The nineteen-vertex model and alternating sign matrices

TL;DR

This paper demonstrates a connection between the nineteen-vertex model's transfer matrix eigenvalues and eigenvectors with alternating sign matrices, providing explicit solutions and linking to combinatorial enumeration.

Contribution

It introduces an explicit solution to the Bethe equations for the nineteen-vertex model and relates the eigenvector properties to alternating sign matrices.

Findings

Eigenvalues have a simple form under certain boundary conditions.

Eigenvectors are explicitly computed using algebraic Bethe ansatz.

Norms and components relate to generating functions for alternating sign matrices.

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.