Nonsymmetric Plethsym in Macdonald Theory

• Nonsymmetric plethsym is an operator calculus that adapts classical plethystic substitution to almost symmetric spaces with finitely many distinguished variables.
• It employs operators such as Πₜ, Πᵣ, and Φᵣ to transform integral-form nonsymmetric Macdonald polynomials and convert signed flagged LLT polynomials to unsigned ones.
• The framework underpins modified nonsymmetric Macdonald theory and supports nonsymmetric formulations of shuffle and compositional Delta theorems.

Nonsymmetric plethsym is a recent nonsymmetric analogue of the classical plethystic transformation $f[X]\mapsto f\!\left[\frac{X}{1-t}\right]$, developed on spaces that are nonsymmetric in finitely many distinguished variables and symmetric in a remaining tail. In the current literature, it is realized by operators such as $\Pi_{t,x}$, its stable limit $\Pi_r$, and the $r$-nonsymmetric plethysm map $\Phi_r$; these operators transport integral-form nonsymmetric Macdonald theory to modified nonsymmetric Macdonald theory, convert signed flagged LLT polynomials to unsigned ones, and support nonsymmetric versions of the shuffle and compositional Delta theorems (Blasiak et al., 10 Jun 2025, Blasiak et al., 28 Sep 2025, Qiu et al., 11 Apr 2026).

1. Definition and ambient spaces

The foundational ambient spaces are “almost symmetric” polynomial spaces. One formulation uses

$$P(r)=(q,t)[x_1,\dots,x_r]\otimes \Lambda_{(q,t)}(x_{r+1},x_{r+2},\dots),$$

so that $r$ variables are distinguished and nonsymmetric while the remaining variables form a symmetric tail (Blasiak et al., 28 Sep 2025). A parallel formulation uses

$$P(\ell)=\mathbb Q(q,t)[x_1,\dots,x_\ell]\otimes [X]_{(q,t)}(x_{\ell+1},x_{\ell+2},\dots),$$

with the same structural feature (Qiu et al., 11 Apr 2026). The basic indexing data for stable nonsymmetric Macdonald theory is a pair $(\eta|\lambda)$, where $\eta\in \mathbb Z_{\ge0}^r$ and $\Pi_{t,x}$0 is a partition; its symmetric shadow is

$\Pi_{t,x}$1

Within this framework, nonsymmetric plethsym is not introduced as a universal binary operation $\Pi_{t,x}$2 on a full nonsymmetric function ring. Instead, it is introduced as an operator calculus that plays, in the nonsymmetric setting, the role of the classical modified-Macdonald plethystic substitution. The construction is therefore basis-sensitive, representation-theoretic, and tied to Weyl symmetrization rather than to an autonomous lambda-ring formalism (Blasiak et al., 28 Sep 2025, Qiu et al., 11 Apr 2026).

2. Operator realizations

A concrete finite-variable realization is the operator $\Pi_{t,x}$3, specialized later to $\Pi_{t,x}$4. It is defined by its action on a flagged basis: $\Pi_{t,x}$5 and, for $\Pi_{t,x}$6,

$\Pi_{t,x}$7

A useful equivalent formula is

$\Pi_{t,x}$8

Its inverse is also explicit (Blasiak et al., 10 Jun 2025).

The stable version is the operator $\Pi_{t,x}$9, characterized by

$\Pi_r$0

This is the precise sense in which the construction tends to the classical substitution $\Pi_r$1 on the symmetric tail (Blasiak et al., 10 Jun 2025).

A later formulation packages the stable operator as the $\Pi_r$2-nonsymmetric plethysm map

$\Pi_r$3

where $\Pi_r$4 is the polynomial truncation operator on Demazure characters (Blasiak et al., 28 Sep 2025). This formulation makes the analogy with symmetric plethystic modification completely explicit.

3. Modified nonsymmetric Macdonald theory

The main algebraic application is the construction of modified nonsymmetric Macdonald polynomials from stable integral forms. In one notation, the stable integral forms are $\Pi_r$5, and the modified objects are defined by

$\Pi_r$6

These polynomials Weyl symmetrize to the symmetric modified Macdonald polynomials: $\Pi_r$7 The same framework defines a nonsymmetric analogue of $\Pi_r$8, acting diagonally by

$\Pi_r$9

and related by conjugation to the unmodified operator: $r$0 The corresponding nonsymmetric compositional Hall–Littlewood functions are

$r$1

The cited work also states that the modified nonsymmetric Macdonald polynomials are monomial positive and conjecturally stable atom positive (Blasiak et al., 28 Sep 2025).

4. Flagged LLT polynomials and combinatorial theorems

The combinatorial side of nonsymmetric plethsym is built from flagged LLT polynomials. The finite-variable theory introduces flagged LLT polynomials $r$2 and proves that they admit both an algebraic and a combinatorial description, that they Weyl symmetrize to the usual symmetric LLT polynomials, and that signed versions are mapped to unsigned ones by the nonsymmetric plethysm operator (Blasiak et al., 10 Jun 2025). The key signed-to-unsigned identity is

$r$3

This mechanism is then used to recast Haglund–Haiman–Loehr type formulas for nonsymmetric Macdonald polynomials as positive sums of signed flagged LLT polynomials and, after stabilization, as positive sums of unsigned flagged LLT polynomials (Blasiak et al., 10 Jun 2025).

In the shuffle-theorem direction, the nonsymmetric operator $r$4 yields a nonsymmetric compositional shuffle theorem: $r$5 The same work formulates stable atom positivity conjectures for flagged LLT polynomials, modified nonsymmetric Macdonald polynomials, and $r$6 (Blasiak et al., 28 Sep 2025).

The nonsymmetric compositional Delta theorem extends this package further. It establishes signed and unsigned nonsymmetric identities in terms of flagged LLT polynomials, introduces nonsymmetric variants of $r$7 and $r$8, and proves that applying Weyl symmetrization to the nonsymmetric identities systematically recovers the original compositional Delta theorem (Qiu et al., 11 Apr 2026).

5. Adjacent notions and non-equivalent usages

Several nearby literatures are relevant but are not themselves the current theory of nonsymmetric plethsym. “Plethysm and orbit harmonics” studies ordinary symmetric-function plethysm $r$9 through graded $\Phi_r$0-modules $\Phi_r$1 and refines Schur expansions by a $\Phi_r$2-separation phenomenon, but it does not define a theory of nonsymmetric plethysm (Zhu, 13 Feb 2026). “Further Pieri-type formulas for the nonsymmetric Macdonald polynomials” develops multiplication by $\Phi_r$3 for $\Phi_r$4, which is a nonsymmetric Pieri calculus rather than a plethystic substitution formalism (Baratta, 2010).

Other works provide methodological or conceptual templates. “Plethysm and fast matrix multiplication” treats only symmetric powers $\Phi_r$5, but its use of $\Phi_r$6, Schur functors, duality, and Littlewood–Richardson rules is explicitly described as a template for more general plethysm questions (Seynnaeve, 2017). “A Superpolynomial Version of Nonsymmetric Jack Polynomials” gives a superspace, operator-theoretic, and representation-valued basis construction for nonsymmetric Jack polynomials in commuting and anti-commuting variables, but it does not use plethystic notation or a plethystic substitution formalism (Dunkl, 2020).

The term “nonsymmetric” is also used in substantially different senses elsewhere. “Partial symmetries of iterated plethysms” studies classical Schur-positive plethysms whose Schur coefficients exhibit a partial involutive symmetry called flip-symmetry; this is a nonclassical symmetry inside ordinary plethysm, not nonsymmetric plethsym (Gutiérrez et al., 2022). “Plethysm is in #BQP” proves complexity results for ordinary plethysm coefficients $\Phi_r$7 defined by

$\Phi_r$8

and explicitly does not formulate a nonsymmetric analogue (Christandl et al., 9 Feb 2026). “Quadratic nonsymmetric quaternary operads” concerns non-$\Phi_r$9 operadic substitution, which is composition-theoretic but not plethysm in the Macdonald-theoretic sense (Bremner et al., 2015). “Transition Matrices between Plethystic Bases of Polysymmetric Functions via Bijective Methods” develops plethystic bases $$P(r)=(q,t)[x_1,\dots,x_r]\otimes \Lambda_{(q,t)}(x_{r+1},x_{r+2},\dots),$$0 in the algebra $$P(r)=(q,t)[x_1,\dots,x_r]\otimes \Lambda_{(q,t)}(x_{r+1},x_{r+2},\dots),$$1, which is a polysymmetric rather than nonsymmetric extension (Khanna, 14 Oct 2025).

6. Present scope, limitations, and open direction

The present theory is best viewed as operator-centered rather than as a closed universal algebra of nonsymmetric plethysm. One recurrent feature is that the plethystic operator transports signed or integral-form objects to unsigned or modified ones: signed flagged LLT polynomials become unsigned flagged LLT polynomials, and stable integral-form nonsymmetric Macdonald polynomials become modified nonsymmetric Macdonald polynomials (Blasiak et al., 10 Jun 2025, Blasiak et al., 28 Sep 2025).

A central limitation is explicit in the current operator theory: $$P(r)=(q,t)[x_1,\dots,x_r]\otimes \Lambda_{(q,t)}(x_{r+1},x_{r+2},\dots),$$2 is not an algebra homomorphism, and the cited work states that there is no simple closed form for $$P(r)=(q,t)[x_1,\dots,x_r]\otimes \Lambda_{(q,t)}(x_{r+1},x_{r+2},\dots),$$3 (Qiu et al., 11 Apr 2026). The same source introduces nonsymmetric $$P(r)=(q,t)[x_1,\dots,x_r]\otimes \Lambda_{(q,t)}(x_{r+1},x_{r+2},\dots),$$4-operators and asks for an analogue of the symmetric identity

$$P(r)=(q,t)[x_1,\dots,x_r]\otimes \Lambda_{(q,t)}(x_{r+1},x_{r+2},\dots),$$5

so the operator calculus is not yet as transparent as in the symmetric theory.

This suggests that “nonsymmetric plethsym” currently names an emerging program rather than a single finished formalism. Its common core is nonetheless clear: an almost symmetric ambient space, a plethystic modification operator such as $$P(r)=(q,t)[x_1,\dots,x_r]\otimes \Lambda_{(q,t)}(x_{r+1},x_{r+2},\dots),$$6, $$P(r)=(q,t)[x_1,\dots,x_r]\otimes \Lambda_{(q,t)}(x_{r+1},x_{r+2},\dots),$$7, or $$P(r)=(q,t)[x_1,\dots,x_r]\otimes \Lambda_{(q,t)}(x_{r+1},x_{r+2},\dots),$$8, modified nonsymmetric Macdonald bases indexed by $$P(r)=(q,t)[x_1,\dots,x_r]\otimes \Lambda_{(q,t)}(x_{r+1},x_{r+2},\dots),$$9, and flagged LLT models that refine the symmetric shuffle- and Delta-theorem packages (Blasiak et al., 10 Jun 2025, Blasiak et al., 28 Sep 2025, Qiu et al., 11 Apr 2026).