FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

Symmetric polynomials and nonfinitely generated $Sym(N)$-invariant ideals

E. A. da Costa
A. N. Krasilnikov

Abstract

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Let $K$ be a field and let $N\; =\; \{1,2,$¼} be the set of all positive integers. Let $R$_n = K[x_ij | 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 $Sym(N)$ be the groups of permutations of the sets $\{1,2,$¼,n} and $N$, respectively. Then $S$_n and $Sym(N)$ act on $R$_n in a natural way: $$t(x_ij) = x_t(i)j and $$s(x_ij) = x_is(j), for all $i$Î {1,2,¼,n} and $j$Î N, $$t Î S_n and $$s Î Sym(N). Let $\$\backslash bar\; R\_n\$$ be the subalgebra of ($S$_n-)symmetric polynomials in $R$_n, i.e.,

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

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Last modified: March 4, 2014