Tensor spaces and the geometry of polynomial representations

Nate Harman; Andrew Snowden

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arXiv · 2407.19132

Tensor spaces and the geometry of polynomial representations

Nate Harman·Andrew Snowden

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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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Nate Harman, Andrew Snowden. 2024-07-27. Tensor spaces and the geometry of polynomial representations. https://arxiv.org/abs/2407.19132

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