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Interpolation Macdonald P polynomials

The interpolation Macdonald polynomials (in type $A$) is 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 [Oko98a]. Let $\lambda \vdash n.$ The interpolation Macdonald polynomial $\macdonaldP^*_\lambda(\xvec;q,t)$ is the unique (up to scalar) polynomial of of degree $n$ that is symmetric in $x_1t^{n-1},x_2 t^{n-2},\dotsc,x_n,$ and fulfills

\[ \macdonaldP^*_\lambda(q^{\mu_1},q^{\mu_2},\dotsc,q^{\mu_n};q,t) = 0 \text{ whenever } \lambda \not\subset \mu. \]

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 [Oko98b] 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 appear in the formula for the classical Macdonald P polynomials $\macdonaldP(\xvec;q,t).$

Type BC interpolation Macdonald polynomials

In [Oko98a], 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.