Fully inhomogeneous spin Hall–Littlewood symmetric rational functions ${\mathsf{F}}_{\lambda }$ 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 ${\mathsf{F}}_{\lambda }$'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-$\frac{1}{2}$ 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.

完全非齐次自旋霍尔-利特伍德对称有理函数 ${\mathsf{F}}_{\lambda }$ 出现在 $\mathfrak{SL}(2)$更高的自旋六顶点模型，并且是经典 Hall-Littlewood 对称多项式的多参数变形。我们获得了一个精炼的柯西恒等式，表示两个乘积的加权和${\mathsf{F}}_{\lambda }$是一个决定因素。行列式是 Izergin-Korepin 类型的：它是具有适当装饰的畴壁边界条件的六顶点模型的配分函数。两个配分函数相等的证明基于 Yang-Baxter 方程。

我们以不同的形式重写我们的 Izergin-Korepin 类型行列式，其中以完全对称的方式包含一组变量。这种行列式恒等式可能具有独立意义，并且还允许将自旋 Hall-Littlewood 有理函数与插值 Macdonald 多项式（的 Hall-Littlewood 特例）直接联系起来。在不同的方向上，我们的 Izergin-Korepin 型行列式的舒尔展开产生舒尔对称多项式的变形。

在自旋——$\frac{1}{2}$专业化，我们改进的柯西恒等式导致 ASEP（非对称简单排除过程）的特征函数的求和恒等式，这是 Kardar-Parisi-Zhang 普适性类中著名的随机相互作用粒子系统。这会为 ASEP 中的某些多次概率生成明确的积分公式。