Translator Disclaimer
15 January 2003 Tableau atoms and a new Macdonald positivity conjecture
L. Lapointe, A. Lascoux, J. Morse
Duke Math. J. 116(1): 103-146 (15 January 2003). DOI: 10.1215/S0012-7094-03-11614-2


Let $\lambda$ be the space of symmetric functions, and let $V\ sb k$ be the subspace spanned by the modified Schur functions $\{S\sb \lambda[X/(1-t)]\}\sb {\lambda\sb 1\leq k}$. We introduce a new family of symmetric polynomials, $\{A\sp {(k)}\sb \lambda[X;t]\}\sp {\lambda\sb 1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A\sp {(k)}\sb \lambda[X;t]$ form a basis for $V\sb k$ 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 also substantially refine it. Our construction of the $A\sp {(k)}\sb \lambda[X;t]$ relies on the use of tableau combinatorics and yields various properties and conjectures on the nature of these polynomials. Another important development following from our investigation is that the $A\sp {(k)}\sb \lambda[X;t]$ seem to play the same role for $V\sb k$ as the Schur functions do for $\lambda$. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri-type and Littlewood-Richardson-type coefficients.


Download Citation

L. Lapointe. A. Lascoux. J. Morse. "Tableau atoms and a new Macdonald positivity conjecture." Duke Math. J. 116 (1) 103 - 146, 15 January 2003.


Published: 15 January 2003
First available in Project Euclid: 26 May 2004

zbMATH: 1020.05069
MathSciNet: MR1950481
Digital Object Identifier: 10.1215/S0012-7094-03-11614-2

Primary: 05E05

Rights: Copyright © 2003 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.116 • No. 1 • 15 January 2003
Back to Top