Tensor invariants for certain subgroups of the orthogonal group

TL;DR

This paper characterizes tensors invariant under specific subgroups of the orthogonal group, providing insights into the structure of vertex models and addressing a question about the rank of edge connection matrices.

Contribution

It offers a combinatorial parameterization of invariant tensors under certain orthogonal subgroups, advancing understanding of vertex models and their associated matrices.

Findings

Provides a combinatorial description of invariant tensors.

Answers a question about the rank of edge connection matrices.

Applicable to vertex models over algebraically closed fields of characteristic zero.

Abstract

Let V be an n-dimensional vector space and let On be the orthogonal group. Motivated by a question of B. Szegedy (B. Szegedy, Edge coloring models and reflection positivity, Journal of the American Mathematical Society Volume 20, Number 4, 2007), about the rank of edge connection matrices of partition functions of vertex models, we give a combinatorial parameterization of tensors in V \otimes k invariant under certain subgroups of the orthogonal group. This allows us to give an answer to this question for vertex models with values in an algebraically closed field of characteristic zero.