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The geometry of polynomial representations

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arxiv 2105.12621 v2 pith:KYZKH6O3 submitted 2021-05-26 math.AG math.RT

Arthur Bik , Jan Draisma , Rob H. Eggermont , Andrew Snowden This is my paper · ORCID
classification math.AGmath.RT
keywords infinitepolynomialactionalgebraicasymptoticbeenchevalleyconjecture
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We define a GL-variety to be a (typically infinite dimensional) algebraic variety equipped with an action of the infinite general linear group under which the coordinate ring forms a polynomial representation. Such varieties have been used to study asymptotic properties of invariants like strength and tensor rank, and played a key role in two recent proofs of Stillman's conjecture. We initiate a systematic study of GL-varieties, and establish a number of foundational results about them. For example, we prove a version of Chevalley's theorem on constructible sets in this setting.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. GL-algebras in positive characteristic III: the divided power algebra

    math.AC 2026-08 conditional novelty 7.0 of 10

    The divided power algebra Div(k^∞) over a field of characteristic p is GL-coherent, and its bounded derived category of finitely presented modules has a semi-orthogonal decomposition into pieces generated by D^(r) ⊗ L_λ.

  2. A remark on automorphisms of tensor spaces

    math.RT 2025-07 accept novelty 7.0 of 10

    Any graph automorphism group is realized as the automorphism group of a symmetric (3,2,2)-tensor space, and the paper constructs spaces where V is irreducible but V⊗2 has infinite length.

  3. The fundamental theorems of invariant theory for linearly oligomorphic groups

    math.RT 2026-07 accept novelty 6.0 of 10

    The G-invariant symmetric forms on V^m for the automorphism group G of a universal homogeneous λ-space are freely generated by the contractions of the defining forms with Schur functors on k^m.

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