Macdonald polynomials at $t=q^k$

Abstract

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

Keywords

• symmetric functions
• macdonald polynomials
• $q$-discriminant
• [math.math-co] mathematics [math]/combinatorics [math.co]
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]