Macdonald symmetric functions of rectangular shapes

TL;DR

This paper develops a vertex operator approach to Macdonald symmetric functions of rectangular shapes, revealing new connections with q-Dyson polynomials and generalizing several classical formulas and conjectures.

Contribution

It introduces a vertex operator realization of Macdonald functions of rectangular shapes and generalizes key formulas and conjectures related to these functions.

Findings

Vertex operator realization of Macdonald functions

Generalized Frobenius formula for Macdonald functions

Extended q-Dyson orthogonality relation and hyperdeterminant formula

Abstract

Using vertex operator we study Macdonald symmetric functions of rectangular shapes and their connection with the q-Dyson Laurent polynomial. We find a vertex operator realization of Macdonald functions and thus give a generalized Frobenius formula for them. As byproducts of the realization, we find a q-Dyson constant term orthogonality relation which generalizes a conjecture due to Kadell in 2000, and we generalize Matsumoto's hyperdeterminant formula for rectangular Jack functions to Macdonald functions.