Fusion hierarchies, $T$-systems and $Y$-systems for the $A_2^{(1)}$ models

TL;DR

This paper explores the fusion hierarchies and functional relations of the $A_2^{(1)}$ models, including loop, vertex, and RSOS models, deriving explicit $T$- and $Y$-systems and their closure properties at roots of unity.

Contribution

It establishes $s ext{l}(3)$-type fusion hierarchies and explicit $T$- and $Y$-systems for $A_2^{(1)}$ models, including closure identities at roots of unity.

Findings

Fusion hierarchies satisfy $s ext{l}(3)$-type relations.

Explicit $T$- and $Y$-systems derived for models.

Closure identities show finite $Y$-system at roots of unity.

Abstract

The family of $A^{(1)}_2$ models on the square lattice includes a dilute loop model, a $15$-vertex model and, at roots of unity, a family of RSOS models. The fused transfer matrices of the general loop and vertex models are shown to satisfy $s\ell(3)$-type fusion hierarchies. We use these to derive explicit $T$- and $Y$-systems of functional equations. At roots of unity, we further derive closure identities for the functional relations and show that the universal $Y$-system closes finitely. The $A^{(1)}_2$ RSOS models are shown to satisfy the same functional and closure identities but with finite truncation.