Symmetric Polynomials and Nonfinitely Generated Sym(ℕ)-Invariant Ideals

Abstract

Abstract. Let K be a field and let N = {1, 2, . . . }. Let Rn = K[xij | 1 ≤ i ≤ n, j ∈ N] be the ring of polynomials in xij (1 ≤ i ≤ n, j ∈ N) over K. Let Sn = Sym({1, 2, . . . , n}) and Sym(N) be the groups of the permutations of the sets {1, 2, . . . , n} and N, respectively. Then Sn and Sym(N) act on Rn in a natural way: τ (xij) = x τ (i)j and σ(xij) = x iσ(j) for all τ ∈ Sn and σ ∈ Sym(N). Let Rn be the subalgebra of the symmetric polynomials in Rn,In 1992 the second author proved that if char(K) = 0 or char(K) = p > n then every Sym(N)-invariant ideal in Rn 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 results.