Modified Nonsymmetric Macdonald Polynomials

• The paper introduces a stable nonsymmetric plethysm operator that transforms integral-form nonsymmetric Macdonald polynomials into a modified r‐nonsymmetric family.
• The framework unifies various modifications—including permuted-basement, interpolation, and partial symmetrization—highlighting deep connections with classical symmetric Macdonald theory.
• Incorporating flagged LLT theory, the construction achieves monomial positivity and conjectured Demazure atom positivity, enriching its combinatorial and algebraic structure.

Modified nonsymmetric Macdonald polynomials do not form a single universally standardized family across the literature. Several works explicitly use other terms and instead study specialized, permuted-basement, interpolation, or partially symmetric variants of the standard nonsymmetric Macdonald polynomials $E_\mu$. The most direct construction carrying the modifier “modified” is the 2025 theory of modified $r$-nonsymmetric Macdonald polynomials $\widetilde{J}_{\eta|\lambda}(x,Y;q,t)$, obtained from right-stable integral-form nonsymmetric Macdonald polynomials by a stable-limit nonsymmetric plethysm operator $\Pi_r$, in close analogy with the symmetric passage $J_\mu(X;q,t)\mapsto H_\mu(X;q,t)$ (Blasiak et al., 10 Jun 2025).

1. Terminology, scope, and competing meanings

The literature suggests that “modified nonsymmetric Macdonald polynomials” is best treated as a family resemblance term rather than the name of a single canonical basis. In particular, several papers explicitly state that they do not define a nonsymmetric analogue of the symmetric modified Macdonald polynomial $\widetilde H_\lambda$, even when they study especially positivity-friendly or renormalized forms of $E_\mu$ (Assaf, 2017). The paper on Demazure crystals for specialized nonsymmetric Macdonald polynomials is explicit that its subject is the usual type $A$ nonsymmetric Macdonald polynomial specialized at $t=0$, not a new two-parameter modified basis (Assaf et al., 2019). The paper on interpolation Macdonald polynomials likewise states that it does not define a standard object called “modified nonsymmetric Macdonald polynomial” analogous to the classical symmetric modified Macdonald polynomial, and instead develops the inhomogeneous interpolation family $E_\mu^*$ and its Hecke transforms (Dali et al., 2 Oct 2025).

A different use of “modified” appears in the $r$0-symmetric framework, where nonsymmetric Macdonald theory is embedded into a partially symmetric ring $r$1, one introduces an integral form $r$2, and then applies a plethystic modification $r$3 in direct analogy with symmetric Macdonald positivity (Lapointe, 2022). A further nearby usage arises in the general-basement theory, where the HHL combinatorial normalization is extended to arbitrary basements $r$4; this is described as a modified or permuted-basement realization of nonsymmetric Macdonald polynomials rather than a separate plethystic basis (Alexandersson, 2016). Against this background, the 2025 flagged-LLT construction is distinctive because it explicitly formulates modified $r$5-nonsymmetric Macdonald polynomials as a stable plethystic image of right-stable integral forms (Blasiak et al., 10 Jun 2025).

Object	Defining feature	Status relative to “modified”
$r$6	Monic triangular nonsymmetric Macdonald basis	Standard reference object
$r$7	$r$8 specialization with positive tabloid model	Specialization, not a separate modified basis
$r$9	General-basement or permuted-basement extension	Modified/permuted realization
$\widetilde{J}_{\eta|\lambda}(x,Y;q,t)$0	Inhomogeneous interpolation polynomial with top part $\widetilde{J}_{\eta|\lambda}(x,Y;q,t)$1	Closest interpolation analogue
$\widetilde{J}_{\eta|\lambda}(x,Y;q,t)$2	$\widetilde{J}_{\eta|\lambda}(x,Y;q,t)$3-image of right-stable integral forms	Explicit modified nonsymmetric family

2. Standard nonsymmetric Macdonald theory and nearby renormalizations

The common starting point is the standard type $\widetilde{J}_{\eta|\lambda}(x,Y;q,t)$4 nonsymmetric Macdonald basis

$\widetilde{J}_{\eta|\lambda}(x,Y;q,t)$5

indexed by weak compositions and characterized by triangularity and Cherednik-type eigenfunction properties. Two integral-form conventions are prominent in the literature. In the HHL setting one uses

$\widetilde{J}_{\eta|\lambda}(x,Y;q,t)$6

while in the 2025 flagged-LLT setting one writes

$\widetilde{J}_{\eta|\lambda}(x,Y;q,t)$7

The two notations reflect the same general principle: the integral form is the product-renormalized version of the monic basis (Assaf et al., 2019).

Two especially important neighboring constructions are the general-basement family and the interpolation family. The general-basement or permuted-basement nonsymmetric Macdonald polynomials are

$\widetilde{J}_{\eta|\lambda}(x,Y;q,t)$8

indexed by a weak composition $\widetilde{J}_{\eta|\lambda}(x,Y;q,t)$9 and a basement permutation $\Pi_r$0. They extend the usual HHL combinatorial model, satisfy monomial triangularity for each fixed basement, and behave naturally under Demazure–Lusztig operators. The ordinary HHL nonsymmetric Macdonald polynomial is recovered at $\Pi_r$1, and the specialization $\Pi_r$2 yields $\Pi_r$3-deformations of Demazure atoms and key polynomials (Alexandersson, 2016).

The interpolation family provides a different modification axis. For $\Pi_r$4, the nonsymmetric interpolation Macdonald polynomial $\Pi_r$5 is the unique inhomogeneous polynomial such that

$\Pi_r$6

and

$\Pi_r$7

The paper also studies the Hecke-transformed basis

$\Pi_r$8

and integral normalizations such as $\Pi_r$9 and $J_\mu(X;q,t)\mapsto H_\mu(X;q,t)$0. This places interpolation polynomials among the closest inhomogeneous analogues of a modified nonsymmetric Macdonald theory (Dali et al., 2 Oct 2025).

3. The $J_\mu(X;q,t)\mapsto H_\mu(X;q,t)$1 specialization as a positivity-friendly nonsymmetric form

A major pre-2025 strand concerns the specialization $J_\mu(X;q,t)\mapsto H_\mu(X;q,t)$2. This object is not a separately named modified basis, but it behaves as a positivity-friendly nonsymmetric refinement of Hall–Littlewood theory. In type $J_\mu(X;q,t)\mapsto H_\mu(X;q,t)$3, one has

$J_\mu(X;q,t)\mapsto H_\mu(X;q,t)$4

where $J_\mu(X;q,t)\mapsto H_\mu(X;q,t)$5 is the set of semistandard key tabloids of shape $J_\mu(X;q,t)\mapsto H_\mu(X;q,t)$6. The same paper proves a nonnegative expansion into fundamental slide polynomials,

$J_\mu(X;q,t)\mapsto H_\mu(X;q,t)$7

and, using weak dual equivalence, a positive graded Demazure expansion

$J_\mu(X;q,t)\mapsto H_\mu(X;q,t)$8

It also proves the stability statement

$J_\mu(X;q,t)\mapsto H_\mu(X;q,t)$9

and interprets the coefficients $\widetilde H_\lambda$0 in

$\widetilde H_\lambda$1

as a nonsymmetric refinement of Kostka–Foulkes polynomials (Assaf, 2017).

The crystal-theoretic refinement of this story constructs a Demazure crystal structure on semistandard key tabloids and proves that

$\widetilde H_\lambda$2

with

$\widetilde H_\lambda$3

The same work emphasizes that this is still the usual nonsymmetric Macdonald polynomial specialized at $\widetilde H_\lambda$4, not a two-parameter modified nonsymmetric Macdonald basis. Its methods are firmly type $\widetilde H_\lambda$5, use $\widetilde H_\lambda$6-crystals, and rely on the combinatorial simplification that occurs only at $\widetilde H_\lambda$7 (Assaf et al., 2019).

These results established a durable misconception-correction. The positivity-friendly object is real and structurally rich, but it is a specialization of $\widetilde H_\lambda$8, not a plethystically modified nonsymmetric basis.

4. Modified $\widetilde H_\lambda$9-nonsymmetric Macdonald polynomials via nonsymmetric plethysm and flagged LLTs

The clearest direct answer to the topic is the 2025 construction of modified $E_\mu$0-nonsymmetric Macdonald polynomials. The starting point is the integral form

$E_\mu$1

followed by stabilization in shapes of the form $E_\mu$2. The stable right-nonsymmetric objects are denoted $E_\mu$3, and more generally $E_\mu$4, obtained as $E_\mu$5-adic limits of specialized nonsymmetric Macdonald polynomials with a long zero block inserted between a nonsymmetric left part and a symmetric right tail (Blasiak et al., 10 Jun 2025).

The central new operator is the nonsymmetric plethysm $E_\mu$6. It is defined by its action on the deformed flagged-complete basis

$E_\mu$7

Its stable limit $E_\mu$8 acts on $E_\mu$9 by

$A$0

The modified $A$1-nonsymmetric Macdonald polynomial is then defined by

$A$2

This is designed to mirror the symmetric transformation $A$3 (Blasiak et al., 10 Jun 2025).

The second ingredient is flagged LLT theory. The paper introduces flagged LLT polynomials $A$4, their signed specializations $A$5, and proves the key transport identity

$A$6

Simultaneously, it rewrites the HHL formula for integral-form nonsymmetric Macdonald polynomials as a positive sum of signed flagged LLTs: $A$7 After stabilization and application of $A$8, the modified nonsymmetric Macdonald polynomials become positive sums of unsigned flagged LLT polynomials. The paper proves monomial positivity for these modified objects and conjectures that they are Demazure atom positive. It also states that $A$9 Weyl symmetrizes to the symmetric modified Macdonald polynomial $t=0$0, so the nonsymmetric theory recovers the symmetric modified theory after symmetrization (Blasiak et al., 10 Jun 2025).

This construction is the most literal nonsymmetric counterpart of the classical modified Macdonald paradigm currently available in the cited literature.

5. Partial symmetrization, prescribed symmetry, and intermediate modified theories

A different route to nonsymmetric modification passes through partial symmetry. In the $t=0$1-symmetric theory, one studies the ring

$t=0$2

whose elements are symmetric only in the tail variables $t=0$3. The $t=0$4-symmetric Macdonald polynomials are obtained by partial $t=0$5-symmetrization of nonsymmetric Macdonald polynomials: $t=0$6 They form a basis of $t=0$7, admit an integral form

$t=0$8

and a plethystic modification

$t=0$9

The central positivity conjecture is

$E_\mu^*$0

When $E_\mu^*$1 is large, this framework yields a positivity conjecture for nonsymmetric Macdonald polynomials expanded in nonsymmetric Hall–Littlewood polynomials modulo a subspace $E_\mu^*$2. This gives an intermediate theory in which modified positivity can be formulated before passing to the fully nonsymmetric limit (Lapointe, 2022).

Another intermediate transformation is the prescribed-symmetry construction. Starting from nonsymmetric Macdonald polynomials $E_\mu^*$3, one applies the Hecke symmetrizer/antisymmetrizer

$E_\mu^*$4

and defines prescribed-symmetry Macdonald polynomials by

$E_\mu^*$5

The paper computes explicit expansion coefficients of $E_\mu^*$6 in the $E_\mu^*$7-basis, the normalizing factor $E_\mu^*$8, and norm formulas under the constant-term inner product. It also proves

$E_\mu^*$9

Although this is not a modified nonsymmetric basis in the plethystic sense, it is a systematic Hecke-algebra transform of nonsymmetric Macdonald polynomials into mixed-symmetry sectors (Baratta, 2010).

6. Structural models, extensions, and current limitations

Several additional frameworks sharpen the meaning of “modified” by changing either normalization, ambient module, or combinatorial model. The interpolation theory gives the inhomogeneous basis $r$00, characterized by vanishing conditions and the top-homogeneous identity $r$01. Its Hecke-transformed forms $r$02, the rescaled objects $r$03, and integral normalizations such as $r$04 provide a precise inhomogeneous analogue of nonsymmetric Macdonald theory, even though the paper explicitly declines the label “modified nonsymmetric Macdonald polynomial” (Dali et al., 2 Oct 2025).

The general-basement theory similarly enlarges the standard HHL family to

$r$05

with triangularity, explicit Demazure–Lusztig basement-permuting formulas, and the specialization $r$06 to $r$07-deformations of key polynomials and Demazure atoms. This gives a combinatorial modified or permuted-basement realization rather than a plethystically modified basis (Alexandersson, 2016).

For the underlying standard $r$08, new structural models also matter. A 2022 path-model paper gives an arbitrary-type formula for nonsymmetric Macdonald polynomials in terms of pseudo-quantum Lakshmibai–Seshadri paths,

$r$09

and constructs a connected pseudo-crystal on the indexing set. The paper is explicit that this is a formula for the standard monic $r$10, not for a separate modified family (Lenart et al., 2022). Likewise, the integrable-vertex-model realization identifies partition functions with the reversed nonsymmetric Macdonald polynomials

$r$11

after multiplication by an explicit normalization factor $r$12, and recovers the HHL combinatorial formula from cylindrical colored path ensembles (Borodin et al., 2019). These models do not define modified nonsymmetric Macdonald polynomials, but they provide comparison tools for any future normalization or plethystic transform.

The present landscape therefore has a sharp asymmetry. The $r$13 specialization has explicit Demazure- and crystal-theoretic positivity but is only a specialization of $r$14 (Assaf et al., 2019). The $r$15-symmetric positivity theory is structurally rich but conjectural (Lapointe, 2022). The interpolation and general-basement theories produce genuine extensions, yet under different conceptual names (Dali et al., 2 Oct 2025). The 2025 flagged-LLT construction supplies the most direct modified nonsymmetric theory presently described in these sources, but its strongest positivity statement remains conjectural: modified $r$16-nonsymmetric Macdonald polynomials are proved monomial positive and are conjectured to be Demazure atom positive (Blasiak et al., 10 Jun 2025).

In that sense, the modern subject is best understood as a convergence of several nonsymmetric Macdonald phenomena—integral-form renormalization, stabilization, nonsymmetric plethysm, flagged LLT expansions, crystal and Demazure positivity, and partial symmetrization—rather than as a single pre-existing canonical notion.