Double Macdonald polynomials as the stable limit of Macdonald superpolynomials

Blondeau-Fournier, O. ; Lapointe, L. ; Mathieu, P.

Journal of Algebraic Combinatorics, Tome 41 (2015) no. 2, pp. 397-459.

Résumé

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, evaluation, 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.

Détail

Classification : 05E05, 05E10, 20F55, 81Q60, 33D52
Keywords: Macdonald polynomials, superpolynomials, Kostka coefficients, Nabla operator, hyperoctahedral group