Specializing Koornwinder polynomials to Macdonald polynomials of type $B,C,D$ and $B C$

TL;DR

This paper explores how Koornwinder polynomials can be specialized to produce Macdonald polynomials for types B, C, D, and BC, providing a systematic framework and verification through Ram-Yip formulas.

Contribution

It introduces a specialization table linking Koornwinder and Macdonald polynomials for various root system types, enhancing understanding of their interrelations.

Findings

Established a systematic specialization table for polynomials.

Verified specializations using Ram-Yip formulas.

Connected Koornwinder and Macdonald polynomials across types B, C, D, BC.

Abstract

We study the specializations of parameters in Koornwinder polynomials to obtain Macdonald polynomials associated to the subsystems of the affine root system of type $(C_n^\vee,C_n)$ in the sense of Macdonald (2003), and summarize them in what we call the specialization table. As a verification of our argument, we check the specializations to type $B,C$ and $D$ via Ram-Yip type formulas of non-symmetric Koornwinder and Macdonald polynomials.