Refined Cauchy identity for spin Hall–Littlewood symmetric rational functions

Abstract

Fully inhomogeneous spin Hall–Littlewood symmetric rational functions Fλ arise in the context of 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 Fλ'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-12 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.