Square ice, alternating sign matrices, and classical orthogonal polynomials

Abstract

The six-vertex model with domain wall boundary conditions, or squareice, is considered for particular values of its parameters, corresponding to1-,2-,and 3-enumerations of alternating sign matrices (ASMs). Using Hankel determinant representationsfor the partition function and the boundary correlator of homogeneous square ice, it is shownhow the ordinary and refined enumerations can be derived in a very simple and straightforwardway. The derivation is based on the standard relationship between Hankel determinantsand orthogonal polynomials. For the particular sets of parameters corresponding to1-,2-,and 3-enumerations of ASMs, the Hankel determinant can be naturally related to ContinuousHahn, Meixner–Pollaczek, and Continuous Dual Hahn polynomials, respectively. Thisobservation allows for a unified and simplified treatment of ASM enumerations.In particular, along the lines of the proposed approach, we provide a completesolution to the long-standing problem of the refined 3-enumeration of ASMs.