# The symmetric functions catalog

An overview of symmetric functions and related topics

2020-09-29

## Whittaker functions

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

T. Lam has a nice intro in https://arxiv.org/abs/1308.5451v2

Iwahori Whittaker functions: https://arxiv.org/pdf/1906.04140.pdf

Spin q-Whittaker: https://arxiv.org/pdf/2003.14260.pdf

(p,q)-Whittaker: https://arxiv.org/pdf/1710.07196.pdf

Metaplectic Whittaker and connection with LLT polynomials https://arxiv.org/pdf/1806.07776.pdf

Whittaker from non-symmetric Macdonalds: dx.doi.org/10.1016/j.jnt.2014.01.001

## 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 , where RSK ans a crystal structure is given.

See 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. http://www.math.lsa.umich.edu/~tfylam/CDM2014talk1.pdf

Geometric RSK https://maths.ucd.ie/~noconnell/pubs/cosz.pdf Reda's thesis: https://arxiv.org/abs/1302.0902