Macdonald polynomials in superspace: conjectural definition and positivity conjectures
arXiv:1112.5188 · doi:10.1007/s11005-011-0542-5
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.
18 pages
References in corpus (2)
Cited by in corpus (10)
- The supersymmetric Ruijsenaars-Schneider model
- The super-Virasoro singular vectors and Jack superpolynomials relationship revisited
- Schur Superpolynomials: Combinatorial Definition and Pieri Rule
- Macdonald polynomials for super-partitions
- Macdonald polynomials in superspace as eigenfunctions of commuting operators
- Symmetric functions in superspace: a compendium of results and open problems (including a SageMath worksheet)
- Super-Hamiltonians for super-Macdonald polynomials
- Bernstein operators and super-Schur functions: combinatorial aspects
- N>=2 symmetric superpolynomials
- Macdonald polynomials as characters of Cherednik algebra modules