A new way to deal with Izergin-Korepin determinant at root of unity

TL;DR

This paper introduces a novel approach to analyzing the Izergin-Korepin determinant at roots of unity, revealing new symmetries and relations in the enumeration of alternating sign matrices using a special functional equation.

Contribution

It demonstrates a new symmetry in the partition function at specific roots of unity and derives relations for refined enumeration of alternating sign matrices.

Findings

Partition function symmetry in the union of spectral parameters at η=2π/3

Reproduction of Mills-Robbins-Rumsey refined ASM distribution

Derivation of equations for top-bottom double refined ASM distribution

Abstract

I consider the partition function of the inhomogeneous 6-vertex model defined on the $n$ by $n$ square lattice. This function depends on 2n spectral parameters $x_i$ and $y_i$ attached to the horizontal and vertical lines respectively. In the case of domain wall boundary conditions it is given by Izergin-Korepin determinant. For $q$ being a root of unity the partition function satisfies to a special linear functional equation. This equation is particularly good when the crossing parameter $\eta=2\pi/3$. In this case it can be used for solving some of the problems related to the enumeration of alternating sign matrices. In particular, it is possible to reproduce the refined ASM distribution discovered by Mills, Robbins and Rumsey and proved by Zeilberger. Further, it is well known that the partition function is symmetric in the $\{x\}$ and as well in the $\{y\}$ variables. I have found that in the case of $\eta=2\pi/3$, the partition function is symmetric in the union $\{x\} \cup \{y\}$! This nice symmetry is used to find some relations between the numbers of such alternating sign matrices of order $n$ whose two '1' are located in fixed positions on the boundary of the matrices. Finally I derive the equation giving `top-bottom double refined' ASM distribution.