Koornwinder polynomials

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#Koornwinder polynomials

The Koornwinder polynomials, also called Macdonald–Koornwinder polynomials, are the Macdonald polynomials for the nonreduced affine root system $(C_n^\vee,C_n)$ [Koo92]. They form the universal classical-type Macdonald family: the Macdonald polynomials for every classical affine root system arise by specializing their parameters [CGR26]. For $n=1,$ they are the Askey–Wilson polynomials.

Unlike type $A$ Macdonald $P$ polynomials, Koornwinder polynomials are Laurent polynomials. They depend on six parameters, customarily written $q,t,a,b,c,d.$ Other common conventions use square-root parameters $q^{1/2},t^{1/2},t_0^{1/2},u_0^{1/2},t_n^{1/2},u_n^{1/2}.$

#Eigenfunction and triangularity definition

Let

\[W_{\mathrm{fin}}=\symS_n\ltimes(\setZ/2\setZ)^n\]

be the signed permutation group. It permutes the variables and replaces individual $x_i$ by $x_i^{-1}.$ For a partition $\lambda=(\lambda_1\geq\dotsb\geq\lambda_n\geq0),$ let

\[m_\lambda^{BC}(\xvec) \coloneqq \sum_{\alpha\in W_{\mathrm{fin}}\lambda}\xvec^\alpha\]

be the corresponding orbit sum, with each distinct orbit element included once.

The symmetric Koornwinder polynomial $P_\lambda^{\mathrm K}$ is the unique $W_{\mathrm{fin}}$-invariant Laurent polynomial which is triangular,

\[P_\lambda^{\mathrm K} = m_\lambda^{BC} +\sum_{\mu\lhd\lambda}c_{\lambda\mu}(q,t;a,b,c,d)m_\mu^{BC},\]

and is a joint eigenfunction of the commuting Koornwinder difference operators. Equivalently, the family is triangular and orthogonal for the Koornwinder inner product. This is the type $BC$ analogue of the eigenfunction characterization of Macdonald polynomials.

Example

The first orbit sums make the shape of the family visible: \[P_{\varnothing}^{\mathrm K}=1, \qquad P_{(1)}^{\mathrm K} =\sum_{i=1}^n(x_i+x_i^{-1})+c,\] where $c$ is a rational function of the six parameters. Similarly, the leading orbit of $P_{(1,1)}^{\mathrm K}$ is \[\sum_{1\leq i\lt{}j\leq n} (x_ix_j+x_ix_j^{-1}+x_i^{-1}x_j+x_i^{-1}x_j^{-1}).\] Lower orbit sums are determined by the eigenvalue equations.

#Electronic and relative polynomials

The nonsymmetric version is indexed by $\mu\in\setZ^n.$ Following the terminology of L. Colmenarejo and A. Ram, it is useful to call $E_\mu^{\mathrm K}$ the electronic Koornwinder polynomial and $P_\lambda^{\mathrm K}$ the bosonic Koornwinder polynomial; compare the corresponding terminology on the Macdonald page [CR22].

The electronic polynomial is characterized by \[Y_jE_\mu^{\mathrm K}=\gamma_{\mu,j}E_\mu^{\mathrm K}, \qquad [\xvec^\mu]E_\mu^{\mathrm K}=1,\] where $Y_1,\dotsc,Y_n$ are the commuting Cherednik operators for the double-affine Hecke algebra of type $(C_n^\vee,C_n),$ and the eigenvalues $\gamma_{\mu,j}$ are explicit monomials in the six parameters.

For a signed permutation $z\in W_{\mathrm{fin}},$ the relative Koornwinder polynomial $E_\mu^z$ is a normalized Hecke operator translate of $E_\mu^{\mathrm K},$ with leading monomial $\xvec^{z\mu}.$ These are the type $BC$ analogues of permuted-basement Macdonald polynomials; they also occur as open-boundary ASEP polynomials. For dominant $\lambda,$ symmetrization gives \[P_\lambda^{\mathrm K} =\sum_{w\in W_{\mathrm{fin}}}E_\lambda^w.\]

#Combinatorial formulas

L. Colmenarejo, L. Gagnon, and A. Ram give three parallel formulas for all relative Koornwinder polynomials [CGR26]:

a creation formula written with the divided-difference operators familiar from Schubert calculus;
an alcove-walk formula reparametrized by uncompressed set-valued tableaux; and
a compressed set-valued tableau formula obtained by across-the-$0$-gap and around-the-end compression.

The relative family is essential here: it plays the same role as a variable basement in type $A.$ The box-greedy reduced word and its coroot sequence replace the arm-and-leg data that organize the usual Macdonald tableau formula.

S. Corteel, O. Mandelshtam, and L. Williams give a different formula using rhombic staircase tableaux [CMW24]. The precise combinatorial relation between the compressed set-valued tableaux and the rhombic staircase tableaux is currently open.

#Specializations and nearby families

For $n=1,$ $P_{(r)}^{\mathrm K}$ is an Askey–Wilson polynomial.
Classical-type Macdonald polynomials are obtained by parameter specialization.
Hall–Littlewood limits of types $B$ and $C$ arise from the corresponding $q=0$ specializations.
Relative Koornwinder polynomials specialize to families related to the open-boundary asymmetric exclusion process.

See the pages on root systems, Coxeter groups, and nonsymmetric Macdonald polynomials for the surrounding type-$A$ and Weyl-group background.

Bibliography

[CGR26]Laura Colmenarejo, Lucas Gagnon and Arun Ram. Formulas for Koornwinder polynomials. arXiv:2608.02810v1, 2026.

.bib

@article{ColmenarejoGagnonRam2026x,
  author = {Laura Colmenarejo and Lucas Gagnon and Arun Ram},
  title = {Formulas for {K}oornwinder polynomials},
  year = {2026},
  eprint = {2608.02810v1},
  url = {https://arxiv.org/abs/2608.02810v1},
  journal = {arXiv e-prints}
}

[CR22]Laura Colmenarejo and Arun Ram. C-functions and Macdonald polynomials. arXiv:2212.03312, 2022.

.bib

@article{ColmenarejoRam2022x,
Author = {Laura Colmenarejo and Arun Ram},
Title = {c-functions and {M}acdonald polynomials},
Year = {2022},
Eprint = {2212.03312},
  url = {https://arxiv.org/abs/2212.03312},
journal = {arXiv e-prints}
}

[CMW24]Sylvie Corteel, Olya Mandelshtam and Lauren Williams. Rhombic staircase tableaux and Koornwinder polynomials. Mathematische Zeitschrift, 308(3), 2024.

.bib

@article{CorteelMandelshtamWilliams2024,
  author = {Corteel, Sylvie and Mandelshtam, Olya and Williams, Lauren},
  title = {Rhombic staircase tableaux and {K}oornwinder polynomials},
  year = {2024},
  journal = {Mathematische Zeitschrift},
  volume = {308},
  number = {3},
  publisher = {Springer Science and Business Media LLC},
  doi = {10.1007/s00209-024-03596-4},
  url = {http://dx.doi.org/10.1007/s00209-024-03596-4},
  issn = {1432-1823},
  eprint = {2312.17469}
}

[Koo92]Tom H. Koornwinder. Askey–wilson polynomials for root systems of type $BC$. Hypergeometric functions on domains of positivity, jack polynomials, and applications, 138:189–204, 1992.

.bib

@incollection{Koornwinder1992BC,
  author = {Tom H. Koornwinder},
  title = {Askey--Wilson polynomials for root systems of type {$BC$}},
  booktitle = {Hypergeometric functions on domains of positivity, Jack
    polynomials, and applications},
  series = {Contemporary Mathematics},
  volume = {138},
  pages = {189--204},
  publisher = {American Mathematical Society},
  address = {Providence, RI},
  year = {1992},
  mrnumber = {1199128}
}