The rank of edge connection matrices and the dimension of algebras of invariant tensors

TL;DR

This paper links the rank of edge connection matrices in vertex models to the dimension of certain invariant tensor algebras, providing a mathematical characterization that resolves a question posed in 2007.

Contribution

It establishes a precise relationship between the rank of edge connection matrices and the dimension of G-invariant tensor algebras, advancing understanding in algebraic graph theory.

Findings

Characterizes the rank of edge connection matrices as the dimension of invariant tensor components.

Provides a mathematical answer to Szegedy's 2007 question.

Connects graph invariants with algebraic structures in tensor spaces.

Abstract

We characterize the rank of edge connection matrices of partition functions of real vertex models, as the dimension of the homogeneous components of the algebra of $G$-invariant tensors. Here $G$ is the sub- group of the real orthogonal group that stabilizes the vertex model. This answers a question of Bal\'azs Szegedy from 2007.