A combinatorial interpretation of the free-fermion condition of the six-vertex model

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Abstract
The free-fermion condition of the six-vertex model provides a five-parameter sub-manifold on which the Bethe ansatz equations for the wavenumbers that enter into the eigenfunctions of the transfer matrices of the model decouple, hence allowing explicit solutions. Such conditions arose originally in early field-theoretic S-matrix approaches. Here we provide a combinatorial explanation for the condition in terms of a generalized Gessel-Viennot involution. By doing so we extend the use of the Gessel-Viennot theorem, originally devised for non-intersecting walks only, to a special weighted type of intersecting walk, and hence express the partition function of N such walks starting and finishing at fixed endpoints in terms of the single-walk partition functions.
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