An Izergin-Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices

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

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 $η=2π/3$. In this case it can be used for solving some of the problems related to the enumeration of alter…

源自Izergin-Korepin 19-vertex模型的有理对称函数

Rational symmetric functions from the Izergin-Korepin 19-vertex model

基于象限中的Izergin-Korepin 19-vertex模型引入两族有理多元函数并证明其对称性与Cauchy恒等式，在谱参数极限下得到稳定对称函数与简化恒等式，并通过对2-排列集合求和给出对称化公式及monodromy矩阵元的显式公式。

Refined Cauchy identity for spin Hall-Littlewood symmetric rational functions

Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $\mathsf{F}_λ$ arise in the context of $\mathfrak{sl}(2)$ higher spin six vertex models, and are multiparameter deformations of the classical Hall-Littlewood symmetric polynomials. We obtain a refined Cauchy identity expressing a weighted sum of the product of two $\mathsf{F}_λ$'s as a determinant. The determinant is of Izergin-Korepin type: it is the partition function of the six vertex model with suitably decorated domain wall boundary conditions. The proof of equality of two partition functions is based on the Yang-Baxter…

Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase

The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite $N$ by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an $N\times N$ Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large $N$ asymptotics of the six-vertex model with DWBC in the disordered phase. The solution is…

Symmetry classes of alternating-sign matrices under one roof

In a previous article [math.CO/9712207], we derived the alternating-sign matrix (ASM) theorem from the Izergin-Korepin determinant for a partition function for square ice with domain wall boundary. Here we show that the same argument enumerates three other symmetry classes of alternating-sign matrices: VSASMs (vertically symmetric ASMs), even HTSASMs (half-turn-symmetric ASMs), and even QTSASMs (quarter-turn-symmetric ASMs). The VSASM enumeration was conjectured by Mills; the others by Robbins [math.CO/0008045]. We introduce several new types of ASMs: UASMs (ASMs with a U-turn side), UUASMs (t…

Multiply-refined enumeration of alternating sign matrices

Four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices (ASMs). Specifically, these statistics are the positions of the 1's in the first and last rows and columns of an ASM, and the numbers of generalized inversions and -1's in an ASM. Previously-known and related results for the exact enumeration of ASMs with prescribed values of some of these statistics are discussed in detail. A quadratic relation which recursively determines the generating function associated with all six statistics is then obtained. This relation also leads to various …