On SO$(N)$ spin vertex models

TL;DR

This paper characterizes the Boltzmann weights of $SO(N)$ spin vertex models, extending previous work on $B_k$ models, and explores their algebraic structures, including quantum algebras and new relations for specific cases.

Contribution

It provides a comprehensive description of $SO(N)$ spin vertex models for all $N$, completing the $B_k$ case and identifying their algebraic properties and new relations.

Findings

$SO(N)$ spin vertex models are described for all $N$

Models obey quantum algebras including BMW algebra and its generalizations

New relations are found for specific cases like $B_4$ and $D_6$

Abstract

We describe the Boltzmann weights of the $D_k$ algebra spin vertex models. Thus, we find the $SO(N)$ spin vertex models, for any $N$, completing the $B_k$ case found earlier. We further check that the real (self-dual) SO$(N)$ models obey quantum algebras, which are the Birman-Murakami-Wenzl (BMW) algebra for three blocks, and certain generalizations, which include the BMW algebra as a sub-algebra, for four and five blocks. In the case of five blocks, the $B_4$ model is shown to satisfy additional twenty new relations, which are given. The $D_6$ model is shown to obey two additional relations.