## Combinatorics of Casselman-Shalika formula in type $A$

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- by Kyu-Hwan Lee, Philip Lombardo and Ben Salisbury PDF
- Proc. Amer. Math. Soc.
**142**(2014), 2291-2301 Request permission

## 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 tableau model for the crystal and obtain the same coefficients using data from each individual tableau; 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.## References

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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
- MR Author ID: 650497
- 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
- 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
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**142**(2014), 2291-2301 - MSC (2010): Primary 17B37; Secondary 05E10
- DOI: https://doi.org/10.1090/S0002-9939-2014-11961-7
- MathSciNet review: 3195754