Relating three combinatorial formulas for type $A$ 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.