Super Macdonald Polynomials and Supersymmetry

• Super Macdonald Polynomials are supersymmetric extensions of Macdonald polynomials that combine bosonic and fermionic variables to form rich algebraic structures.
• They encompass multiple formalisms—such as Sergeev–Veselov, superspace, half-box, and vector-valued constructions—each with distinct variables and combinatorial rules.
• Characteristic features like triangularity, orthogonality, eigenoperator structure, and Pieri rules make them essential in quantum algebra, BPS state counting, and integrable systems.

Super Macdonald polynomials are supersymmetric extensions of Macdonald polynomials in which the classical symmetric-function variables are replaced or supplemented by fermionic data, additional alphabets, or module-valued coefficients, while preserving Macdonald-type structures such as triangularity, orthogonality, and joint eigenfunction properties for commuting operators. The literature uses the term for several closely related but non-identical constructions: the Sergeev–Veselov family in two commuting alphabets, Macdonald polynomials in superspace with commuting $x_i$ and anticommuting $\theta_i$, half-box or super-Young-diagram polynomials in power sums $p_k$ and Grassmann times $\theta_k$, and vector-valued nonsymmetric superpolynomials attached to Hecke modules (Atai et al., 2021, Blondeau-Fournier et al., 2012, Galakhov et al., 2024, Dunkl, 2020).

1. Principal formalisms

Taken together, the literature presents several supersymmetric Macdonald theories rather than a single universally fixed definition. In the Sergeev–Veselov framework, super Macdonald polynomials are images of ordinary Macdonald symmetric functions under a homomorphism from the ring of symmetric functions to an algebra $A_{N,M,q,t}$ of polynomials in two commuting alphabets $x_1,\dots,x_N$ and $y_1,\dots,y_M$ satisfying symmetry and $q,t$-shift constraints; for $\lambda\in H_{N,M}$, one writes

$$SP_\lambda(x_1,\dots,x_N;y_1,\dots,y_M;q,t) = \varphi_{q,t^{-1}}\bigl(P_\lambda(z;q,t)\bigr).$$

These functions are the super Macdonald polynomials in the sense used in the deformed Macdonald–Ruijsenaars and quantum-corner-VOA literature (Atai et al., 2021, Cheewaphutthisakun et al., 24 Apr 2025).

A second formalism, developed in the superspace literature, works with commuting variables $\theta_i$0 and anticommuting variables $\theta_i$1, and indexes symmetric superpolynomials by superpartitions $\theta_i$2. Macdonald superpolynomials $\theta_i$3 are characterized by monomial triangularity together with orthogonality for a superspace Macdonald scalar product, and they admit a construction from nonsymmetric Macdonald polynomials via a non-standard combination of $\theta_i$4-symmetrization, antisymmetrization, and Grassmann dressing (Blondeau-Fournier et al., 2012, Blondeau-Fournier et al., 2011).

A third formalism, introduced through super-Young diagrams built from half-boxes, uses bosonic times $\theta_i$5 together with Grassmann times $\theta_i$6. In that setting, super-Macdonald polynomials $\theta_i$7 are $\theta_i$8-deformations of super-Schur polynomials, indexed by super-partitions with half-integer parts, and defined by super-Schur triangularity plus a $\theta_i$9-deformed scalar product; a later spectral reformulation characterizes them as common eigenfunctions of four super-Hamiltonians (Galakhov et al., 2024, Galakhov et al., 24 Jan 2025).

A fourth line of work studies nonsymmetric Macdonald superpolynomials as vector-valued objects. Here the polynomial variables are commuting $p_k$0, while the fermionic sector furnishes an explicit Hecke module. The resulting eigenfunctions $p_k$1 are indexed by a composition $p_k$2 together with a hook-tableau label $p_k$3, and symmetric or antisymmetric super Macdonald polynomials are recovered by Hecke symmetrization and antisymmetrization (Dunkl, 2020).

Formalism	Variables	Indexing data
Sergeev–Veselov	$p_k$4, $p_k$5	partitions $p_k$6
Superspace	$p_k$7, $p_k$8	superpartitions $p_k$9
Half-box / super-Young	$\theta_k$0, $\theta_k$1 or $\theta_k$2	half-integer super-partitions
Vector-valued nonsymmetric	$\theta_k$3 with fermionic Hecke-module coefficients	$\theta_k$4

This multiplicity of definitions is a structural feature of the subject, not a contradiction. The common theme is the replacement of the ordinary partition combinatorics of Macdonald theory by data carrying bosonic and fermionic sectors.

2. Combinatorics and indexing

In the superspace approach, a superpartition $\theta_k$5 is described either as a pair $\theta_k$6, where $\theta_k$7 has $\theta_k$8 distinct parts and $\theta_k$9 is an ordinary partition, or as a pair $A_{N,M,q,t}$0 of ordinary partitions with $A_{N,M,q,t}$1 and $A_{N,M,q,t}$2 an $A_{N,M,q,t}$3-rook strip. Diagrammatically, one draws the Ferrers diagram of $A_{N,M,q,t}$4 and marks the cells of $A_{N,M,q,t}$5 by circles. The dominance order relevant for triangularity is the simultaneous dominance condition on $A_{N,M,q,t}$6 and $A_{N,M,q,t}$7 (Blondeau-Fournier et al., 2012).

In the half-box formalism, super-partitions are weakly decreasing sequences of half-integers,

$A_{N,M,q,t}$8

with the extra rule that adjacent half-integers must be strictly decreasing. These are represented by super-Young diagrams or crystal Young diagrams built from full boxes and half-boxes. Ordinary partitions are recovered as the special case with only integral parts. The supersymmetric polynomial algebra is generated by commuting $A_{N,M,q,t}$9 of degree $x_1,\dots,x_N$0 and anticommuting $x_1,\dots,x_N$1 of degree $x_1,\dots,x_N$2, and basis monomials $x_1,\dots,x_N$3 are in bijection with super-Young diagrams (Galakhov et al., 2024).

The Sergeev–Veselov family uses a different combinatorial constraint. Super Macdonald polynomials $x_1,\dots,x_N$4 are indexed by ordinary partitions in the fat hook

$x_1,\dots,x_N$5

Their tableaux formula uses reverse semi-standard Young bitableaux filled by ordinary labels $x_1,\dots,x_N$6 and super labels $x_1,\dots,x_N$7, with weak row and column monotonicity plus strictness conditions separating the ordinary and super parts (Cheewaphutthisakun et al., 24 Apr 2025).

The stable-limit theory of double Macdonald polynomials replaces a superpartition by a pair of ordinary partitions $x_1,\dots,x_N$8. In the stable regime of fermionic degree $x_1,\dots,x_N$9, the dependence on $y_1,\dots,y_M$0 disappears, and the Macdonald superpolynomials map to bisymmetric polynomials $y_1,\dots,y_M$1 indexed by this pair (Blondeau-Fournier et al., 2012).

These indexing conventions are not interchangeable. A half-box superpartition, a circled Ferrers diagram, a fat-hook partition, and a pair $y_1,\dots,y_M$2 encode different supersymmetric extensions, even when the resulting functions play analogous roles.

3. Eigenoperators and Hamiltonian characterizations

A defining feature inherited from ordinary Macdonald theory is characterization by commuting operators. In the Sergeev–Veselov setting, the relevant operators are the deformed Macdonald–Ruijsenaars operators $y_1,\dots,y_M$3 acting on functions of two alphabets $y_1,\dots,y_M$4 and $y_1,\dots,y_M$5. They involve $y_1,\dots,y_M$6-shifts in the $y_1,\dots,y_M$7-variables and $y_1,\dots,y_M$8-shifts in the $y_1,\dots,y_M$9-variables, and the homomorphism $q,t$0 intertwines the infinite-variable Macdonald operator with $q,t$1. Consequently,

$q,t$2

so the super Macdonald polynomials are eigenfunctions of deformed Macdonald–Ruijsenaars operators (Atai et al., 2021).

In the superspace construction of Macdonald superpolynomials, commuting operator families $q,t$3 and $q,t$4 are built from Cherednik operators and fermionic projection operators. Their common eigenfunctions are the superpolynomials $q,t$5, with eigenvalues determined separately by $q,t$6 and $q,t$7. This yields a spectral proof of existence and uniqueness compatible with the orthogonality–triangularity definition (Blondeau-Fournier et al., 2012).

The half-box theory makes the spectral viewpoint primary. Super-Macdonald polynomials $q,t$8 are common eigenfunctions of four super-Hamiltonians

$q,t$9

whose eigenvalues are sums over upper or lower half-boxes weighted by $\lambda\in H_{N,M}$0. For even diagrams the first two Hamiltonians reduce to the ordinary Macdonald Hamiltonian, while for generic odd diagrams the four eigenvalues are independent. The explicit operators are vertex-operator-type difference–differential operators in $\lambda\in H_{N,M}$1 coupled to auxiliary fermions, and an infinite commuting hierarchy is generated from Pieri-type half-box creation and annihilation operators by anticommutators (Galakhov et al., 24 Jan 2025).

In the shifted quantum toroidal $\lambda\in H_{N,M}$2 approach, super Macdonald polynomials form a basis of the level-zero super Fock module of $\lambda\in H_{N,M}$3. The action of the supercharges implies the Pieri rule, and the corresponding supersymmetric Hamiltonians are recovered as anticommutators of those supercharges. The paper emphasizes that a shifted quantum toroidal algebra is required in this realization (Kanno et al., 16 May 2026).

Across these frameworks, the analogue of the Macdonald–Ruijsenaars commuting family survives, but the operator content is enriched by fermionic grading, extra alphabets, or explicit supercharges.

4. Orthogonality, scalar products, and norms

The superspace theory attaches a Macdonald scalar product directly to the power-sum superfunctions. For superpartitions $\lambda\in H_{N,M}$4,

$\lambda\in H_{N,M}$5

with $\lambda\in H_{N,M}$6 involving the bosonic partition $\lambda\in H_{N,M}$7 and the factor $\lambda\in H_{N,M}$8. The operators used in the spectral construction are self-adjoint for this scalar product, and the resulting $\lambda\in H_{N,M}$9 are orthogonal. The same paper also introduces a second constant-term-type scalar product and computes norms matching, up to a $$SP_\lambda(x_1,\dots,x_N;y_1,\dots,y_M;q,t) = \varphi_{q,t^{-1}}\bigl(P_\lambda(z;q,t)\bigr).$$0-power, the conjectured norm in the original scalar product (Blondeau-Fournier et al., 2012).

In the Sergeev–Veselov theory, orthogonality is proved by a Hermitian form defined by integration over a product of tori with weight

$$SP_\lambda(x_1,\dots,x_N;y_1,\dots,y_M;q,t) = \varphi_{q,t^{-1}}\bigl(P_\lambda(z;q,t)\bigr).$$1

The super-Macdonald polynomials $$SP_\lambda(x_1,\dots,x_N;y_1,\dots,y_M;q,t) = \varphi_{q,t^{-1}}\bigl(P_\lambda(z;q,t)\bigr).$$2 are orthogonal, but their norms may vanish. More precisely, the norm is nonzero exactly when the diagram $$SP_\lambda(x_1,\dots,x_N;y_1,\dots,y_M;q,t) = \varphi_{q,t^{-1}}\bigl(P_\lambda(z;q,t)\bigr).$$3 contains the rectangle $$SP_\lambda(x_1,\dots,x_N;y_1,\dots,y_M;q,t) = \varphi_{q,t^{-1}}\bigl(P_\lambda(z;q,t)\bigr).$$4, and in that case the norm factorizes through two ordinary Macdonald norms attached to the east and south components $$SP_\lambda(x_1,\dots,x_N;y_1,\dots,y_M;q,t) = \varphi_{q,t^{-1}}\bigl(P_\lambda(z;q,t)\bigr).$$5 and $$SP_\lambda(x_1,\dots,x_N;y_1,\dots,y_M;q,t) = \varphi_{q,t^{-1}}\bigl(P_\lambda(z;q,t)\bigr).$$6: $$SP_\lambda(x_1,\dots,x_N;y_1,\dots,y_M;q,t) = \varphi_{q,t^{-1}}\bigl(P_\lambda(z;q,t)\bigr).$$7 Passing to the quotient by the kernel yields a Hilbert space with orthonormal basis given by the nonzero super-Macdonald polynomials (Atai et al., 2021).

The half-box approach defines super-Macdonald polynomials by orthogonality with respect to a $$SP_\lambda(x_1,\dots,x_N;y_1,\dots,y_M;q,t) = \varphi_{q,t^{-1}}\bigl(P_\lambda(z;q,t)\bigr).$$8-deformed scalar product on the basis $$SP_\lambda(x_1,\dots,x_N;y_1,\dots,y_M;q,t) = \varphi_{q,t^{-1}}\bigl(P_\lambda(z;q,t)\bigr).$$9,

$\theta_i$00

and the associated Cauchy kernel is written explicitly. This reproduces the ordinary Macdonald scalar product on the purely bosonic sector (Galakhov et al., 2024).

For vector-valued nonsymmetric superpolynomials $\theta_i$01, the inner product extends the fermionic Hecke-module form and makes both the Demazure–Lusztig operators and the Cherednik operators self-adjoint. Distinct $\theta_i$02 are orthogonal, and explicit norm formulas are available, including a closed product formula in the dominant case and a concise norm formula for the minimal symmetric polynomial (Dunkl, 2020).

The norm problem is therefore resolved differently in different branches of the subject: explicitly in the Sergeev–Veselov, superspace, and vector-valued theories, and by direct $\theta_i$03-orthogonality in the half-box theory.

5. Pieri rules, symmetrization, and higher hierarchies

Pieri-type operators play an organizing role throughout super Macdonald theory. In the half-box formalism, multiplication by $\theta_i$04 and differentiation $\theta_i$05 add or remove a half-box: $\theta_i$06 By commuting these operators with the basic super-Hamiltonians, one obtains higher creation and annihilation operators $\theta_i$07 and $\theta_i$08, and then a commuting family of Hamiltonians $\theta_i$09. The appearance of anticommutators, rather than commutators, is specific to the Grassmann-odd nature of the half-box operators (Galakhov et al., 24 Jan 2025).

In the shifted quantum toroidal $\theta_i$10 realization, the supercharges $\theta_i$11 and $\theta_i$12 act on the superpartition basis by changing the parity of a row endpoint, and the resulting coefficients are exactly the Pieri coefficients of the super Macdonald polynomials after the substitution $\theta_i$13. The paper also expresses the Pieri rule in terms of differential operators in the bosonic power sums $\theta_i$14 and fermionic power sums $\theta_i$15, providing free-boson/free-fermion operators on Fock space (Kanno et al., 16 May 2026).

A geometric realization of the same mechanism appears in BPS state counting on the blow-up of $\theta_i$16. There, torus fixed points are labeled by super partitions $\theta_i$17, and the equivariant character of the tangent space defines a super Nekrasov factor $\theta_i$18. Ratios of these Nekrasov factors, together with a diagonal normalization $\theta_i$19, reproduce the Pieri coefficients of super Macdonald polynomials and therefore match the action of the super DIM currents on the fixed-point basis (Kanno et al., 2 Jun 2025).

In the vector-valued Hecke-module theory, the nonsymmetric eigenfunctions $\theta_i$20 are the basic objects, and the symmetric and antisymmetric super Macdonald polynomials are constructed by applying explicit Hecke symmetrization and antisymmetrization operators $\theta_i$21 and $\theta_i$22. Existence of Hecke-symmetric vectors is governed by column-strictness of the associated hook tableau, and the labels of the symmetric objects are naturally identified with superpartitions (Dunkl, 2020).

The Pieri problem is therefore not auxiliary. It is the mechanism through which super Macdonald bases interact with integrable hierarchies, toroidal superalgebras, and geometric correspondences.

6. Geometric, VOA, and integrable-system realizations

A representation-theoretic realization of the Sergeev–Veselov polynomials is provided by the quantum corner VOA $\theta_i$23. In this setting, properly normalized and specialized vacuum correlators of the Miura currents $\theta_i$24 coincide with the super Macdonald polynomials $\theta_i$25. The proof passes through a sum over reverse semi-standard Young bitableaux whose weights match the Sergeev–Veselov combinatorial formula exactly (Cheewaphutthisakun et al., 24 Apr 2025).

A geometric realization arises in the BPS counting problem on the blow-up of $\theta_i$26, modeled by framed stable perverse coherent sheaves. Torus fixed points are labeled by super partitions, and the resulting $\theta_i$27-theoretic fixed-point basis carries an action of the quantum toroidal algebra of type $\theta_i$28. The super Nekrasov factor extracted from the tangent-space character controls both localization formulas and the matrix elements of this toroidal action, with the super Macdonald Pieri rule emerging as the compatibility condition (Kanno et al., 2 Jun 2025).

The integrable-systems interpretation is sharpest in the Sergeev–Veselov branch. There the deformed Macdonald–Ruijsenaars operators admit a Hilbert-space realization, and the super-Macdonald polynomials provide orthogonal wave functions for a relativistic model with two kinds of excitations. The paper proposes that the two alphabets $\theta_i$29 and $\theta_i$30 should be interpreted as particles and anti-particles, thereby extending the trigonometric Ruijsenaars model to a setting with two species (Atai et al., 2021).

These realizations connect super Macdonald theory to quantum toroidal algebras, VOAs, gauge theory, BPS algebras, and supersymmetric integrable systems. A plausible implication is that the various super Macdonald families should be regarded not only as deformations of symmetric functions but also as preferred bases in several different representation categories.

7. Limits, stable sectors, and scope of the subject

Several degeneration patterns organize the subject. In the half-box theory, super-Macdonald polynomials reduce to super-Schur polynomials in the $\theta_i$31 limit followed by $\theta_i$32, and they reduce to ordinary Macdonald polynomials on the purely integral sector. The paper also records explicit low-level examples showing that some lowest elements in each block remain undeformed, just as ordinary Macdonald polynomials coincide with Schur polynomials on single-column diagrams (Galakhov et al., 2024).

In the superspace theory of 2011, the limits $\theta_i$33 and $\theta_i$34 lead to two Hall–Littlewood families in superspace, and the specializations $\theta_i$35 and $\theta_i$36 produce two Schur-type superbases $\theta_i$37 and $\theta_i$38. The paper formulates positivity conjectures for the corresponding Hall–Littlewood–Schur transition coefficients and for the superspace $\theta_i$39-Kostka coefficients defined from a modified Schur superbasis (Blondeau-Fournier et al., 2011).

A major structural simplification occurs in the stable regime of Macdonald superpolynomials when the fermionic degree is sufficiently large. Then the superspace theory maps to bisymmetric polynomials indexed by pairs $\theta_i$40, called double Macdonald polynomials, and these factorize as

$\theta_i$41

From this factorization follow explicit norms, kernels, duality, and positivity results, as well as a connection to irreducible representations of the hyperoctahedral group $\theta_i$42 and a type-$\theta_i$43 Nabla operator (Blondeau-Fournier et al., 2012).

A recurring source of confusion is the scope of the phrase “super Macdonald polynomials.” The term does not denote a single universally standardized object. It can refer to superspace polynomials indexed by superpartitions, to Sergeev–Veselov polynomials in two commuting alphabets, to half-box super-Young-diagram polynomials in $\theta_i$44, or to vector-valued nonsymmetric constructions. Closely related extensions also exist outside this terminology: for example, the four-parameter family $\theta_i$45 reduces to ordinary Macdonald polynomials at $\theta_i$46 and to Borodin–Petrov functions at $\theta_i$47, but that paper does not explicitly use the terminology “super Macdonald polynomials” (Garbali et al., 2016).

The subject is therefore best understood as a cluster of supersymmetric Macdonald theories sharing common Macdonald features—triangularity, orthogonality, Pieri rules, commuting Hamiltonians, and rich representation theory—while differing substantially in variables, indexing objects, and ambient algebraic structures.