Izergin-Korepin Determinant at a Third Root of Unity

We consider the partition function of the inhomogeneous six-vertex model defined on an n×n square lattice. This function depends on 2n spectral parameters xi and yi 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 Nth root of unity, the partition function satisfies a special linear functional equation. This equation is particularly simple and useful when the crossing parameter is η = 2π/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 η = 2π/3, it is symmetric in the union {x} ∪ {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.