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 [Ber20a]. The $q$-Whittaker function $\qWhittaker_\lambda(\xvec;q)$ can be defined as any of the quantities:
- the modified Macdonald polynomial $[t^{\partitionN(\lambda)}] \macdonaldH_\lambda(\xvec;q,t),$
- the Macdonald $P$ polynomial $\macdonaldP_\lambda(\xvec;q,0)$
- the transformed Hall–Littlewood polynomial $\omega \hallLittlewoodT_{\lambda'}(\xvec;q),$
- the non-symmetric Macdonald polynomial $\macdonaldE_\lambda(\xvec;q,0).$
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. 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
References
- [AG18] Sami Assaf and Nicolle S. González. Crystal graphs, key tabloids, and nonsymmetric Macdonald polynomials. 30th International Conference on Formal Power Series and Algebraic Combinatorics. Séminaire Lotharingien de Combinatoire, 80B(90) 2018. 12 pages
- [Ass18a] Sami Assaf. Nonsymmetric Macdonald polynomials and a refinement of Kostka–Foulkes polynomials. Transactions of the American Mathematical Society, 370(12):8777–8796, July 2018.
- [AU20] Per Alexandersson and Joakim Uhlin. Cyclic sieving, skew Macdonald polynomials and Schur positivity. Algebraic Combinatorics, 3(4):913-939, 2020.
- [Ber20a] François Bergeron. A survey of $q$-Whittaker polynomials. arXiv e-prints, 2020.
- [Ber20b] François Bergeron. $(gl_k\times s_n)$-modules of multivariate diagonal harmonics. arXiv e-prints, 2020.
- [Jac67] Hervé Jacquet. Fonctions de Whittaker associées aux groupes de Chevalley. Bulletin de la Société Mathématique de France, 95:243-309, 1967.
- [Uhl19] Joakim Uhlin. Combinatorics of Macdonald polynomials and cyclic sieving. 2019.