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From Macdonald polynomials to their hyperoctahedral extension: the superspace bridge | | O. Blondeau-Fournier
; L. Lapointe
; P. Mathieu
; | | Date: |
14 Nov 2012 | | Abstract: | 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. | | Source: | arXiv, 1211.3186 | | Services: | Forum | Review | PDF | Favorites | |
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