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

Publication date: 20 September 2018

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 e l l (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.

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