Relating three combinatorial formulas for type $A$ Whittaker functions

TL;DR

This paper explores the connections between three combinatorial formulas for type $A$ Whittaker functions, revealing how each can be derived from the previous through a process called compression, a novel insight in this context.

Contribution

It establishes the relationship and transformation process between three different combinatorial formulas for type $A$ Whittaker functions, a previously unexplored area.

Findings

Demonstrates how alcove walk formulas relate to Young diagram fillings.

Shows the connection between fillings and semistandard Young tableaux.

Introduces the concept of compression in the context of Whittaker functions.

Abstract

In this work we study the relationship between several combinatorial formulas for type $A$ spherical Whittaker functions. These are spherical functions on $p$-adic groups, which arise in the theory of automorphic forms. They depend on a parameter $t$, and are a specialization of Macdonald polynomials, and further specialize to Schur polynomials upon setting $t=0$. There are three types of formulas for these polynomials. The first formula is in terms of so-called alcove walks, works in arbitrary Lie type, and is derived from the Ram-Yip formula for Macdonald polynomials. The second one is in terms of certain fillings of Young diagrams, and is derived from, or is analogous to the Haglund-Haiman-Loehr formula for Macdonald polynomials. The third formula is in terms of the classical semistandard Young tableaux. We study the way in which each such formula is obtained from the previous one by combining terms $-$ a phenomenon called compression. No such results existed in the case of Whittaker functions.