Symmetry and Pieri rules for the bisymmetric Macdonald polynomials

TL;DR

This paper explores bisymmetric Macdonald polynomials, revealing their symmetry properties and deriving Pieri rules, which extend classical results and deepen understanding of their algebraic structure.

Contribution

It introduces a symmetry property for bisymmetric Macdonald polynomials and derives Pieri rules using the double affine Hecke algebra framework, extending known theories.

Findings

Bisymmetric Macdonald polynomials satisfy a generalized symmetry property.

Pieri rules for these polynomials involve sums over specific vertical strips.

The results extend classical Macdonald polynomial theory to a bisymmetric context.

Abstract

Bisymmetric Macdonald polynomials can be obtained through a process of antisymmetrization and $t$-symmetrization of non-symmetric Macdonald polynomials. Using the double affine Hecke algebra, we show that the evaluation of the bisymmetric Macdonald polynomials satisfies a symmetry property generalizing that satisfied by the usual Macdonald polynomials. We then obtain Pieri rules for the bisymmetric Macdonald polynomials where the sums are over certain vertical strips.