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 TMF, 2006, Volume 146, Number 1, Pages 65–76 (Mi tmf2009)

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: https://doi.org/10.4213/tmf2009

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English version:
Theoretical and Mathematical Physics, 2006, 146:1, 53–62

Bibliographic databases:

Citation: Yu. G. Stroganov, “Izergin–Korepin Determinant at a Third Root of Unity”, TMF, 146:1 (2006), 65–76; Theoret. and Math. Phys., 146:1 (2006), 53–62

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
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