Some Properties of Macdonald Polynomials with Prescribed Symmetry

TL;DR

This paper explores properties of Macdonald polynomials with prescribed symmetry, deriving expansion formulas, normalization, and connections to conjectured identities using eigenoperator methods.

Contribution

It introduces new expansion and normalization formulas for prescribed symmetry Macdonald polynomials and relates symmetric and antisymmetric cases through eigenoperator techniques.

Findings

Derived an explicit normalization formula.

Established relations between symmetric and antisymmetric Macdonald polynomials.

Provided a derivation of a special case of a conjectured q-constant term identity.

Abstract

The Macdonald polynomials with prescribed symmetry are obtained from the nonsymmetric Macdonald polynomials via the operations of $t$-symmetrisation, $t$-antisymmetrisation and normalisation. Motivated by corresponding results in Jack polynomial theory we proceed to derive an expansion formula and a related normalisation. Eigenoperator methods are used to relate the symmetric and antisymmetric Macdonald polynomials, and we discuss how these methods can be extended to special classes of the prescribed symmetry polynomials in terms of their symmetric counterpart. We compute the explicit form of the normalisation with respect to the constant term inner product. Surpassing our original motivation, this is used to provide a derivation of a special case of a conjectured $q$-constant term identity.