Remarks on $A^{(1)}_n$ face weights

TL;DR

This paper provides elementary proofs for the factorization of elliptic Boltzmann weights and the sum-to-1 property in the trigonometric limit for the $A^{(1)}_n$ face model, extending recent vertex model results.

Contribution

It offers new elementary proofs for key properties of $A^{(1)}_n$ face weights, generalizing previous vertex model findings.

Findings

Factorization of elliptic Boltzmann weights proved

Sum-to-1 property established in the trigonometric limit

Results extend to face models from vertex models

Abstract

Elementary proofs are presented for the factorization of the elliptic Boltzmann weights of the $A^{(1)}_n$ face model, and for the sum-to-1 property in the trigonometric limit, at a special point of the spectral parameter. They generalize recent results obtained in the context of the corresponding trigonometric vertex model.