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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Combinatorics of Casselman-Shalika formula in type $ A$


Authors: Kyu-Hwan Lee, Philip Lombardo and Ben Salisbury
Journal: Proc. Amer. Math. Soc. 142 (2014), 2291-2301
MSC (2010): Primary 17B37; Secondary 05E10
Published electronically: March 11, 2014
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Abstract: In the recent works of Brubaker-Bump-Friedberg, Bump-Nakasuji, and others, the product in the Casselman-Shalika formula is written as a sum over a crystal. The coefficient of each crystal element is defined using the data coming from the whole crystal graph structure. In this paper, we adopt the tableaux model for the crystal and obtain the same coefficients using data from each individual tableaux; i.e., we do not need to look at the graph structure. We also show how to combine our results with tensor products of crystals to obtain the sum of coefficients for a given weight. The sum is a $ q$-polynomial which exhibits many interesting properties. We use examples to illustrate these properties.


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Additional Information

Kyu-Hwan Lee
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009 – and – Korea Institute for Advanced Study, Seoul, Korea
Email: khlee@math.uconn.edu

Philip Lombardo
Affiliation: Department of Mathematics and Computer Science, St. Joseph’s College, Patchogue, New York 11772
Email: plombardo@sjcny.edu

Ben Salisbury
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Address at time of publication: Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan 48859
Email: ben.salisbury@cmich.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11961-7
PII: S 0002-9939(2014)11961-7
Keywords: Casselman-Shalika formula, crystals, Young tableaux
Received by editor(s): November 3, 2011
Received by editor(s) in revised form: July 24, 2012
Published electronically: March 11, 2014
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.