Monday, February 20, 2012

1202.3922 (O. Blondeau-Fournier et al.)

Macdonald polynomials in superspace as eigenfunctions of commuting

O. Blondeau-Fournier, P. Desrosiers, L. Lapointe, P. Mathieu
A generalization of the Macdonald polynomials depending upon both commuting
and anticommuting variables has been introduced recently. The construction
relies on certain orthogonality and triangularity relations. Although many
superpolynomials were constructed as solutions of highly over-determined
system, the existence issue was left open. This is resolved here: we
demonstrate that the underlying construction has a (unique) solution. The proof
uses, as a starting point, the definition of the Macdonald superpolynomials in
terms of the Macdonald non-symmetric polynomials via a non-standard
(anti)symmetrization and a suitable dressing by anticommuting monomials. This
relationship naturally suggests the form of two family of commuting operators
that have the defined superpolynomials as their common eigenfunctions. These
eigenfunctions are then shown to be triangular and orthogonal. Up to a
normalization, these two conditions uniquely characterize these
superpolynomials. Moreover, the Macdonald superpolynomials are found to be
orthogonal with respect to a second (constant-term-type) scalar product and its
norm is evaluated. The latter is shown to match (up to a q-power) the
conjectured norm with respect to the original scalar product. Finally, we
recall the super-version of the Macdonald positivity conjecture and present two
new conjectures which both provide a remarkable relationship between the new
(q,t)-Kostka coefficients and the usual ones.
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