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

Abstract

The family of $A_2^{(1)}$ 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 $A_2^{(1)}$ RSOS models are shown to satisfy the same functional and closure identities but with finite truncation.

36 pages