Macdonald Polynomials in Superspace: Conjectural Definition and Positivity Conjectures

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 = a, 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 = a 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.