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

2025/06/18 by Karthik Ganapathy, Ganapathy, Karthik
Mathematics · #05E05 #13A50 (Primary) 13A35 #18A25 (Secondary) #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2506.15134

openalex publication_date 2025/06/18 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

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 × ∞ 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.

Citations

• Non-noetherian GL-algebras in characteristic two
• Tensor spaces and the geometry of polynomial representations
• The geometry of polynomial representations in positive characteristic
• Uniformity for limits of tensors
• Vector invariants in arbitrary characteristic

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