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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Refinements of the Littlewood-Richardson rule


Authors: J. Haglund, K. Luoto, S. Mason and S. van Willigenburg
Journal: Trans. Amer. Math. Soc. 363 (2011), 1665-1686
MSC (2000): Primary 05E05; Secondary 05E10, 33D52
Published electronically: October 15, 2010
MathSciNet review: 2737282
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Abstract: In the prequel to this paper, we showed how results of Mason involving a new combinatorial formula for polynomials that are now known as Demazure atoms (characters of quotients of Demazure modules, called standard bases by Lascoux and Schützenberger) could be used to define a new basis for the ring of quasisymmetric functions we call ``Quasisymmetric Schur functions'' (QS functions for short). In this paper we develop the combinatorics of these polynomials further, by showing that the product of a Schur function and a Demazure atom has a positive expansion in terms of Demazure atoms. We use these techniques, together with the fact that both a QS function and a Demazure character have explicit expressions as a positive sum of atoms, to obtain the expansion of a product of a Schur function with a QS function (Demazure character) as a positive sum of QS functions (Demazure characters). Our formula for the coefficients in the expansion of a product of a Demazure character and a Schur function into Demazure characters is similar to known results and includes in particular the famous Littlewood-Richardson rule for the expansion of a product of Schur functions in terms of the Schur basis.


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Additional Information

J. Haglund
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: jhaglund@math.upenn.edu

K. Luoto
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
Email: kwluoto@math.ubc.ca

S. Mason
Affiliation: Department of Mathematics, University of California at San Diego, San Diego, California 92093
Address at time of publication: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109
Email: skmason@math.ucsd.edu, masonsk@wfu.edu

S. van Willigenburg
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
Email: steph@math.ubc.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05244-4
PII: S 0002-9947(2010)05244-4
Keywords: Key polynomials, non-symmetric Macdonald polynomials, Littlewood-Richardson rule, quasisymmetric functions, Schur functions, tableaux
Received by editor(s): July 10, 2009
Received by editor(s) in revised form: November 10, 2009
Published electronically: October 15, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.