The nineteen-vertex model and 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.

arXiv comment28 pages, 6 figures; v2: corrected misprint in arXiv metadata (title); v3: revised version

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Cited by in corpus (8)

• Quasi-local conserved charges and spin transport in spin-$1$ integrable chains
• Lattice Supersymmetry in the Open XXZ Model: An Algebraic Bethe Ansatz Analysis
• Spectral flow for an integrable staggered superspin chain
• Coupled Fredkin and Motzkin chains from quantum six- and nineteen-vertex models
• The nineteen-vertex model and alternating sign matrices
• Symmetry classes of alternating sign matrices in the nineteen-vertex model
• The bounded 15-vertex model
• The bounded 19-vertex model