Combinatorics of the Casselman-Shalika formula in type A

Details Combinatorics of the Casselman-Shalika formula in type A Combinatorics of the Casselman-Shalika formula in type A

2011-11-04

arxiv.org/abs/1111.1134v1

Mathematics

Representation Theory

10 pages

Scientific paper

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. As Bump et al. showed in their earlier work, one can use Gelfand-Tsetlin patterns to obtain a similar result. Contrasting those results, our approach may naturally be generalized to other types of root systems through Kashiwara-Nakashima tableaux, which we hope to address in future work. We also show how to combine our results with tensor product 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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