Square ice, alternating sign matrices and classical orthogonal polynomials

TL;DR

This paper links the enumeration of Alternating Sign Matrices to classical orthogonal polynomials through Hankel determinants, providing a unified approach and solving the longstanding refined 3-enumeration problem.

Contribution

It introduces a unified method connecting ASM enumeration with orthogonal polynomials via Hankel determinants, simplifying derivations and solving the refined 3-enumeration problem.

Findings

Hankel determinants relate ASM enumeration to classical orthogonal polynomials.

Unified treatment of ASM enumeration using orthogonal polynomial theory.

Complete solution to the refined 3-enumeration of ASMs.

Abstract

The six-vertex model with Domain Wall Boundary Conditions, or square ice, is considered for particular values of its parameters, corresponding to 1-, 2-, and 3-enumerations of Alternating Sign Matrices (ASMs). Using Hankel determinant representations for the partition function and the boundary correlator of homogeneous square ice, it is shown how the ordinary and refined enumerations can be derived in a very simple and straightforward way. The derivation is based on the standard relationship between Hankel determinants and orthogonal polynomials. For the particular sets of parameters corresponding to 1-, 2-, and 3-enumerations of ASMs, the Hankel determinant can be naturally related to Continuous Hahn, Meixner-Pollaczek, and Continuous Dual Hahn polynomials, respectively. This observation allows for a unified and simplified treatment of ASMs enumerations. In particular, along the lines of the proposed approach, we provide a complete solution to the long standing problem of the refined 3-enumeration of AMSs.