The nineteen-vertex model and alternating sign matrices
(2015) Journal of Statistical Mechanics: Theory and Experiment — p. P01017 (2015)
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- Abstract
- It is shown that the transfer matrix of the inhomogeneous nineteen-vertex model with certain diagonal twisted boundary conditions possesses a simple eigenvalue. This is achieved through the identification of a simple and completely explicit solution of its Bethe equations. The corresponding eigenvector is computed by means of the algebraic Bethe ansatz, and both a simple component and its square norm are expressed in terms of the Izergin-Korepin determinant. In the homogeneous limit, the vector coincides with a supersymmetry singlet of the twisted spin-one XXZ chain. It is shown that in a natural polynomial normalisation scheme its square norm and the simple component coincide with generating functions for weighted enumeration of alternating sign matrices.
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Walmsley Hagendorf, C. (2015). The nineteen-vertex model and alternating sign matrices. Journal of Statistical Mechanics: Theory and Experiment, P01017. https://doi.org/10.1088/1742-5468/2015/01/P01017 (Original work published 2015)