Refined Littlewood identity for spin Hall–Littlewood symmetric rational functions

Algebraic Combinatorics. 2023. Vol. 6. No. 1. P. 37–51.

Gavrilova S.

Fully inhomogeneous spin Hall–Littlewood symmetric rational functions $F_{\lambda}$ are multiparameter deformations of the classical Hall–Littlewood symmetric polynomials and can be viewed as partition functions in 𝔰𝔩(2) higher spin six vertex models.

We obtain a refined Littlewood identity expressing a weighted sum of $F_{\lambda}$’s over all signatures $\lambda$ with even multiplicities as a certain Pfaffian. This Pfaffian can be derived as a partition function of the six vertex model in a triangle with suitably decorated domain wall boundary conditions. The proof is based on the Yang–Baxter equation.

Keywords: симметрические функции, Hall-Littlewood symmetric functions, шестивершинная модель, Six vertex model