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.

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

References in corpus (6)

• The eight-vertex model and Painleve VI
• Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain
• Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Delta = -1/2
• Spin chains with dynamical lattice supersymmetry
• Bethe roots and refined enumeration of alternating-sign matrices
• The nineteen-vertex model and alternating sign matrices

Cited by in corpus (3)

• Lattice Supersymmetry in the Open XXZ Model: An Algebraic Bethe Ansatz Analysis
• The nineteen-vertex model and alternating sign matrices
• The bounded 19-vertex model