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

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

E. A. da Costa
A. N. Krasilnikov


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Let K be a field and let N = {1,2,¼} be the set of all positive integers. Let Rn = K[xij | 1 £ i £ n, j Î N] be the ring of polynomials in xij (1 £ i £ n, j Î N) over K. Let Sn = Sym({1,2,¼,n}) and Sym(N) be the groups of permutations of the sets {1,2,¼,n} and N, respectively. Then Sn and Sym(N) act on Rn 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 (Sn-)symmetric polynomials in Rn, i.e.,

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

An ideal I in $\bar R_n$ 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 $\bar {R}_n$ 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