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

TL;DR

This paper generalizes Razumov-Stroganov sum rules for higher rank open IRF models using boundary qKZ equations, revealing connections to symplectic characters and combinatorial structures.

Contribution

It introduces higher rank generalizations of sum rules at specific q-values for open boundary models via polynomial solutions of boundary qKZ equations.

Findings

Derived sum rules as symplectic characters

Connected sum rules to combinatorial objects like symmetric matrices

Explored q=-1 case related to nilpotent matrix varieties

Abstract

We find higher rank generalizations of the Razumov--Stroganov sum rules at $q=-e^{i\pi\over k+1}$ for $A_{k-1}$ models with open boundaries, by constructing polynomial solutions of level one boundary quantum Knizhnik--Zamolodchikov equations for $U_q(\frak{sl}(k))$. 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.