Macdonald polynomials at $t=q^k$

TL;DR

This paper studies Macdonald polynomials at the specialization t=q^k, establishing identities and operators that characterize these polynomials, thereby advancing understanding of their structure and properties.

Contribution

It introduces a new identity linking specialized Macdonald polynomials and describes an operator with eigenvalues that characterize these polynomials.

Findings

Derived an identity relating P_lambda(X;q,q^k) and P_lambda((1-q)/(1-q^k)X;q,q^k)

Identified an operator whose eigenvalues uniquely determine P_lambda(X;q,q^k)

Enhanced understanding of Macdonald polynomials at t=q^k

Abstract

We investigate the homogeneous symmetric Macdonald polynomials $P_\lambda(\X;q,t)$ for the specialization $t=q^k$. We show an identity relying the polynomials $P_\lambda(\X;q,q^k)$ and $P_\lambda(\frac{1-q}{1-q^k}\X;q,q^k)$. As a consequence, we describe an operator whose eigenvalues characterize the polynomials $P_\lambda(\X;q,q^k)$.