Tableau atoms and a new Macdonald positivity conjecture
| dc.creator | Lapointe, L. | |
| dc.creator | Lascoux, A. | |
| dc.creator | Morse, J. | |
| dc.date | 2000-08-09 | |
| dc.date | 2001-11-17 | |
| dc.date.accessioned | 2026-07-07T04:36:43Z | |
| dc.date.available | 2026-07-07T04:36:43Z | |
| dc.description | Let $Λ$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_λ[X/(1-t)]\}_{λ_1\leq k}$. We introduce a new family of symmetric polynomials, $\{A_λ^{(k)}[X;t]\}_{λ_1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A_λ^{(k)}[X;t]$ form a basis for $V_k$ and that the Macdonald polynomials indexed by partitions whose first part is not larger than $k$ expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but would substantially refine it. Our construction of the $A_λ^{(k)}[X;t]$ relies on the use of tableaux combinatorics and yields various properties and conjectures on the nature of these polynomials. Another important development following from our investigation is that the $A_λ^{(k)}[X;t]$ seem to play the same role for $V_k$ as the Schur functions do for $Λ$. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri and Littlewood-Richardson type coefficients. | |
| dc.description | 38 pages, 7 figures. New version, with minor modifications, of "A filtration of the symmetric function space and a refinement of the Macdonald positivity conjecture" | |
| dc.identifier | https://arxiv.org/abs/math/0008073 | |
| dc.identifier | http://arxiv.org/abs/math/0008073 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/59702 | |
| dc.subject | Quantum Algebra | |
| dc.subject | Combinatorics | |
| dc.subject | 05E05;05E10 | |
| dc.title | Tableau atoms and a new Macdonald positivity conjecture | |
| dc.type | text |