Laurent polynomial solutions of the boundary quantum Knizhnik--Zamolodchikov equation

TL;DR

This paper constructs Laurent polynomial solutions to the boundary quantum Knizhnik--Zamolodchikov equation for a specific quantum algebra, utilizing non-symmetric Koornwinder polynomials, and identifies solutions of minimal degree.

Contribution

It introduces a novel method to explicitly construct Laurent polynomial solutions using non-symmetric Koornwinder polynomials for the boundary quantum KZ equation.

Findings

Constructed Laurent polynomial solutions for the boundary quantum KZ equation.

Characterized solutions using specialized non-symmetric Koornwinder polynomials.

Obtained minimal degree solutions as a special case.

Abstract

We construct Laurent polynomial solutions of the boundary quantum Knizhnik--Zamolodchikov equation for $U_{q}(\widehat{\mathfrak{sl}}_{2})$ on the parabolic Kazhdan--Lusztig bases. They are characterized by non-symmetric Koornwinder polynomials with the specialized parameters. As a special case, we obtain the solution of the minimal degree.