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Tensor spaces and the geometry of polynomial representations

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arxiv 2407.19132 v1 pith:MEZGEWKE submitted 2024-07-27 math.RT math.AGmath.LO

Nate Harman , Andrew Snowden This is my paper · ORCID
classification math.RTmath.AGmath.LO
keywords tensorspacespacesherehomogeneousisomorphismpolynomialunique
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abstract

A "tensor space" is a vector space equipped with a finite collection of multi-linear forms. In previous work, we showed that (for each signature) there exists a universal homogeneous tensor space, which is unique up to isomorphism. Here we generalize that result: we show that each Zariski class of tensor spaces contains a weakly homogeneous space, which is unique up to isomorphism; here, we say that two tensor spaces are "Zariski equivalent" if they satisfy the same polynomial identities. Our work relies on the theory of $\mathbf{GL}$-varieties developed by Bik, Draisma, Eggermont, and Snowden.

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Cited by 2 Pith papers

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

  1. 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.

  2. 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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