Combinatorics of the Casselman-Shalika formula in type A

TL;DR

This paper simplifies the combinatorial understanding of the Casselman-Shalika formula in type A by deriving coefficients directly from tableaux, avoiding the need for crystal graph analysis, and explores properties of resulting q-polynomials.

Contribution

It introduces a tableaux-based method to compute coefficients in the Casselman-Shalika formula, bypassing the crystal graph structure, and analyzes the properties of associated q-polynomials.

Findings

Coefficients can be computed directly from tableaux data.

The sum of coefficients forms a q-polynomial with interesting properties.

Examples illustrate the polynomial properties and combinatorial structure.

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.