Factorised solutions of Temperley-Lieb $qKZ$ equations on a segment

Abstract

We study the $q$-deformed Knizhnik–Zamolodchikov ($qKZ$) equation in path representations of the Temperley–Lieb algebras. We consider two types of open boundary conditions, and in both cases we derive factorized expressions for the solutions of the $qKZ$ equation in terms of Baxterized Demazurre–Lusztig operators. These expressions are alternative to known integral solutions for tensor product representations. The factorized expressions reveal the algebraic structure within the $qKZ$ equation, and effectively reduce it to a set of truncation conditions on a single scalar function. The factorized expressions allow for an efficient computation of the full solution once this single scalar function is known. We further study particular polynomial solutions for which certain additional factorized expressions give weighted sums over components of the solution. In the homogeneous limit, we formulate positivity conjectures in the spirit of Di Francesco and Zinn-Justin. We further conjecture relations between weighted sums and individual components of the solutions for larger system sizes.