Double Macdonald polynomials as the stable limit of Macdonald superpolynomials

TL;DR

This paper introduces stable double Macdonald polynomials derived from Macdonald superpolynomials, providing explicit formulas, properties, and connections to representation theory of hyperoctahedral groups, thus extending classical Macdonald polynomial theory.

Contribution

It establishes the factorization of stable bisymmetric Macdonald polynomials into products of Macdonald polynomials with plethystic transformations, and explores their algebraic and representation-theoretic properties.

Findings

Stable superpolynomials become independent of m for large m.

Double Macdonald polynomials factor into products of Macdonald polynomials.

q,t-Kostka coefficients relate to hyperoctahedral group representations.

Abstract

Macdonald superpolynomials provide a remarkably rich generalization of the usual Macdonald polynomials. The starting point of this work is the observation of a previously unnoticed stability property of the Macdonald superpolynomials when the fermionic sector m is sufficiently large: their decomposition in the monomial basis is then independent of m. These stable superpolynomials are readily mapped into bisymmetric polynomials, an operation that spoils the ring structure but drastically simplifies the associated vector space. Our main result is a factorization of the (stable) bisymmetric Macdonald polynomials, called double Macdonald polynomials and indexed by pairs of partitions, into a product of Macdonald polynomials (albeit subject to non-trivial plethystic transformations). As an off-shoot, we note that, after multiplication by a t-Vandermonde determinant, this provides explicit formulas for a large class of Macdonald polynomials with prescribed symmetry. The factorization of the double Macdonald polynomials leads immediately to the generalization of basically every elementary properties of the Macdonald polynomials to the double case (norm, kernel, duality, positivity, etc). When lifted back to superspace, this validates various previously formulated conjectures in the stable regime. The q,t-Kostka coefficients associated to the double Macdonald polynomials are shown to be q,t-analogs of the dimensions of the irreducible representations of the hyperoctahedral group B_n. Moreover, a Nabla operator on the double Macdonald polynomials is defined and its action on a certain bisymmetric Schur function can be interpreted as the Frobenius series of a bigraded module of dimension (2n+1)^n, a formula again characteristic of the Coxeter group of type B_n. Finally, as a side result, we obtain a simple identity involving products of four Littlewood-Richardson coefficients.