A Generalization of Tokuyama's Formula to the Hall-Littlewood Polynomials

Abstract

A theorem due to Tokuyama expresses Schur polynomials in terms of Gelfand-Tsetlin patterns, providing a deformation of the Weyl character formula and two other classical results, Stanley's formula for the Schur q-polynomials and Gelfand's parametrization for the Schur polynomial. We generalize Tokuyama's formula to the Hall-Littlewood polynomials by extending Tokuyama's statistics. Our result, in addition to specializing to Tokuyama's result and the aforementioned classical results, also yields connections to the monomial symmetric function and a new deformation of Stanley's formula.

Relating three combinatorial formulas for type $A$ Whittaker functions

“…In this section we review what we know for the spherical Whittaker function. There are two formulas of relevance when studying the spherical Whittaker function, the Ram-Yip formula derived in [BBL15] and [OS18], and Tokuyama's formula derived in the 1980's and stated in [GRV15] and [BrBF11, Chapter V]. The Ram-Yip formula for the spherical Whittaker function has a x ρ monomial factor.…”

“…The Casselman-Shalika formula [CS80,S76] expresses the spherical Whittaker function, for a dominant weight λ, as the corresponding irreducible character times a scaling factor in K. Upon specializing to type A, the mentioned product becomes a t-deformation of the Vandermonde determinant times the Schur polynomial for the given dominant weight λ [BrBF11, Chapter V]. In the 1980's it was discovered that this product can be expanded as a sum over certain combinatorial objects; this identity is known as Tokuyama's formula [GRV15]. The combinatorial objects in question are known as Gelfand-Tsetlin patterns and are in bijection with SSYT [BrBF11].…”

A note on corollaries to Tokuyama's Identity for symplectic Schur $Q$-Functions

“…This is intended as a brief note for those already familiar with the ideas and notations. However, for background information and motivation on Tokuyama's identity, see, for example, Tokuyama's original paper [9], Okada's proof and variations [7,8], and related work of Hamel and King [3,4], Brubaker, Bump, and Friedberg [1], Gupta, Roy, and Van Peski [5], and Brubaker and Schultz [2], as well as the references therein.…”

Resonant Mirković–Vilonen polytopes and formulas for highest-weight characters

“…where G std , known as the standard contribution function, is defined in terms of the crystal graph structure of B(λ + ρ). This formula has ties to several areas in representation theory [GRVP15,BBF11a] and may be viewed as a discrete analogue of the connection between archimedean Whittaker functions and geometric crystals [Chh13,Lam13]. Finding generalizations of this formula for other groups is a difficult problem in combinatorial representation theory that has seen much recent interest [HK02a,FGG15,DeF18].…”