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

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.

arXiv comment15 pages, an unimportant, if bad, mistake is corrected (see the first page of Section 2)

References in corpus (4)

• Alternating sign matrices and domino tilings
• Another proof of the alternating sign matrix conjecture
• The Importance of being Odd
• Proof of the Refined Alternating Sign Matrix Conjecture

Cited by in corpus (26)

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• Izergin-Korepin determinant reloaded
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• The Wave Functions for the Free-Fermion Part of the Spectrum of the $S{U}_q(N)$ Quantum Spin Models
• New differential equations in the six-vertex model
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• Doubly-refined enumeration of Alternating Sign Matrices and determinants of 2-staircase Schur functions
• Studies in Integrable Quantum Lattice Models and Classical Hierarchies
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• Three-coloring statistical model with domain wall boundary conditions. II. Trigonometric limit
• On the refined 3-enumeration of alternating sign matrices
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