The symmetric functions catalog

An overview of symmetric functions and related topics


Whittaker functions

The Whittaker functions where introduced by H. Jacquet in 1967 [Jac67].

T. Lam has a nice intro in

Iwahori Whittaker functions:

Spin q-Whittaker:


Metaplectic Whittaker and connection with LLT polynomials

Whittaker from non-symmetric Macdonalds:

q-Whittaker polynomials

$q$-Whittaker functions

The $q$-Whittaker functions $\qWhittaker_\lambda(\xvec;q)$ are eigenfunctions of the quantum Toda lattice.

For a recent survey on this topic, see [Ber20]. The $q$-Whittaker function $\qWhittaker_\lambda(\xvec;q)$ can be defined as any of the quantities:

We have that

\[ \qWhittaker_\mu(\xvec;q) = \sum_{\lambda} K_{\lambda'\mu'}(q) \schurS_\lambda, \]

where the coefficients are given by the Kostka–Foulkes polynomials. Proofs of this can be found in [Ass18aAG18], where RSK ans a crystal structure is given.

See [Uhl19AU20] and the cyclic sieving page for a cyclic sieving phenomena on non-attacking fillings associated with $q$-Whittaker polynomials.

Skew $q$-Whittaker functions

In [AU20], we introduce a skew version, $\qWhittaker_{\lambda/\mu}(\xvec;q)$ which is symmetric and Schur positive for partitions $\mu \subseteq \lambda.$

Relation with geometric RSK and crystals

The Whittaker functions show up when considering a geometric lift of RSK. There is also a notion of geometric crystals.

Geometric RSK Reda's thesis: