Abstract
We introduce a conjectural construction for an extension to superspace of the Macdonald polynomials. The construction, which depends on certain orthogonality and triangularity relations, is tested for high degrees. We conjecture a simple form for the norm of the Macdonald polynomials in superspace and a rather non-trivial expression for their evaluation. We study the limiting cases q = 0 and q = ∞, which lead to two families of Hall–Littlewood polynomials in superspace. We also find that the Macdonald polynomials in superspace evaluated at q = t = 0 or q = t = ∞ seem to generalize naturally the Schur functions. In particular, their expansion coefficients in the corresponding Hall–Littlewood bases appear to be polynomials in t with nonnegative integer coefficients. More strikingly, we formulate a generalization of the Macdonald positivity conjecture to superspace: the expansion coefficients of the Macdonald superpolynomials expanded into a modified version of the Schur superpolynomial basis (the q = t = 0 family) are polynomials in q and t with nonnegative integer coefficients.
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Blondeau-Fournier, O., Desrosiers, P., Lapointe, L. et al. Macdonald Polynomials in Superspace: Conjectural Definition and Positivity Conjectures. Lett Math Phys 101, 27–47 (2012). https://doi.org/10.1007/s11005-011-0542-5
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DOI: https://doi.org/10.1007/s11005-011-0542-5