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.

arXiv comment36 pages

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Cited by in corpus (3)

• Extended T-systems, Q matrices and T-Q relations for $s\ell (2)$ models at roots of unity
• Fusion hierarchies, $T$-systems and $Y$-systems for the dilute $A_2^{(2)}$ loop models on a strip
• Integrability of planar-algebraic models