---
title: Interpolation Macdonald Polynomials
url: https://www.emergentmind.com/topics/interpolation-macdonald-polynomials
type: topic
---

# Interpolation Macdonald Polynomials

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 [1712.08018]. Closely related theories exist for nonsymmetric polynomials indexed by compositions, for \(BC\)-type interpolation Laurent polynomials, and for more recent wreath variants [1201.4450] [1408.5993] [2505.01732]. 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 [1712.08018].

## 1. Defining characterization and normalizations

For a partition \(\mu\) of length at most \(N\), one type-\(A\) finite-variable interpolation Macdonald polynomial is characterized by
\[
\deg I_\mu=|\mu|,\qquad
I_\mu(X_N(\lambda);q,t)=0 \quad \text{for } \lambda\neq \mu,\ |\lambda|\le |\mu|,
\]
together with
\[
I_\mu(x_1,\dots,x_N;q,t)=P_\mu(x_1,\dots,x_N;q,t)+\text{lower degree terms},
\]
where
\[
X_N(\lambda)_i=q^{-\lambda_i}t^{\,i-1}.
\]
A crucial strengthening is the extra vanishing property
\[
I_\mu(X_N(\lambda);q,t)=0 \quad \text{if the diagram }\lambda\text{ does not contain }\mu
\]
[1712.08018]. In the shifted notation used elsewhere, the type-\(A\) symmetric interpolation polynomial \(P_\lambda^*(x;q,t)\) is the unique \(S_n\)-invariant, \(\lambda\)-monic polynomial of degree \(|\lambda|\) such that
\[
P_\lambda^*(q^\mu t^\delta;q,t)=0
\quad\text{for every partition }\mu\text{ with }|\mu|<|\lambda|,
\]
and it also satisfies the stronger condition
\[
P_\lambda^*(q^\mu t^\delta;q,t)=0 \quad\text{if }\mu\not\supseteq\lambda
\]
[1408.5993].

The same pattern persists under other normalizations. Sahi’s interpolation Macdonald polynomials \(J_\lambda^{(q,t)}(\mathbf{x})\) are the unique symmetric polynomials of degree \(|\lambda|\) whose top-degree homogeneous component is the ordinary Macdonald polynomial \(J_\lambda^{(q,t)}\) and which vanish at the interpolation points
\[
\widetilde{\mu}=\bigl(q^{\mu_1}t^{N-1},\,q^{\mu_2}t^{N-2},\,\dots,\,q^{\mu_N}\bigr)
\]
for every partition \(\mu\neq \lambda\) with \(|\mu|\le |\lambda|\) [1609.09686]. In the 2024 notation \(h_\mu\), one also distinguishes unital and monic normalizations; the unital normalization satisfies \(h_\mu(\mu)=1\), while the monic normalization is fixed by the top homogeneous term [2403.02490].

Nonsymmetric interpolation Macdonald polynomials are indexed by compositions rather than partitions. One family, \(E_\eta^*(z;q,t)\), is inhomogeneous and is generated recursively from \(E^*_{(0,\ldots,0)}=1\) [1201.4450]. Another family, \(G_\alpha(x;q,t)\), is characterized as the unique polynomial of degree \(\le |\alpha|\) with leading monomial coefficient \(1\) and vanishing on all interpolation points of smaller or equal total degree except the point indexed by \(\alpha\) itself [1901.04852]. In \(BC_n\)-type, Okounkov’s interpolation Macdonald polynomials are \(W_n\)-invariant Laurent polynomials \(P_\lambda^*(x;q,t,a)\) characterized by
\[
P_\lambda^*(q^\mu t^\delta a;q,t,a)=0 \quad \text{if }\mu\not\supseteq \lambda,
\]
with \(a\) a generic parameter [1408.5993].

## 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
\[
I_\mu(x_1,\dots,x_N,t^N;q,t)=I_\mu(x_1,\dots,x_{N-1};q,t)
\]
when \(\mu\in\mathbb Y(N-1)\), and zero otherwise [1712.08018]. This is not the ordinary specialization \(x_N=0\); the stable theory is built from the shifted specialization \(x_N=t^N\), and this distinction governs the projective limit construction [1712.08014]. The resulting interpolation Macdonald symmetric functions \(I_\mu(\cdot;q,t)\) form an inhomogeneous basis of the algebra of symmetric functions over \(\mathbb Q(q,t)\) [1712.08014].

The finite theory is diagonalized by commuting \(q\)-difference operators. Okounkov’s hierarchy
\[
D_N(z;q,t)=1+\sum_{k=1}^N D_N^k z^k
\]
satisfies
\[
D_N(z;q,t)I_{\mu|N}=\prod_{i=1}^N\bigl(1+q^{\mu_i}t^{1-i}z\bigr)\,I_{\mu|N},
\]
so the coefficients \(D_N^1,\dots,D_N^N\) commute pairwise [1712.08014]. After renormalization, one obtains operators \(A_N(u;q,t)\) with the cleaner eigenrelation
\[
A_N(u;q,t)\,I_{\mu|N}
=
\prod_{i=1}^N \frac{q^{\mu_i}-t^{i-1}u}{1-t^{i-1}u}\,I_{\mu|N}.
\]
Because the eigenvalue equals \(1\) for \(\mu_i=0\), these operators are compatible with shifted stability and define a projective-limit operator \(A_\infty(u;q,t)\) satisfying
\[
A_\infty(u;q,t)I_\mu(\cdot;q,t)
=
\prod_{i=1}^\infty \frac{q^{\mu_i}-t^{i-1}u}{1-t^{i-1}u}\,I_\mu(\cdot;q,t)
\]
[1712.08014].

The explicit description of the coefficients in
\[
A_\infty(u;q,t)=1+\frac{A^1}{(u;t)_1}+\frac{A^2}{(u;t)_2}+\cdots
\]
is one of the structural results of the operator theory. The operators \(A^k\) 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 \(q\to 0\) degeneration of interpolation Macdonald functions with inverted parameters [1712.08014]. 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
\[
I_\mu(x_1,\dots,x_N;q,t)
=
\sum_{T\in \operatorname{RTab}(\mu,N)}
\psi_T(q,t)\prod_{(i,j)\in\mu}
\Bigl(x_{T(i,j)}-q^{\,1-j}t^{\,T(i,j)+i-2}\Bigr),
\]
where \(\operatorname{RTab}(\mu,N)\) denotes reverse semistandard tableaux of shape \(\mu\) with entries in \(\{1,\dots,N\}\) [1712.08018]. In the shifted notation \(P_\lambda^*(x;q,t)\), analogous tableau formulas are available in both type \(A\) and \(BC\), with the \(BC\)-type factors exhibiting the symmetry under \(x_i\mapsto x_i^{-1}\) [1408.5993].

Nonsymmetric interpolation polynomials admit a direct recursive generation mechanism. The family \(E_\eta^*(z;q,t)\) is generated from \(1\) by a Hecke-type switching operator
\[
H_i:=t+\frac{z_i-tz_{i+1}}{z_i-z_{i+1}}(s_i-1)
\]
and a raising operator
\[
\Phi_q^*:=(z_n-t^{\,n-1})\Delta,
\qquad
\Delta f(z_1,\ldots,z_n):=f(z_2/q,\ldots,z_n/q,z_1),
\]
with
\[
\Phi_q^*E_\eta^*(z)=q^{-\eta_1}E^*_{\Phi\eta}(z)
\]
[1201.4450]. The combinatorial input is a minimal operator sequence that constructs the indexing composition from \((0,\ldots,0)\); the polynomial recursion follows the same sequence [1201.4450].

The nonsymmetric theory also contains several equivalent versions. The three families \(G_\alpha\), \(G'_\alpha\), and \(K_\alpha\) are related by explicit double affine Hecke algebra operators, and the normalized family satisfies the duality theorem
\[
K_u(ae_v)=K_v(ae_u)
\]
for \(u,v\in \mathbb Z^n\) [1901.04852]. The same paper derives binomial and dual-binomial formulas for these normalized polynomials, recovering symmetric duality by symmetrization [1901.04852]. In the \(BC_n\) setting, nonsymmetric interpolation Laurent polynomials \(G_\alpha(x;q,\tau)\) are defined on a signed \(q\)-grid, and the symmetric Okounkov polynomials arise by \(W_n\)-symmetrization:
\[
R_\lambda(x;q,\tau)=\sum_{\alpha\in W_n\lambda} G_\alpha(x;q,\tau)
\]
[1808.01221].

A more recent combinatorial model replaces tableaux by signed multiline queues. For interpolation ASEP polynomials \(f_\mu^*\) and symmetric interpolation Macdonald polynomials \(P_\lambda^*\), one has
\[
P_\lambda^*(x;q,t)=\sum_{\mu\in S_n(\lambda)} f_\mu^*(x;q,t)
\]
and, in the signed multiline queue model,
\[
F_\mu^*=\sum_{Q\in \SMLQ(\mu)}\wt(Q),\qquad
Z_\lambda^*=\sum_{\mu\in S_n(\lambda)}F_\mu^*=P_\lambda^*
\]
[2510.02587]. 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 \(\langle\cdot,\cdot\rangle_{q,t}\) under which \(P_\mu\) and \(Q_\mu\) are dual:
\[
\langle P_\mu,Q_\nu\rangle_{q,t}=\delta_{\mu\nu}.
\]
Interpolation Macdonald functions are not orthogonal in this scalar product, but they do admit a biorthogonal dual basis. Specifically, one defines \(H_\mu(\cdot;q,t)\) by
\[
\langle I_\mu(\cdot;q,t),H_\nu(\cdot;q,t)\rangle_{q,t}=\delta_{\mu\nu},
\]
with \(H_\mu\) living in a completion of the symmetric-function algebra and satisfying
\[
H_\mu(y_1,y_2,\dots;q,t)=Q_\mu(y_1,y_2,\dots;q,t)+\text{higher degree terms}
\]
[1712.08018].

The resulting Cauchy-type identity has exactly the classical Macdonald kernel:
\[
\sum_{\mu\in\mathbb Y} I_\mu(x_1,x_2,\dots;q,t)\, H_\mu(y_1,y_2,\dots;q,t)
=
\prod_{i,j=1}^\infty
\frac{(x_i y_j t;q)_\infty}{(x_i y_j;q)_\infty}.
\]
Thus the replacement
\[
P_\mu \longrightarrow I_\mu,\qquad Q_\mu \longrightarrow H_\mu
\]
preserves the reproducing kernel while changing orthogonality into biorthogonality [1712.08018]. A modified dual family \(\widetilde H_\mu\) is obtained by inverting variables and multiplying by \(\prod_j (u_j^{-1};q)_\infty\), so that the duals become formal power series and the Cauchy identity acquires a fully formal version [1712.08018].

The shifted-symmetric interpretation provides another bridge between interpolation theory and ordinary Macdonald polynomials. Shifted Macdonald polynomials \(J_\mu^*\) are characterized by shifted symmetry, degree \(|\mu|\), normalization at \(\mu\), and vanishing on diagrams \(\lambda\) not containing \(\mu\) [2404.03904]. In the operator formalism built around the creation operator \(\Gamma(u,v)\), one has
\[
f^*(\lambda)
=
\left\langle P_{\frac{1}{1-q}f},\; t^{-n(\lambda)}J_\lambda\right\rangle_{q,t},
\]
which gives an explicit realization of the isomorphism between ordinary and shifted symmetric functions [2404.03904]. The same framework defines Macdonald characters as
\[
{}_{\mu,k}(v_1,\dots,v_k)
=
\left\langle p_\mu,\,(1,v_1)(t^{-1},v_2)\cdots(t^{-(k-1)},v_k)\cdot 1\right\rangle_{q,t},
\]
with \({}_\mu=p_\mu^*\) forming a basis of shifted symmetric functions [2404.03904].

## 5. Degenerations, factorization, and positivity phenomena

Interpolation Macdonald polynomials admit several standard degenerations. In the Jack limit
\[
(q,t)=(q,q^k),\qquad q\to 1,
\]
the interpolation Macdonald functions degenerate to interpolation Jack polynomials:
\[
I_\mu(x_1,\dots,x_N;k)
=
\lim_{q\to 1}(1-q)^{|\mu|}\,
I_\mu(q^{-1}x_1,\dots,q^{-1}x_N;q,q^k),
\]
and the dual functions \(H_\mu\) admit a parallel limit [1712.08018]. In the \(BC\)-type theory, the \(q\to1\) limit of Okounkov’s \(BC\)-type interpolation Macdonald polynomials yields \(BC\)-type interpolation Jack polynomials, including the explicit tableau formula
\[
P_\lambda^*(x;\tau,a)
=
\sum_T \psi_T(\tau)
\prod_{s\in\lambda}
\left(x_{T(s)}^2-\bigl(a'(s)+\tau(n-T(s)-\ell'(s))+a\bigr)^2\right)
\]
[1408.5993]. Other specializations discussed in the symmetric-function setting include \(t=0\), giving \(q\)-Whittaker-type interpolation functions, \(q=0\), giving Hall–Littlewood-type limits, and the \(q\to0\) degeneration to inhomogeneous Hall–Littlewood functions used in the operator-at-infinity theory [1712.08018] [1712.08014].

The regime \(q\to1\) is also the setting of a strong factorization theorem. For partitions \(\lambda^1,\dots,\lambda^r\), write \(\lambda^I=\bigoplus_{i\in I}\lambda^i\). Then the interpolation Macdonald polynomials satisfy
\[
\prod_{I\subset [r]}
\left(J_{\lambda^I}^{(q,t)}\right)^{(-1)^{|I|}}
=
1+O_1\!\left((q-1)^{r-1}\right),
\]
equivalently, their cumulants are \(O_1((q-1)^{r-1})\) [1609.09686]. Passing to the top homogeneous component transfers the same small-cumulant property to ordinary Macdonald polynomials, which in turn yields integrality of multivariate \(q,t\)-Kostka numbers [1609.09686].

Interpolation coefficients themselves exhibit a positive combinatorics. In the 2024 treatment, the generalized binomial coefficients are the evaluations
\[
b_{\lambda\mu}=h_\mu(\lambda),
\]
and they satisfy weighted-sum formulas over chains of partitions, both in type \(A\) and in type \(BC\) [2403.02490]. The same work proves positivity and monotonicity: \(b_{\lambda\mu}\in F_{>0}\) precisely when \(\lambda\supseteq\mu\), and if \(\lambda\supseteq\mu\) then \(b_{\lambda\nu}-b_{\mu\nu}\in F_{>0}\) [2403.02490]. As applications, the paper derives inequalities for power sums, Jack polynomials, and their monomial, Schur, zonal, and elementary specializations [2403.02490].

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

At the specialization \(t=q\), interpolation Macdonald polynomials become a basis adapted to the center of the quantum group \(U_q(\mathfrak{gl}_N)\). Under the Harish–Chandra isomorphism, central elements are determined by evaluations at the points
\[
\bigl(q^{-\mu_1-N+1},\,q^{-\mu_2-N+2},\,\dots,\,q^{-\mu_N}\bigr),
\]
and the specialized interpolation polynomials \(F_\lambda\) are tailored to vanish unless \(\mu\supseteq\lambda\) [2101.08243]. Transporting them through Harish–Chandra produces central elements \(\sigma_\lambda\), while Okounkov’s inverse interpolation theorem yields a dual basis for the Hopf pairing. This is the mechanism behind cyclotomic expansions of \(\mathfrak{gl}_N\) Reshetikhin–Turaev invariants, universal knot invariants, and explicit Kirby colors for integral homology \(3\)-spheres [2101.08243].

A probabilistic interpretation appears at \(q=1\). For a partition \(\lambda\), the interpolation \(t\)-Push TASEP is a Markov chain on the reorderings \(S_n(\lambda)\), and its stationary distribution is
\[
\pi_\lambda^*(\mu)=\frac{F_\mu^*(x;1,t)}{P_\lambda^*(x;1,t)}.
\]
Thus the steady-state weight of a state \(\mu\) is the interpolation ASEP polynomial \(F_\mu^*(x;1,t)\), and the partition function is the interpolation Macdonald polynomial \(P_\lambda^*(x;1,t)\) [2602.13492]. The proof uses signed multiline queues and the factorization
\[
P_\lambda^*(x;1,t)=\prod_{i=1}^{\lambda_1} e^*_{\lambda_i'}(x;t)
\]
at \(q=1\) [2602.13492].

The interpolation paradigm also extends beyond type \(A\) and beyond ordinary symmetric functions. In the wreath setting for \(\Sigma_n\wr \mathbb Z/r\mathbb Z\), the interpolation polynomial \(\mathfrak H_\mu\in \Lambda_{q,t}^{\otimes r}\) is characterized by highest degree \(|\mathrm{quot}(\mu)|\), vanishing at colored spectral points \(\iota D_\lambda^\bullet\) for smaller partitions with the same core, and normalization \(\mathfrak H_\mu[\iota D_\mu^\bullet]=1\) [2505.01732]. The explicit formula
\[
\mathfrak{H}_\mu = \frac{1}{N_\mu}\,\nabla_{w_0\,core(\mu)}^{-1}\,H_\mu^\dagger[X^\bullet-1]
\]
shows that interpolation phenomena survive in a colored, core-sensitive setting controlled by quantum toroidal and shuffle algebra operators [2505.01732].

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 [1712.08018].

Source: https://www.emergentmind.com/topics/interpolation-macdonald-polynomials