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Izergin–Korepin Determinant at a Third Root of Unity
Yu. G. Stroganov Institute for High Energy Physics
Abstract:
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.
Keywords:
alternating-sign matrices, enumeration, square-ice model.
DOI:
10.4213/tmf2009
Fulltext:
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