Symmetry classes of alternating sign matrices in the nineteen-vertex model

TL;DR

This paper explores the symmetry classes of alternating sign matrices through the nineteen-vertex model with anti-diagonal twists, revealing eigenstates with combinatorial features and deriving new determinant formulas for ASM enumeration.

Contribution

It explicitly constructs eigenstates for the nineteen-vertex model with anti-diagonal twists and links them to symmetry classes of ASMs, providing new determinant and pfaffian formulas.

Findings

Eigenstates exhibit combinatorial features related to ASM symmetry classes

Sum rules are expressed via Kuperberg's determinants for the six-vertex model

New formulas for weighted enumeration of ASM symmetry classes are derived

Abstract

The nineteen-vertex model on a periodic lattice with an anti-diagonal twist is investigated. Its inhomogeneous transfer matrix is shown to have a simple eigenvalue, with the corresponding eigenstate displaying intriguing combinatorial features. Similar results were previously found for the same model with a diagonal twist. The eigenstate for the anti-diagonal twist is explicitly constructed using the quantum separation of variables technique. A number of sum rules and special components are computed and expressed in terms of Kuperberg's determinants for partition functions of the inhomogeneous six-vertex model. The computations of some components of the special eigenstate for the diagonal twist are also presented. In the homogeneous limit, the special eigenstates become eigenvectors of the Hamiltonians of the integrable spin-one XXZ chain with twisted boundary conditions. Their sum rules and special components for both twists are expressed in terms of generating functions arising in the weighted enumeration of various symmetry classes of alternating sign matrices (ASMs). These include half-turn symmetric ASMs, quarter-turn symmetric ASMs, vertically symmetric ASMs, vertically and horizontally perverse ASMs and double U-turn ASMs. As side results, new determinant and pfaffian formulas for the weighted enumeration of various symmetry classes of alternating sign matrices are obtained.