# 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 ≤ in, j ∈ ℕ] be the ring of polynomials in x ij (1 ≤ in, 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 (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) = pn. We also survey some results on Sym(ℕ)-invariant ideals in polynomial algebras and some related topics.

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Correspondence to E. A. da Costa.