Higher Rank Macdonald Polynomials

TL;DR

This paper introduces higher rank generalizations of Macdonald polynomials, extending their algebraic and combinatorial properties, and explores their applications in representation theory and algebraic structures.

Contribution

It develops higher rank non-symmetric and symmetric Macdonald polynomials, proves their key relations, and constructs new algebraic representations, advancing the theory of Macdonald polynomials.

Findings

Higher rank non-symmetric Macdonald polynomials satisfy generalized Knop--Sahi relations.

Higher rank symmetric Macdonald polynomials form eigenbases with stability properties.

Construction of higher rank polynomial representations of the double Dyck path algebra.

Abstract

In this paper, we introduce higher rank generalizations of Macdonald polynomials. The higher rank non-symmetric Macdonald polynomials are Laurent polynomials in several sets of variables which form weight bases for higher rank polynomial representations of double affine Hecke algebras with respect to higher rank Cherednik operators. We prove that these polynomials satisfy generalized versions of the classical Knop--Sahi relations and we give combinatorial descriptions of their weights. The higher rank symmetric Macdonald polynomials are defined as Hecke-symmetrizations of the higher rank non-symmetric Macdonald polynomials and form eigenbases for the spaces of Hecke-invariant higher rank polynomials with respect to generalized finite variable Macdonald operators. We prove that the higher rank symmetric Macdonald polynomials satisfy stability properties allowing for the construction of infinite variable limits. These higher rank symmetric Macdonald functions form eigenbases for certain representations of the (positive) elliptic Hall algebra with respect to generalized infinite variable Macdonald operators. Lastly, we show that the higher rank polynomial representations may be used to construct higher rank polynomial representations of the double Dyck path algebra. This is a copy of the author's accepted extended abstract for FPSAC2025.