2011/12/21 by Olivier Blondeau-Fournier, O. Blondeau-Fournier, P. Desrosiers +5 · 20 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebra over a field #Algebraic structures and combinatorial models #Mathematical economics #Mathematical physics #Mathematics #Pure mathematics #Superspace #hep-th #math-ph #math.CO #math.MP
paper · pdf · doi:10.1007/s11005-011-0542-5
published in Letters in Mathematical Physics 101(1), 27-47 (Springer Science+Business Media) · 18 pages
arxiv created 2011/12/21 · openalex publication_date 2012/01/07 · arxiv updated 2012/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.