Linear preservers of secant varieties and other varieties of tensors

TL;DR

This paper characterizes linear transformations that preserve specific algebraic varieties of tensors, including secant and other tensor varieties, using geometric methods with applications in quantum information.

Contribution

It provides a comprehensive geometric characterization of linear preservers for secant varieties and various tensor varieties, extending previous results and introducing new techniques.

Findings

Characterization of linear preservers of secant varieties of Segre varieties.

Identification of linear preservers for subspace and slice rank one tensor varieties.

Applications discussed in quantum information theory.

Abstract

We study the problem of characterizing linear preserver subgroups of algebraic varieties, with a particular emphasis on secant varieties and other varieties of tensors. We introduce a number of techniques built on different geometric properties of the varieties of interest. Our main result is a simple characterization of the linear preservers of secant varieties of Segre varieties in many cases, including $\sigma_r((\mathbb{P}^{n-1})^{\times k})$ for all $r \leq n^{\lfloor k/2 \rfloor}$. We also characterize the linear preservers of several other sets of tensors, including subspace varieties, the variety of slice rank one tensors, symmetric tensors of bounded Waring rank, the variety of biseparable tensors, and hyperdeterminantal surfaces. Computational techniques and applications in quantum information theory are discussed. We provide geometric proofs for several previously known results on linear preservers.