Interpolation Macdonald Polynomials

• Interpolation Macdonald polynomials are inhomogeneous analogues of Macdonald polynomials defined by precise interpolation and vanishing conditions on q,t-grids, with the leading term replicating the ordinary Macdonald form.
• They are constructed through diverse methodologies including combinatorial tableau expansions, Hecke algebra actions, and recursive operators, ensuring stability and compatibility with symmetric function theory.
• These polynomials unify various aspects of symmetric functions, linking operator theory, probabilistic models, and representation theory through explicit biorthogonality and commuting q-difference operators.

Interpolation Macdonald polynomials are inhomogeneous analogues of Macdonald polynomials in which orthogonality is replaced, at the level of definition, by interpolation and vanishing conditions on special $q,t$-grids. In type $A$, their top homogeneous components are the ordinary Macdonald polynomials, while lower-degree terms are fixed by prescribed zeros; finite-variable versions were studied by Knop, Okounkov, and Sahi and admit stable lifts to symmetric functions in infinitely many variables (Olshanski, 2017). Closely related theories exist for nonsymmetric polynomials indexed by compositions, for $BC$-type interpolation Laurent polynomials, and for more recent wreath variants (Baratta, 2012, Koornwinder, 2014, Romero et al., 3 May 2025). The literature uses several normalizations and notations—among them $I_\mu$, $P_\lambda^*$, $J_\lambda^{(q,t)}$, $E_\eta^*$, and $G_\alpha$—but the common principle is that the interpolation object is inhomogeneous, is characterized by vanishing at a finite collection of spectral points, and has the ordinary Macdonald polynomial as its highest-degree part (Olshanski, 2017).

1. Defining characterization and normalizations

For a partition $\mu$ of length at most $N$, one type-$A$0 finite-variable interpolation Macdonald polynomial is characterized by

$A$1

together with

$A$2

where

$A$3

A crucial strengthening is the extra vanishing property

$A$4

(Olshanski, 2017). In the shifted notation used elsewhere, the type-$A$5 symmetric interpolation polynomial $A$6 is the unique $A$7-invariant, $A$8-monic polynomial of degree $A$9 such that

$BC$0

and it also satisfies the stronger condition

$BC$1

(Koornwinder, 2014).

The same pattern persists under other normalizations. Sahi’s interpolation Macdonald polynomials $BC$2 are the unique symmetric polynomials of degree $BC$3 whose top-degree homogeneous component is the ordinary Macdonald polynomial $BC$4 and which vanish at the interpolation points

$BC$5

for every partition $BC$6 with $BC$7 (Dołęga, 2016). In the 2024 notation $BC$8, one also distinguishes unital and monic normalizations; the unital normalization satisfies $BC$9, while the monic normalization is fixed by the top homogeneous term (Chen et al., 2024).

Nonsymmetric interpolation Macdonald polynomials are indexed by compositions rather than partitions. One family, $I_\mu$0, is inhomogeneous and is generated recursively from $I_\mu$1 (Baratta, 2012). Another family, $I_\mu$2, is characterized as the unique polynomial of degree $I_\mu$3 with leading monomial coefficient $I_\mu$4 and vanishing on all interpolation points of smaller or equal total degree except the point indexed by $I_\mu$5 itself (Sahi et al., 2019). In $I_\mu$6-type, Okounkov’s interpolation Macdonald polynomials are $I_\mu$7-invariant Laurent polynomials $I_\mu$8 characterized by

$I_\mu$9

with $P_\lambda^*$0 a generic parameter (Koornwinder, 2014).

2. Finite-variable theory, stability, and commuting operators

The finite-variable interpolation polynomials admit a stable passage to symmetric functions. A key property is the shifted stability

$P_\lambda^*$1

when $P_\lambda^*$2, and zero otherwise (Olshanski, 2017). This is not the ordinary specialization $P_\lambda^*$3; the stable theory is built from the shifted specialization $P_\lambda^*$4, and this distinction governs the projective limit construction (Cuenca, 2017). The resulting interpolation Macdonald symmetric functions $P_\lambda^*$5 form an inhomogeneous basis of the algebra of symmetric functions over $P_\lambda^*$6 (Cuenca, 2017).

The finite theory is diagonalized by commuting $P_\lambda^*$7-difference operators. Okounkov’s hierarchy

$P_\lambda^*$8

satisfies

$P_\lambda^*$9

so the coefficients $J_\lambda^{(q,t)}$0 commute pairwise (Cuenca, 2017). After renormalization, one obtains operators $J_\lambda^{(q,t)}$1 with the cleaner eigenrelation

$J_\lambda^{(q,t)}$2

Because the eigenvalue equals $J_\lambda^{(q,t)}$3 for $J_\lambda^{(q,t)}$4, these operators are compatible with shifted stability and define a projective-limit operator $J_\lambda^{(q,t)}$5 satisfying

$J_\lambda^{(q,t)}$6

(Cuenca, 2017).

The explicit description of the coefficients in

$J_\lambda^{(q,t)}$7

is one of the structural results of the operator theory. The operators $J_\lambda^{(q,t)}$8 are expressed through Hall–Littlewood data and a new family of inhomogeneous Hall–Littlewood functions; moreover, those inhomogeneous Hall–Littlewood functions are identified as the $J_\lambda^{(q,t)}$9 degeneration of interpolation Macdonald functions with inverted parameters (Cuenca, 2017). This places interpolation Macdonald theory in direct analogy with the ordinary Macdonald “operators at infinity” framework of Nazarov and Sklyanin, but in an inhomogeneous setting.

3. Combinatorial, Hecke-theoretic, and recursive constructions

A principal explicit formula in the symmetric theory is the Okounkov–Sahi tableau expansion

$E_\eta^*$0

where $E_\eta^*$1 denotes reverse semistandard tableaux of shape $E_\eta^*$2 with entries in $E_\eta^*$3 (Olshanski, 2017). In the shifted notation $E_\eta^*$4, analogous tableau formulas are available in both type $E_\eta^*$5 and $E_\eta^*$6, with the $E_\eta^*$7-type factors exhibiting the symmetry under $E_\eta^*$8 (Koornwinder, 2014).

Nonsymmetric interpolation polynomials admit a direct recursive generation mechanism. The family $E_\eta^*$9 is generated from $G_\alpha$0 by a Hecke-type switching operator

$G_\alpha$1

and a raising operator

$G_\alpha$2

with

$G_\alpha$3

(Baratta, 2012). The combinatorial input is a minimal operator sequence that constructs the indexing composition from $G_\alpha$4; the polynomial recursion follows the same sequence (Baratta, 2012).

The nonsymmetric theory also contains several equivalent versions. The three families $G_\alpha$5, $G_\alpha$6, and $G_\alpha$7 are related by explicit double affine Hecke algebra operators, and the normalized family satisfies the duality theorem

$G_\alpha$8

for $G_\alpha$9 (Sahi et al., 2019). The same paper derives binomial and dual-binomial formulas for these normalized polynomials, recovering symmetric duality by symmetrization (Sahi et al., 2019). In the $\mu$0 setting, nonsymmetric interpolation Laurent polynomials $\mu$1 are defined on a signed $\mu$2-grid, and the symmetric Okounkov polynomials arise by $\mu$3-symmetrization: $\mu$4 (Disveld et al., 2018).

A more recent combinatorial model replaces tableaux by signed multiline queues. For interpolation ASEP polynomials $\mu$5 and symmetric interpolation Macdonald polynomials $\mu$6, one has

$\mu$7

and, in the signed multiline queue model,

$\mu$8

(Dali et al., 2 Oct 2025). This gives a combinatorial formula in which the inhomogeneous corrections are encoded by signed rows, sign-sensitive weights, and modified pairing rules.

4. Biorthogonality, Cauchy kernels, and shifted-symmetric realizations

Ordinary Macdonald theory is governed by the scalar product $\mu$9 under which $N$0 and $N$1 are dual: $N$2 Interpolation Macdonald functions are not orthogonal in this scalar product, but they do admit a biorthogonal dual basis. Specifically, one defines $N$3 by

$N$4

with $N$5 living in a completion of the symmetric-function algebra and satisfying

$N$6

(Olshanski, 2017).

The resulting Cauchy-type identity has exactly the classical Macdonald kernel: $N$7 Thus the replacement

$N$8

preserves the reproducing kernel while changing orthogonality into biorthogonality (Olshanski, 2017). A modified dual family $N$9 is obtained by inverting variables and multiplying by $A$00, so that the duals become formal power series and the Cauchy identity acquires a fully formal version (Olshanski, 2017).

The shifted-symmetric interpretation provides another bridge between interpolation theory and ordinary Macdonald polynomials. Shifted Macdonald polynomials $A$01 are characterized by shifted symmetry, degree $A$02, normalization at $A$03, and vanishing on diagrams $A$04 not containing $A$05 (Dali et al., 2024). In the operator formalism built around the creation operator $A$06, one has

$A$07

which gives an explicit realization of the isomorphism between ordinary and shifted symmetric functions (Dali et al., 2024). The same framework defines Macdonald characters as

$A$08

with $A$09 forming a basis of shifted symmetric functions (Dali et al., 2024).

5. Degenerations, factorization, and positivity phenomena

Interpolation Macdonald polynomials admit several standard degenerations. In the Jack limit

$A$10

the interpolation Macdonald functions degenerate to interpolation Jack polynomials: $A$11 and the dual functions $A$12 admit a parallel limit (Olshanski, 2017). In the $A$13-type theory, the $A$14 limit of Okounkov’s $A$15-type interpolation Macdonald polynomials yields $A$16-type interpolation Jack polynomials, including the explicit tableau formula

$A$17

(Koornwinder, 2014). Other specializations discussed in the symmetric-function setting include $A$18, giving $A$19-Whittaker-type interpolation functions, $A$20, giving Hall–Littlewood-type limits, and the $A$21 degeneration to inhomogeneous Hall–Littlewood functions used in the operator-at-infinity theory (Olshanski, 2017, Cuenca, 2017).

The regime $A$22 is also the setting of a strong factorization theorem. For partitions $A$23, write $A$24. Then the interpolation Macdonald polynomials satisfy

$A$25

equivalently, their cumulants are $A$26 (Dołęga, 2016). Passing to the top homogeneous component transfers the same small-cumulant property to ordinary Macdonald polynomials, which in turn yields integrality of multivariate $A$27-Kostka numbers (Dołęga, 2016).

Interpolation coefficients themselves exhibit a positive combinatorics. In the 2024 treatment, the generalized binomial coefficients are the evaluations

$A$28

and they satisfy weighted-sum formulas over chains of partitions, both in type $A$29 and in type $A$30 (Chen et al., 2024). The same work proves positivity and monotonicity: $A$31 precisely when $A$32, and if $A$33 then $A$34 (Chen et al., 2024). As applications, the paper derives inequalities for power sums, Jack polynomials, and their monomial, Schur, zonal, and elementary specializations (Chen et al., 2024).

6. Representation theory, probability, and higher-rank extensions

At the specialization $A$35, interpolation Macdonald polynomials become a basis adapted to the center of the quantum group $A$36. Under the Harish–Chandra isomorphism, central elements are determined by evaluations at the points

$A$37

and the specialized interpolation polynomials $A$38 are tailored to vanish unless $A$39 (Beliakova et al., 2021). Transporting them through Harish–Chandra produces central elements $A$40, while Okounkov’s inverse interpolation theorem yields a dual basis for the Hopf pairing. This is the mechanism behind cyclotomic expansions of $A$41 Reshetikhin–Turaev invariants, universal knot invariants, and explicit Kirby colors for integral homology $A$42-spheres (Beliakova et al., 2021).

A probabilistic interpretation appears at $A$43. For a partition $A$44, the interpolation $A$45-Push TASEP is a Markov chain on the reorderings $A$46, and its stationary distribution is

$A$47

Thus the steady-state weight of a state $A$48 is the interpolation ASEP polynomial $A$49, and the partition function is the interpolation Macdonald polynomial $A$50 (Dali et al., 13 Feb 2026). The proof uses signed multiline queues and the factorization

$A$51

at $A$52 (Dali et al., 13 Feb 2026).

The interpolation paradigm also extends beyond type $A$53 and beyond ordinary symmetric functions. In the wreath setting for $A$54, the interpolation polynomial $A$55 is characterized by highest degree $A$56, vanishing at colored spectral points $A$57 for smaller partitions with the same core, and normalization $A$58 (Romero et al., 3 May 2025). The explicit formula

$A$59

shows that interpolation phenomena survive in a colored, core-sensitive setting controlled by quantum toroidal and shuffle algebra operators (Romero et al., 3 May 2025).

Taken together, these developments identify interpolation Macdonald polynomials as a unifying inhomogeneous layer inside Macdonald theory: they retain Macdonald leading terms, replace orthogonality by spectral vanishing, support dual Cauchy kernels and commuting operator hierarchies, and interact naturally with Hecke algebras, quantum groups, probabilistic models, and higher-rank generalizations (Olshanski, 2017).