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

Abstract

Let K be a field and let ℕ = {1, 2, . . .} be the set of all positive integers. Let R _n = K[x _ij | 1 ≤ i ≤ n, j ∈ ℕ] be the ring of polynomials in x _ij (1 ≤ i ≤ n, j ∈ ℕ) over K. Let S _n = Sym({1, 2, . . . , n}) and Sym(ℕ) be the groups of permutations of the sets {1, 2, . . . , n} and ℕ, respectively. Then S _n and Sym(ℕ) act on R _n in a natural way: τ (x _ij ) = x _τ(i)j and σ(x _ij ) = x _iσ(j), for all i ∈ {1, 2, . . . , n} and j ∈ ℕ, τ ∈ S _n and σ ∈ Sym(ℕ). Let $\overline{R}$ _n be the subalgebra of (S _n -)symmetric polynomials in R _n , i.e.,

$$ {\overline{R}}_n=\left\{f\in {R}_n\left|\tau (f)=f\;\mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{each}\ \tau \right.\in {\mathrm{S}}_n\right\}. $$

An ideal I in $\overline{R}$ _n is called Sym(ℕ)-invariant if σ(I) = I for each σ ∈ 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 $\overline{R}$ _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 on Sym(ℕ)-invariant ideals in polynomial algebras and some related topics.