论文标题
旋转霍尔 - 小木对称合理功能的精制凯奇（Cauchy）身份
Refined Cauchy identity for spin Hall-Littlewood symmetric rational functions
论文作者
论文摘要
完全不均匀的旋转Hall-littlewood对称合理功能$ \ MATHSF {f}_λ$在$ \ Mathfrak {sl}（2）的上下文中出现。我们获得了一种精致的cauchy身份，表达了两个$ \ m rathsf {f}_λ$作为决定因素的加权总和。决定因素是Izergin-Korepin类型的：它是具有适当装饰的域壁边界条件的六个顶点模型的分区函数。两个分区函数的平等证明是基于杨巴克斯特方程。 我们以不同的形式重写izergin-korepin类型的决定因素，该类型包括一个完全对称的变量集之一。这种决定性的身份可能具有独立的兴趣，并且还允许将旋转霍尔 - 小木有理函数与（Hall-littlewood特定情况的）插值直接连接起来。在不同的方向上，我们的Izergin-Korepin类型的Schur扩展产生了Schur对称多项式的变形。 在旋转 - $ \ frac12 $专业化中，我们精致的凯奇（Cauchy）身份导致了ASEP本征征的总和（不对称简单排除过程），这是Kardar-Parisi-Parisi-Parisi-parisi-Zhang普遍性的著名随机相互作用的粒子系统。这为ASEP中的某些多时间概率产生了明确的积分公式。
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 equation. We rewrite our Izergin-Korepin type determinant in a different form which includes one of the sets of variables in a completely symmetric way. This determinantal identity might be of independent interest, and also allows to directly link the spin Hall-Littlewood rational functions with (the Hall-Littlewood particular case of) the interpolation Macdonald polynomials. In a different direction, a Schur expansion of our Izergin-Korepin type determinant yields a deformation of Schur symmetric polynomials. In the spin-$\frac12$ specialization, our refined Cauchy identity leads to a summation identity for eigenfunctions of the ASEP (Asymmetric Simple Exclusion Process), a celebrated stochastic interacting particle system in the Kardar-Parisi-Zhang universality class. This produces explicit integral formulas for certain multitime probabilities in ASEP.
