Interpolation Macdonald P polynomials

2026-07-04 Improve this page

#Interpolation Macdonald P polynomials

The interpolation Macdonald polynomials (in type $A$) are a family of non-homogeneous polynomials which generalize the shifted Schur polynomials. They are also known under the name shifted Macdonald polynomials or inhomogeneous Macdonald polynomials.

This definition is taken from [Oko98]. Let $\lambda \vdash n.$ The interpolation Macdonald polynomial $\macdonaldP^*_\lambda(\xvec;q,t)$ is the unique (up to scalar) polynomial of degree $n$ that is symmetric in $x_1t^{n-1},x_2 t^{n-2},\dotsc,x_n,$ and satisfies \[\macdonaldP^*_\lambda(q^{\mu_1},q^{\mu_2},\dotsc,q^{\mu_n};q,t) = 0 \text{ whenever } \lambda \not\subset \mu.\]

The shifted Schur functions obtained as the limit \[\schurS^*_\lambda(\xvec) = \lim_{q \to 1} \frac{\macdonaldP^*_\lambda(q^{x_1},\dotsc,q^{x_n} ;q,q)}{ (q-1)^n }.\] Moreover, the shifted Jack polynomials are obtained as the limit \[\jackShifted_\mu(\xvec;a) = \lim_{q \to 1} \frac{\macdonaldP^*_\lambda(q^{x_1},\dotsc,q^{x_n} ;q,q^a)}{ (q-1)^n }.\]

#RSSYT formula

In [Oko98] A. Okounkov proves the following combinatorial formula for the interpolation Macdonald polynomials: \[\macdonaldP^*_\mu(\xvec;q,t) = \sum_T \psi_{T}(q,t) \prod_{\square \in \mu} t^{1-T(\square)} \left( \xvec_{T(\square)} - q^{\arm'(\square)} t^{-\leg'(\square) } \right)\] Here, the sum is over all reverse-tableaux of shape $\mu,$ and $\psi_{T}(q,t)$ is the same weight which appears in the formula for the classical Macdonald $P$ polynomials $\macdonaldP(\xvec;q,t).$

H. B. Dali and L. Williams give a combinatorial formula for interpolation Macdonald polynomials in terms of signed multiline queues [DW25]. Their formula generalizes the multiline-queue formula for ordinary Macdonald polynomials of S. Corteel, O. Mandelshtam, and L. Williams.

#Type BC interpolation Macdonald polynomials

In [Oko98], a type BC family is introduced. These are polynomials in the variables $x_1^{\pm},\dotsc,x_n^{\pm},$ with coefficients in $\setQ(q,t,s).$ Moreover, as $s\to \infty,$ the polynomials $\macdonaldP^*_\lambda(\xvec;q,t)$ are recovered.

Bibliography

[DW25]Houcine Ben Dali and Lauren Williams. A combinatorial formula for Interpolation Macdonald polynomials. arXiv:2510.02587, 2025.

.bib

@article{DaliWilliams2025x,
  author = {Houcine Ben Dali and Lauren Williams},
  title = {A combinatorial formula for {I}nterpolation {M}acdonald
    polynomials},
  year = {2025},
  eprint = {2510.02587},
  url = {https://arxiv.org/abs/2510.02587},
  journal = {arXiv e-prints}
}

[Oko98]A. Okounkov. BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials. Transformation Groups, 3(2):181–207, June 1998.

.bib

@article{Okounkov1998,
  doi = {10.1007/bf01236432},
  url2 = {https://doi.org/10.1007/bf01236432},
  year = {1998},
  month = jun,
  publisher = {Springer Science and Business Media {LLC}},
  volume = {3},
  number = {2},
  pages = {181--207},
  author = {A. Okounkov},
  title = {{BC}-type interpolation {M}acdonald polynomials and binomial formula for {K}oornwinder polynomials},
  journal = {Transformation Groups}
}

[Oko98]Andrei Okounkov. (Shifted) Macdonald polynomials: $q$-integral representation and combinatorial formula. Compositio Mathematica, 112(2):147–182, 1998.

.bib

@article{Okounkov1998shifted,
  doi = {10.1023/a:1000436921311},
  url2 = {https://doi.org/10.1023/a:1000436921311},
  year = {1998},
  publisher = {Cambridge University Press ({CUP})},
  volume = {112},
  number = {2},
  pages = {147--182},
  title = {(Shifted) {M}acdonald polynomials: $q$-Integral representation and combinatorial formula},
  author = {Andrei Okounkov},
  journal = {Compositio Mathematica}
}