Ideal-theoretic non-noetherianity of polynomial functors in positive characteristic

Abstract

A long-standing open problem in representation stability is whether every finitely generated commutative algebra in the category of strict polynomial functors satisfies the noetherian property. In this paper, we resolve this problem negatively over fields of positive characteristic using ideas from invariant theory. Specifically, we consider the algebra $P$ of polarizations of elementary symmetric polynomials inside the ring of all multisymmetric polynomials in $p \times \infty$ variables. We show $P$ is not noetherian based on two key facts: (1) the $p$-th power of every multisymmetric polynomial is in $P$ (our main technical result) and (2) the ring of multisymmetric polynomials is Frobenius split.