Triangular tensor networks, pencils of matrices and beyond

TL;DR

This paper characterizes tensor network varieties related to triangular graphs, especially for tensors with a physical dimension of 2, by linking them to pencils of matrices and their Kronecker invariants, and explores their geometric properties.

Contribution

It provides a complete characterization of these tensor network varieties in terms of Kronecker invariants and explores their geometric and dimensional properties.

Findings

Complete classification of tensor network varieties for triangular graphs.

Identification of cases with smaller-than-expected dimension.

Necessary conditions for membership based on classical algebraic geometry.

Abstract

We study tensor network varieties associated with the triangular graph, with a focus on the case where one of the physical dimensions is 2. This allows us to interpret the tensors as pencils of matrices. We provide a complete characterization of these varieties in terms of the Kronecker invariants of pencils. We determine their dimension, identifying the cases for which the dimension is smaller than the expected parameter count. We provide necessary conditions for membership in these varieties, in terms of the geometry of classical determinantal varieties, coincident root loci and plane cubic curves. We address some extensions to arbitrary graphs.