sn}.

An ideal $I$ in $\bar R_n$ is called $Sym\left(N\right)$-invariant if s(I) = I for each s Î Sym(N). In 1992, the second author proved that if $char\left(K\right) = 0$ or $char\left(K\right) = p > n$, then every $Sym\left(N\right)$-invariant ideal in $\bar \left\{R\right\}_n$ is finitely generated (as such). In this note, we prove that this is not the case if $char\left(K\right) = p$£ n. We also survey some results about $Sym\left(N\right)$-invariant ideals in polynomial algebras and some related topics.

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