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

TL;DR

This paper demonstrates that certain polynomial functor algebras in positive characteristic are not noetherian, resolving a long-standing open problem in representation stability using invariant theory techniques.

Contribution

It proves that the algebra of polarizations of elementary symmetric polynomials is not noetherian in positive characteristic, providing a negative answer to a key open question.

Findings

The algebra $P$ is not noetherian in positive characteristic.

The $p$-th power of multisymmetric polynomials lies in $P$.

The ring of multisymmetric polynomials is Frobenius split.

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.