Macdonald polynomials in superspace: conjectural definition and positivity conjectures

TL;DR

This paper proposes a conjectural extension of Macdonald polynomials to superspace, explores their properties, and formulates a positivity conjecture, with implications for Hall-Littlewood and Schur functions.

Contribution

It introduces a conjectural construction for Macdonald polynomials in superspace, including their norm, evaluation, and positivity properties, extending classical symmetric function theory.

Findings

Tested the construction for high degrees

Conjectured a simple norm formula and evaluation expression

Formulated a positivity conjecture for expansion coefficients

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=\infty, 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=\infty 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.