Tensor spaces and the geometry of polynomial representations

TL;DR

This paper generalizes the concept of universal tensor spaces by showing that each Zariski class contains a unique (up to isomorphism) weakly homogeneous tensor space, expanding the understanding of tensor space classifications.

Contribution

It extends previous results by demonstrating the existence and uniqueness of weakly homogeneous tensor spaces within each Zariski class, using the theory of GL-varieties.

Findings

Existence of weakly homogeneous tensor spaces in each Zariski class.

Uniqueness of these spaces up to isomorphism.

Application of GL-varieties theory to tensor space classification.

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.