Whittaker functions

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#Whittaker functions

The Whittaker functions were introduced by H. Jacquet in 1967 [Jac67]. For an introduction to Whittaker functions, geometric crystals, and quantum Schubert calculus, see [Lam13]. B. Brubaker, V. Buciumas, D. Bump, and H. P. A. Gustafsson study colored vertex models for Iwahori Whittaker functions [BBBG24], as well as vertex-operator models for metaplectic Whittaker functions [BBBG20]. See also [BBL15] for the relation between Whittaker functions and Demazure operators.

Several deformations are useful in nearby combinatorics and representation theory. M. Mucciconi and L. Petrov introduce spin $q$-Whittaker polynomials and a deformed quantum Toda system [MP22]. G. Rahman, S. Mubeen, K. S. Nisar, and J. Choi define a $(p,q)$-Whittaker function and record its basic identities [RMNC17].

#$q$-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 [Ber24]. The $q$-Whittaker function $\qWhittaker_\lambda(\xvec;q)$ can be defined in any of the following equivalent ways:

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 $\macdonaldE_\lambda(\xvec;q,0).$

Example

For the one-box partition, the Kostka–Foulkes expansion gives \[\qWhittaker_{(1)}(\xvec;q)=\schurS_{(1)}.\]

We have \[\qWhittaker_\mu(\xvec;q) = \sum_{\lambda} K_{\lambda'\mu'}(q) \schurS_\lambda,\] where the coefficients are given by the Kostka–Foulkes polynomials. Proofs of this can be found in [Ass18, AG18], where RSK and a crystal structure are given.

See [Uhl19, AU20] and the cyclic sieving page for a cyclic sieving phenomenon on non-attacking fillings associated with $q$-Whittaker polynomials.

A. Borodin and S. Korotkikh introduce inhomogeneous spin $q$-Whittaker polynomials [BK21]. These symmetric functions generalize the $t=0$ Macdonald polynomials and have vertex-model formulas, Cauchy-type identities, and interpolation characterizations. J. He and M. Wheeler study free-boundary $q$-Whittaker and Hall–Littlewood processes via six-vertex partition functions [HW25]. Their formulas include free-boundary Cauchy-type identities and probabilistic specializations of the associated vertex models. B. Brubaker, D. Bump, A. Hardt, and H. Spink introduce lattice models for double Whittaker polynomials and motivic Chern classes [BBHS25]. One boundary condition recovers motivic Chern classes and deformed Kazhdan–Lusztig $R$-polynomials, while another specialization recovers colored lattice models for Iwahori Whittaker functions.

#Skew $q$-Whittaker functions

In [AU20], a skew $q$-Whittaker function $\qWhittaker_{\lambda/\mu}(\xvec;q)$ is introduced. It is symmetric and Schur-positive for partitions $\mu \subseteq \lambda.$

#Cauchy identities and combinatorial models

There are several recent combinatorial and probabilistic models for $q$-Whittaker functions. T. Imamura, M. Mucciconi, and T. Sasamoto prove a restricted Cauchy identity relating $q$-Whittaker and skew Schur sums in [IMS21]. In a companion paper [IMS23], they use skew RSK dynamics, Greene invariants, and affine crystals to give bijective proofs of Cauchy and Littlewood identities involving $q$-Whittaker polynomials. Together with T. Scrimshaw, they later introduce skew-column RSK dynamics on pairs of skew semistandard tableaux [IMSS26]. The dynamics linearizes the box-ball system, carries an affine-crystal structure, and gives Greene-type invariants and transformed Hall–Littlewood Cauchy and Littlewood identities.

P. D. Francesco and H. T. Vu derive rank recursions for $q$-Whittaker and Macdonald difference operators [FV26]. Powers of the $q$-Whittaker operator are expressed through a $q$-binomial distribution, while the Macdonald recursion follows from a Cauchy-determinant identity.

C. Lenart and J. Sidoli [LS21] compare three type $A$ Whittaker formulas: an alcove-walk formula, a filling formula of Haglund–Haiman–Loehr type, and Tokuyama’s formula in terms of semistandard Young tableaux. Their comparison is organized by compression maps between the indexing objects.

S. Korotkikh [Kor22] later gives a representation theoretic interpretation and further interpolation properties for this family. Finally, T. V. Ratheesh [Rat24] constructs bijections between combinatorial models for the monomial expansions of $q$-Whittaker and modified Hall–Littlewood polynomials, preserving the relevant content and major-index statistics.

A. Bhattacharya explains the equality between the Haglund–Haiman–Loehr inv formula and the Ayyer–Mandelshtam–Martin quinv formula for $q$-Whittaker functions and modified Hall–Littlewood functions [Bha25]. The proof uses weighted Dyck path symmetric functions of E. Carlsson and A. Mellit; the two Dyck paths attached to a partition are related by the $\zeta$ map, its inverse, and path reversal [Cor. 2.5 and Thm. 2.8, Bha25]. T. Basu and A. Bhattacharya give a standard-tableau formula for the Garsia–Remmel $q$-rook numbers [BB25]. They relate this formula to the coefficients appearing in $q$-Whittaker expansions of unicellular LLT functions.

A. Bhattacharya, T. V. Ratheesh, and S. Viswanath compare the column-strict filling model for $q$-Whittaker polynomials with partition-overlaid patterns indexing the Chari–Loktev basis of local Weyl modules [BRV24]. They construct weight-preserving bijections between these models which are compatible with projection, branching, and direct limits, and reinterpret the resulting data using coloured lattice paths [Thm. 2, BRV24].

#Subspace profiles over finite fields

A recent application of the $q$-Whittaker functions appears in [Ram26], which solves a finite-field enumeration problem posed by E. Bender, R. Coley, D. Robbins, and H. Rumsey [BCRR92]. Let $\Delta$ be an $n \times n$ matrix over the finite field $\mathbb{F}_q.$ A subspace $W \subseteq \mathbb{F}_q^n$ has $\Delta$-profile $\mu=(\mu_1,\mu_2,\dotsc)$ if \[\dim(W+\Delta W+\cdots+\Delta^{j-1}W)=\mu_1+\cdots+\mu_j \qquad (j \geq 1).\] Let $\sigma(\mu,\Delta)$ be the number of subspaces with $\Delta$-profile $\mu.$

The diagonalizable case was treated by S. Ram and M. J. Schlosser [RS26]. Their solution leads to polynomials $b_{\mu,\nu}(q)$ with positive semistandard-tableau formulas, connections to set partition statistics, and a $q$-rook-theoretic interpretation. They also express the Touchard–Riordan crossing polynomial for chord diagrams in terms of $q$-Whittaker functions.

The key symmetric function attached to $\Delta$ is the invariant flag generating function $F_\Delta(\xvec).$ Suppose the $\mathbb{F}_q[t]$-module defined by the action of $\Delta$ on $\mathbb{F}_q^n$ decomposes as \[\mathbb{F}_q^n \cong \bigoplus_{i=1}^r \bigoplus_{j=1}^{\ell_i} \frac{\mathbb{F}_q[t]}{(g_i^{\lambda^i_j})},\] where the $g_i$ are distinct monic irreducible polynomials of degrees $d_i$ and $\lambda^i=(\lambda^i_1,\lambda^i_2,\dotsc)$ are partitions. Then \[F_\Delta(\xvec)= \prod_{i=1}^r \hallLittlewoodH_{\lambda^i}(x_1^{d_i},x_2^{d_i},\dotsc;q^{d_i}) = \prod_{i=1}^r \powerSum_{d_i}\circ \hallLittlewoodH_{\lambda^i}(\xvec;q^{d_i}).\] Thus the primary decomposition data of $\Delta,$ equivalently the similarity class type data of J. A. Green [Gre55], is packaged into a symmetric function built from modified Hall–Littlewood functions.

For example, a matrix with irreducible characteristic polynomial gives $F_\Delta(\xvec)=\powerSum_n(\xvec)$; an elementary nilpotent Jordan block of size $n$ gives $F_\Delta(\xvec)=\completeH_n(\xvec)$; a nilpotent matrix of Jordan type $\lambda$ gives $F_\Delta(\xvec)=\hallLittlewoodH_\lambda(\xvec;q)$; and a matrix diagonalizable over $\mathbb{F}_q$ with eigenspace dimensions $\nu_1,\dotsc,\nu_r$ gives \[F_\Delta(\xvec)=\prod_{i=1}^r \qWhittaker_{(\nu_i)}(\xvec;q),\] a product of single-row $q$-Whittaker functions.

Let $\widetilde{\qWhittaker}_\lambda(\xvec;q)$ denote the basis dual to the $q$-Whittaker basis with respect to the Hall scalar product $\langle -,- \rangle.$ The main theorem of [Ram26] is \[\sigma(\mu,\Delta) = (-1)^{\sum_{j \geq 2}\mu_j} q^{\sum_{j \geq 2}\binom{\mu_j}{2}} \bigl\langle F_\Delta(\xvec), \widetilde{\qWhittaker}_\mu(\xvec;q)\completeH_{n-|\mu|}(\xvec) \bigr\rangle.\] When $|\mu|=n,$ this gives a direct finite-field interpretation of the coefficients in the $q$-Whittaker expansion of $F_\Delta(\xvec).$ In particular, the coefficients in the $q$-Whittaker expansions of $\powerSum_n,$ $\completeH_n,$ products of modified Hall–Littlewood functions, and products of one-row $q$-Whittaker functions count subspaces, up to sign and a power of $q,$ with specified profiles for suitable operators.

For the special case $F_\Delta(\xvec)=\powerSum_n(\xvec),$ the operator $\Delta$ is simple, equivalently its characteristic polynomial is irreducible. S. Ram gives a symmetric-function-free proof of the corresponding subspace count, thereby rederiving the $q$-Whittaker expansion coefficients of $\powerSum_n$ [Ram24]. This also gives another route to the Chen–Tseng solution of Niederreiter’s splitting-subspace problem [Cor. 2.9, Ram24].

The paper also applies this perspective to Krylov subspace methods. These methods have a long history going back to Lagrange, Euler, Gauss, Hilbert, and von Neumann, among others; see [LS13] for an overview. For an ordered set $S=\{v_1,\dotsc,v_k\}\subseteq \mathbb{F}_q^n,$ the truncated Krylov subspace of order $\ell$ generated by $S$ is \[\operatorname{Kry}(\Delta,S,\ell) = \left\{ \sum_{i=1}^k f_i(\Delta)v_i : f_i\in \mathbb{F}_q[x],\ \deg f_i\lt{}\ell \right\}.\] [Thm. 7.1, Ram26] gives a Hall scalar-product formula involving dual $q$-Whittaker functions for the probability that a randomly chosen ordered set $S$ of size $k$ satisfies $\operatorname{Kry}(\Delta,S,\ell)=\mathbb{F}_q^n.$

#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. I. Corwin, N. O'Connell, T. Seppäläinen, and N. Zygouras relate tropical combinatorics, geometric RSK, and Whittaker functions [COSZ14]. R. Chhaibi develops the Littelmann path model for geometric crystals and connects it with Whittaker functions and Brownian motion [Chh13].

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[LS21]Cristian Lenart and James Sidoli. Relating three combinatorial formulas for type A Whittaker functions. arXiv:2109.12908, 2021.

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@article{LenartSidoli2021x,
  author = {Cristian Lenart and James Sidoli},
  title = {Relating three combinatorial formulas for type {A} {W}hittaker functions},
  year = {2021},
  eprint = {2109.12908},
  url = {https://arxiv.org/abs/2109.12908},
  journal = {arXiv e-prints}
}

[LS13]Jörg Liesen and Zdeněk Strakoš. Krylov subspace methods: Principles and analysis. Numerical mathematics and scientific computation. Oxford University Press, 2013.

.bib

@book{LiesenStrakos2013,
  author    = {J{\"o}rg Liesen and Zden{\v e}k Strako{\v s}},
  title     = {Krylov Subspace Methods: Principles and Analysis},
  publisher = {Oxford University Press},
  year      = {2013},
  series    = {Numerical Mathematics and Scientific Computation},
  isbn      = {978-0-19-965541-0}
}

[MP22]Matteo Mucciconi and Leonid Petrov. Spin q-Whittaker polynomials and deformed quantum Toda. Communications in Mathematical Physics, 389(3):1331–1416, 2022.

.bib

@article{MucciconiPetrov2022,
  author = {Mucciconi, Matteo and Petrov, Leonid},
  title = {Spin q-{W}hittaker polynomials and deformed quantum {T}oda},
  year = {2022},
  journal = {Communications in Mathematical Physics},
  volume = {389},
  number = {3},
  pages = {1331--1416},
  publisher = {Springer Science and Business Media LLC},
  doi = {10.1007/s00220-021-04279-5},
  url = {http://dx.doi.org/10.1007/s00220-021-04279-5},
  issn = {1432-0916}
}

[RMNC17]G. Rahman, S. Mubeen, K. S. Nisar and J. Choi. $(p,q)$-Whittaker function and associated properties and formulas. arXiv:1710.07196, 2017.

.bib

@article{RahmanMubeenNisarChoi2017x,
  author = {G. Rahman and S. Mubeen and K. S. Nisar and J. Choi},
  title = {$(p,q)$-{W}hittaker function and associated properties and
    formulas},
  year = {2017},
  eprint = {1710.07196},
  url = {https://arxiv.org/abs/1710.07196},
  journal = {arXiv e-prints}
}

[Ram24]Samrith Ram. Simple operators and $q$-Whittaker coefficients of power sum symmetric functions. arXiv:2411.16485, 2024.

.bib

@article{Ram2024x,
  author = {Samrith Ram},
  title = {Simple operators and $q$-{W}hittaker coefficients of power sum symmetric functions},
  year = {2024},
  eprint = {2411.16485},
  url = {https://arxiv.org/abs/2411.16485},
  journal = {arXiv e-prints}
}

[Ram26]Samrith Ram. Subspace profiles over finite fields and $q$-Whittaker expansions of symmetric functions. Forum of Mathematics, Sigma, 14:e48, 2026.

.bib

@article{Ram2026,
  author  = {Samrith Ram},
  title   = {Subspace profiles over finite fields and $q$-{W}hittaker expansions of symmetric functions},
  journal = {Forum of Mathematics, Sigma},
  volume  = {14},
  year    = {2026},
  pages   = {e48},
  doi     = {10.1017/fms.2026.10193},
  url     = {https://doi.org/10.1017/fms.2026.10193}
}

[RS26]Samrith Ram and Michael J. Schlosser. Diagonal operators, $q$-Whittaker functions and rook theory. Combinatorial Theory, 5(4), 2026.

.bib

@article{RamSchlosser2026,
  author = {Ram, Samrith and Schlosser, Michael J.},
  title = {Diagonal operators, $q$-{W}hittaker functions and rook theory},
  year = {2026},
  journal = {Combinatorial Theory},
  volume = {5},
  number = {4},
  publisher = {California Digital Library (CDL)},
  doi = {10.5070/c65465677},
  url = {http://dx.doi.org/10.5070/C65465677},
  issn = {2766-1334}
}

[Rat24]T. V. Ratheesh. Bijections between different combinatorial models for $q$-Whittaker and modified Hall-Littlewood polynomials. arXiv:2401.07481, 2024.

.bib

@article{Ratheesh2024x,
  author = {T. V. Ratheesh},
  title = {Bijections between different combinatorial models for $q$-{W}hittaker and modified {H}all-{L}ittlewood polynomials},
  year = {2024},
  eprint = {2401.07481},
  url = {https://arxiv.org/abs/2401.07481},
  journal = {arXiv e-prints}
}

[Uhl19]Joakim Uhlin. Combinatorics of Macdonald polynomials and cyclic sieving. TRITA-SCI-GRU. KTH, Mathematics (Div.), 2019.

.bib

@mastersthesis{Uhlin2019,
   author = {Joakim Uhlin},
   school = {KTH, Mathematics (Div.)},
   title = {Combinatorics of {M}acdonald polynomials and cyclic sieving},
   series = {TRITA-SCI-GRU},
   type = {M.S Thesis},
   url = {http://kth.diva-portal.org/smash/record.jsf?pid=diva2%3A1282825},
   urn = {urn:nbn:se:kth:diva-241919},
   number = {2019:008},
   year = {2019},
   month = jan
}

The symmetric functions catalog