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Please use this identifier to cite or link to this item: http://hdl.handle.net/1860/1942

Title: Tableau atoms and a new Macdonald positivity conjecture
Authors: Morse, Jennifer
Lapointe, Luc
Lascoux, Alain
Issue Date: 17-Nov-2001
Citation: Retrieved November 2, 2007 from http://xxx.lanl.gov/find/grp_math/1/au:+Morse_J/0/1/0/all/0/1.
Abstract: Let A be the space of symmetric functions and Vk be the subspace spanned by the modified Schur functions {Sy[X/(1 − t)]}1k. We introduce a new family of symmetric polynomials, {A(k) [X; t]}1<k, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials Ay(k) [X; t] form a basis for Vk 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 Ay(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 Ay(k) [X; t] seem to play the same role for Vk as the Schur functions do for . 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.
URI: http://lanl.arxiv.org/abs/math/0008073v2
http://hdl.handle.net/1860/1942
Appears in Collections:Faculty Research and Publications (Mathematics)

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