Izergin–Korepin Determinant at a Third Root of Unity

We consider the partition function of the inhomogeneous six-vertex model defined on an $(n\times n)$ square lattice. This function depends on $2n$ spectral parameters $x_i$ and $y_i$ attached to the respective horizontal and vertical lines. In the case of the domain-wall boundary conditions, it is given by the Izergin–Korepin determinant. For $q$ being an $N$-th root of unity, the partition function satisfies a special linear functional equation. This equation is particularly simple and useful when the crossing parameter is $\eta=2\pi/3$, i. e., $N = 3$. It is well known, for example, that the partition function is symmetric in both the $\{x\}$ and the $\{y\}$ variables. Using the abovementioned equation, we find that in the case of $\eta=2\pi/3$, it is symmetric in the union $\{x\}\cup\{y\}$. In addition, this equation can be used to solve some of the problems related to enumerating alternating-sign matrices. In particular, we reproduce the refined alternating-sign matrix enumeration discovered by Mills, Robbins, and Rumsey and proved by Zeilberger, and we obtain formulas for the doubly refined enumeration of these matrices.