The Rational qKZ Equation and Shifted Non-Symmetric Jack Polynomials

TL;DR

This paper constructs special solutions to the rational qKZ equation for $gl_N$ using shifted non-symmetric Jack polynomials, linking them to matrix elements of vertex operators and extending Dunkl's singular polynomials.

Contribution

It introduces a new class of solutions to the rational qKZ equation based on shifted non-symmetric Jack polynomials, connecting them to vertex operator matrix elements.

Findings

Solutions include scaling limits of vertex operator matrix elements

Introduces shifted non-symmetric Jack polynomials as key tools

Links solutions to Dunkl's singular polynomials

Abstract

We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra $gl_N$. The main ingredient is a special class of the shifted non-symmetric Jack polynomials. It may be regarded as a shifted version of the singular polynomials studied by Dunkl. We prove that our solutions contain those obtained as a scaling limit of matrix elements of the vertex operators of level one.