O. Blondeau-Fournier, L. Lapointe, P. Mathieu
Macdonald superpolynomials provide a remarkably rich generalization of the usual Macdonald polynomials. Our starting point 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. Our first 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 huge class of Macdonald polynomials with prescribed symmetry. The double (plethystically-)deformed Macdonald polynomials are then shown to have a positive expansion in the corresponding bisymmetric Schur basis (with coefficients being generalized Kostka's). Two points of contact with the hyperoctahedral group are then uncovered. First, the q=t=1 specialization of the Kostka coefficients corresponds to the dimensions of the irreducible representations of the hyperoctahedral group. Second, the action of a generalized Nabla operator on the bisymmetric Schur polynomial is found to be Schur positive, with a possible interpretation as that of the Frobenius series of a certain bigraded module of dimension (2n+1)^n, a formula characteristic of the Coxeter group of type B_n. The aforementioned factorization of the double Macdonald polynomials leads immediately to the generalization of basically every elementary properties of the Macdonald polynomials to the double case. When lifted back to superspace, these results provide proofs of various previously formulated conjectures in the stable regime.
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http://arxiv.org/abs/1211.3186
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