Boundary qKZ equation and generalized Razumov-Stroganov sum rules for open IRF models

Abstract

We find higher rank generalizations of the Razumov--Stroganov sum rules at $q=-e^{\frac{i\pi }{k+1}}$ for $A_{k-1}$ models with open boundaries, by constructing polynomial solutions of level one boundary quantum Knizhnik--Zamolodchikov equations for $U_q(\mathfrak{sl}(k))$. The result takes the form of a character of the symplectic group, that leads to a generalization of the number of vertically symmetric alternating sign matrices. We also investigate the other combinatorial point $q=-1$, presumably related to the geometry of nilpotent matrix varieties.

21 pages, 7 figures, 1 table, uses epsf and lanlmac conclusion slightly modified

References in corpus (4)

• Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials
• Inhomogeneous loop models with open boundaries
• Enumeration of Symmetry Classes of Alternating Sign Matrices and Characters of Classical Groups
• A scheme related to the Brauer loop model

Cited by in corpus (15)

• Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics
• Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain
• Combinatorial point for higher spin loop models
• Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations
• Loop model with mixed boundary conditions, qKZ equation and alternating sign matrices
• Correlation Functions and the Boundary qKZ Equation in a Fractured XXZ Chain
• Boundary quantum Knizhnik-Zamolodchikov equations and fusion
• Factorised solutions of Temperley-Lieb $q$KZ equations on a segment
• A_k Generalization of the O(1) Loop Model on a Cylinder: Affine Hecke Algebra, q-KZ Equation and the Sum Rule
• Connection problems for quantum affine KZ equations and integrable lattice models
• Fully packed loop models on finite geometries
• Finite size lattice results for the two-boundary Temperley--Lieb loop model
• Inhomogeneous discrete-time exclusion processes
• Laurent polynomial solutions of the boundary quantum Knizhnik--Zamolodchikov equation
• The Rotor Model with spectral parameters and enumerations of Alternating Sign Matrices