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

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 antidiagonal 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.

Diagonally and antidiagonally symmetric alternating sign matrices of odd order

“…. = u n = 1 in (33), and using ( 21) and (28). The second equality (which was previously obtained by Okada [37, Conj.…”

“…For the case of unrestricted ASMs see, for example, Ayyer and Romik [5], Behrend [7], and references therein. For cases involving certain other classes of ASMs see, for example, Cantini [14], de Gier, Pyatov and Zinn-Justin [22], Fischer and Riegler [26], Hagendorf and Morin-Duchesne [28], Kuperberg [32], Okada [37], Razumov and Stroganov [38,40], Robbins [43], and Stroganov [48].…”

“…• In Section 3, the main notation and conventions of the paper are outlined, the three-statistic DSASM generating function which will be studied in much of the paper is introduced, and some enumerative results for DSASMs are obtained using simple combinatorial methods in Propositions 1 and 2. • In Section 4, the main results for the three-statistic DSASM generating function and the number of DSASMs are stated, in Theorem 3, Corollary 4 and Theorem 5, and alternative versions of some of these results are stated in ( 36)- (38) and (43). • In Section 5, the case of the six-vertex model which is related to DSASMs is introduced, the bijection between its configurations and DSASMs is identified in (45), some important properties of its partition function are identified in Propositions 6-10, and a Pfaffian expression for this partition function is obtained in Theorem 11.…”

Lattice supersymmetry in the open XXZ model: an algebraic Bethe Ansatz analysis

“…The type of lattice SUSY displayed by (1.1) has now been found and studied, primarily by Fendley and Hagendorf and collaborators, in a growing range of open and closed quantum spin chains [3,[10][11][12][13][14][15]. In particular, the manifestation of SUSY as pairings of sets of roots of the Bethe Equations was observed and studied for the closed 8-vertex model along its combinatorial line in [11].…”

Vertex Models and Spin Chains in Formulas and Pictures

“…Rather developed graphical methods are used in the quantum theory of angular momentum [6,74,76], the general relativity [58,59,60], and physical applications of the group theory [27]. The graphical methods used in the theory of quantum integrable models of statistical physics [5] were successfully applied to the problems of enumerative combinatorics [3,4,14,34,35,47,48,62,63,64].…”