Super-Hamiltonians for super-Macdonald polynomials

Alexei Morozov; Dmitry Galakhov; Nikita Tselousov

REVIEW 3 major objections 5 minor 36 references

Super-Hamiltonians for super-Macdonald polynomials

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Four explicit super-Hamiltonians, with conjectural vertex-operator formulas checked up to order 25/2, make super-Macdonald polynomials their eigenfunctions.

desk verdict A neat, clearly written conjectural construction of super-Hamiltonians for super-Macdonald polynomials, but the central eigenfunction claim rests on a finite check that is partly fitted to the target eigenvalues, so it is a plausible conjecture rather than an established result. read the letter →

arxiv 2501.14714 v2 pith:J76JPDFG submitted 2025-01-24 hep-th math-phmath.MPmath.QAmath.RT

Dmitry Galakhov , Alexei Morozov , Nikita Tselousov This is my paper · ORCID

classification hep-thmath-phmath.MPmath.QAmath.RT MSC 05E0517B8081R12

keywords super-Macdonaldpolynomialssuper-HamiltoniansGrassmanntimevariablessuper-YoungdiagramssumoverboxesPierrirulesvertexoperatorscommutingHamiltonians

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Super-Macdonald polynomials are symmetric functions in both ordinary variables $p_k$ and Grassmann variables $\theta_k$, labeled by half-integer super-Young diagrams. This paper tries to establish that these polynomials are eigenfunctions of a set of four explicitly written super-Hamiltonians, two with positive and two with negative powers of the parameters, whose eigenvalues are sums over the diagram of $q^{2j}t^{-2i}$. The paper writes closed vertex-operator formulas for the Hamiltonians and reports verification up to diagrams of order $|\lambda|=25/2$; the general statement is left as a conjecture. If correct, the construction gives super-Macdonald polynomials an operator definition analogous to the classical operator characterization of ordinary Macdonald polynomials.

What carries the argument

The operative machinery is the 'sum over boxes' principle combined with vertex-operator building blocks. The Hamiltonians are contour integrals of two exponentials: one in the bosonic times $p_k$ and Grassmann times $\theta_k$, the other in the derivatives $\partial/\partial p_k$ and $\partial/\partial\theta_k$. Auxiliary fermionic oscillators $\psi,\psi^\dagger$ for the first pair, and an invertible boson $s$ with a fermionic pair $\nu,\nu^\dagger$ for the second pair, project the vertex operators onto the sector that acts on super-Macdonald polynomials; the paper calls $s$ and $\nu$ a mere simplification trick. The commuting set of higher Hamiltonians (106)-(107) is then generated from box-adding and box-removing operators whose commutators and anticommutators cancel off-diagonal terms, following the Pierri-rule logic set out in Section 2.

What would settle it

Compute the action of the operator in (98) on a super-Macdonald polynomial labeled by a diagram of order $|\lambda|=27/2$, for instance $\lambda=[13,1/2]$, and compare with the eigenvalue formula (97); a single mismatch would disprove the conjectural Hamiltonian. A counterexample to the assumed Pierri rules (102)-(103) would likewise invalidate the commuting family (106)-(107).

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Extended reading notes

Core claim

The central claim is that super-Macdonald polynomials $M^{q,t}_\lambda$ are common eigenfunctions of the four super-Hamiltonians defined by (97) and given in closed form by (98) and (100). For the two positive Hamiltonians the eigenvalues read $$ 1+($q^{2}$-1)(1-$t^{{-2}}$)\sum_{\square\in\$\lambda$} $q^{{2j}}$$t^{{-2i}}$, $$ where the sum runs over the half-boxes of the super-Young diagram $\lambda$; the two negative Hamiltonians have the same form with $q\to q^{-1}$, $t\to t^{-1}$. The paper states that the four eigenvalues are generically distinct on odd diagrams and that two of the Hamiltonians suffice to determine all super-Macdonald polynomials. The closed formulas are labeled conjectural and have been checked up to $|\lambda|=25/2$.

Load-bearing premise

The closed vertex-operator formulas (98) and (100) are conjectural: they have been checked only up to $|\lambda|=25/2$, and the check uses a correlator chosen to reproduce the target eigenvalues, so the general case is not independently confirmed.

Editorial extensions

If this is right

Super-Macdonald polynomials become computable as common eigenfunctions of two explicitly written super-Hamiltonians, without relying on Cauchy formulas or triangularity conditions.
The four eigenvalues are independent on generic odd super-diagrams, so the two positive Hamiltonians separate all super-Macdonald polynomials.
The construction predicts a full tower of commuting higher super-Hamiltonians, with one family explicitly built and the second family left to future work.
The broken $q,t$ inversion symmetry of super-Macdonald polynomials implies the super-case carries additional commuting operators, which the paper links to a difference in the commuting structures of two infinite-dimensional algebras.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the conjectural formulas are verified for all orders, the auxiliary $s,\nu$ correlator is most naturally a remnant of a supersymmetric many-body integrable system, a connection the paper leaves open.
A proof of the closed formulas without assuming the eigenvalues would resolve their conjectural status; the natural route is an independent evaluation of the vertex-operator action on arbitrary super-Macdonald polynomials.
The broken $q\leftrightarrow q^{-1}$, $t\leftrightarrow t^{-1}$ symmetry suggests that super-generalizations of the relevant infinite-dimensional algebras carry two independent commuting families of Cartan-like generators, one for each sign, a representational prediction not stated in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the quick take: the paper gives an explicit conjectural construction of four super-Hamiltonians whose eigenfunctions are supposed to be the super-Macdonald polynomials. The formulas (98) and (100) are concrete and attractive, and the paper does a good job of situating them in the Schur/Macdonald framework. But the evidence for the central eigenfunction claim is thin: the formulas are checked only up to |λ|=25/2, and that check is not independent because the auxiliary bosonic operator s and fermionic pair ν,ν† with correlator (101) are chosen to reproduce the target eigenvalues. In that sense the construction is reverse-engineered, and the finite verification doesn't establish the general statement.

What's genuinely new: the Grassmann-time extension of the Macdonald vertex-operator Hamiltonian, the explicit four-Hamiltonian structure, and the observation that the q,t inversion symmetry is broken for super-Macdonald polynomials. That last point is interesting and, as far as I know, not in the earlier literature. The paper is also honest: it labels the formulas conjectural and describes the correlator as a 'mere simplification trick.'

The soft spots, in order of severity. First, the eigenvalue claim for (98) and (100) is load-bearing and unproved. The finite check is a useful sanity test but, because the ansatz has free parameters that are tuned to the eigenvalues, it doesn't rule out a failure at higher degree. Second, the Pierri rules (102)-(103) for super-Macdonald polynomials are assumed without proof, and the commuting family (106)-(107) is derived from them. Third, the paper does not explain how the new Hamiltonians relate to the existing supersymmetric Ruijsenaars operators of [5,29]; a comparison would help readers judge whether this is a genuinely different construction or a repackaging.

The paper is worth engaging with. I would send it to a serious referee, with the expectation that the referee will ask for either a proof of the eigenvalue claim, a more extensive check that is not fitted to the answer, or at least a clear statement that the central result is a conjecture whose only evidence is the finite check. For a reader working on super-Macdonald polynomials or Yangian/DIM representations, it's a useful pointer to a plausible structure. But I wouldn't treat the Hamiltonians as established until the verification is independent or a proof appears.

Referee Report

3 major / 5 minor

Summary. The paper proposes a supersymmetric generalization of the Macdonald finite-difference Hamiltonian. After reviewing the Schur, super-Schur, and Macdonald cases, the authors introduce Grassmann time variables θ_k and define four super-Hamiltonians whose claimed eigenfunctions are super-Macdonald polynomials. The eigenvalue formula (97) has the expected sum-over-boxes form with weights q^{2j}t^{-2i}. Closed vertex-operator expressions for the four Hamiltonians are given in Eqs. (98) and (100); the authors state that these formulas are conjectural and have been checked only up to diagrams of order |λ| = 25/2. The paper also constructs a commuting family of higher Hamiltonians in Eqs. (106)-(107), based on Pieri rules for super-Macdonald polynomials stated in Eqs. (102)-(103). Sections 2-4 provide background and set up the general "sum-over-boxes plus Pieri rules" framework.

Significance. If the conjectural formulas are correct, the paper provides an explicit Hamiltonian characterization of super-Macdonald polynomials in a vertex-operator form, with potential applications to super-Yangian and DIM representations. The paper's strengths are its explicit formulas, the consistency checks at order |λ| = 25/2, and the reduction to ordinary Macdonald Hamiltonians on even diagrams. However, the main eigenfunction statement is not proven at all orders, and the verification of the H^{-} operators relies on an auxiliary correlator that appears chosen to reproduce the target eigenvalues; the result is therefore conditional. The paper is internally consistent, but the load-bearing claims require either a proof or a substantially stronger and more independent body of evidence.

major comments (3)

[Section 5, Eqs. (98), (100)] The central claim of the paper is that the operators defined by (98) and (100) have super-Macdonald polynomials as eigenfunctions with the eigenvalues (97). The authors explicitly label these formulas as conjectural and report a check only up to |λ| = 25/2, with no all-orders derivation. Since this eigenfunction statement is the paper's main result, the absence of a proof is a load-bearing gap: the finite check does not by itself establish the general statement. I recommend either supplying a proof by a normal-ordering or contraction analysis, or, if the paper is intended as a conjecture, stating this status more prominently and substantially extending the numerical evidence.
[Section 5, Eq. (101)] The auxiliary correlator ⟨∅|νν†s^b|∅⟩_B is introduced ad hoc and described by the authors as a "mere simplification trick." Because the H^{-} operators in (100) are defined through this correlator, and the correlator is chosen so that the desired eigenvalue terms survive, the finite-order check of (100) is not independent of the target eigenvalues in (97). This weakens the evidential value of the check for the all-orders identity. A concrete way to address this is to derive (101) from a principled algebraic requirement such as a free-field realization, or to verify (100) at orders beyond |λ| = 25/2 with a comparison that does not use the desired eigenvalues as input.
[Section 5, Eqs. (102)-(103)] The Pieri rules for super-Macdonald polynomials are stated without proof and without a precise citation to a place where they are established. The commuting family (106)-(107) and the recursive operators (104)-(105) rest on these rules; as written, this part of the construction is conditional on an additional unproved assumption. Please provide a proof of (102)-(103) or point to the specific statement in the cited literature where these rules are proven.

minor comments (5)

[Section 5, Eq. (97)] In the fourth eigenvalue expression, the summation variables x and y are undefined; they should be the box coordinates i and j as in the preceding lines of (97).
[Throughout] The name "Pierri" is a misspelling of "Pieri" and should be corrected throughout the manuscript.
[Section 3.2] The notation for the two families of super-Schur Hamiltonians, with two labels and multiple hats, is difficult to follow; a short glossary or table of the operators would improve readability.
[Section 3.3] The sentence beginning "Operators as being split in brackets in the r.h.s." in the discussion after Eq. (64) is grammatically incomplete and should be rephrased.
[Eq. (100)] The typesetting of the second Hamiltonian in (100) contains an anomalous vertical bar adjacent to the vacuum ket; please fix the formatting.

Circularity Check

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No significant circularity: the super-Hamiltonian formulas are explicitly conjectural and reverse-engineered from the target eigenvalues, which the authors acknowledge; the finite check is presented as evidence, not as a derived prediction.

full rationale

The paper does not claim to derive the eigenvalues (97) from the explicit formulas (98) and (100). Instead, it defines the four super-Hamiltonians through the eigenvalue equations (97) and then proposes closed contour-integral formulas (98) and (100), stating: 'Based on super-Macdonald polynomials [2–5] and eigenvalues (97) we compute explicit form of the super-Hamiltonians and propose the following conjectural formulas (98) and (100) for super-Hamiltonians. We have checked our formulas for super-Hamiltonians up to diagrams of order |λ| = 25/2.' Reverse engineering an operator from a desired spectrum is not circular as long as the resulting operator formula is genuinely new content, and here it is explicitly labeled conjectural. The auxiliary correlator (101) is admittedly a 'mere simplification trick', which limits the evidential weight of the finite check, but it does not make the conjecture equivalent to its input: the closed operator expression is an independent mathematical assertion whose general validity is not established. The Pieri rules (102)–(103) are quoted from prior literature, including independent sources [4,5], rather than derived from the target result. The commuting family (106)–(107) depends on those rules, so if the rules were unproved that would be a rigor gap, not a circular step. The 'by definition' sum-over-boxes remark refers only to the initial eigenvalue definition, not to a derived prediction. Overall, the paper is honest about the conjectural status of its central formulas, and no load-bearing step reduces by construction to its own inputs.

Assumptions & free parameters 1 free parameters · 3 assumptions · 1 invented entities

The construction assumes the full algebraic package of super-Macdonald polynomials from prior work, adds an ad hoc auxiliary correlator to close the vertex-operator formulas, and transfers the bosonic commutativity argument to the super case. The central mathematical input is therefore largely assumed or reverse-engineered rather than derived inside the paper.

free parameters (1)

Auxiliary correlator ⟨∅|νν†s^b|∅⟩_B = 1-(qt)^{2b+2} / (q^{2b}(1-(qt)^2)) for b ≥ 0, 0 otherwise
This correlator in Eq. (101) is an ad hoc choice that determines which terms survive in the super-Hamiltonians (100). It is chosen to reproduce the target super-Macdonald eigenvalues rather than derived from independent principles.

assumptions (3)

domain assumption Super-Macdonald polynomials as defined in [2-4] exist and satisfy the stated Pierri rules (102)-(103) with rational coefficients.
The commuting-family construction and the eigenvalue statements depend on these rules, but the paper states them without proof or a specific reference to where they are established.
ad hoc to paper The auxiliary bosonic operator s and fermionic pair ν,ν† with correlator (101) correctly encode the super-Macdonald eigenvalue data.
The authors describe these fields as a 'mere simplification trick' and give no independent justification for the correlator, so the closed forms (100) stand or fall with this choice.
domain assumption The higher Hamiltonians H'_a+b = {E_a,F_b} commute for the super-Macdonald case, as asserted in (107).
Only the base anti-commutators (56)-(57) are verified; the general commutativity is transferred from the bosonic Schur and Macdonald reasoning without a super-specific proof.

invented entities (1)

Auxiliary invertible bosonic operator s and fermionic pair ν,ν†
purpose: They are used in (100) to build the second pair of super-Hamiltonians and to suppress unwanted terms involving products of p_k or θ derivatives.
The authors explicitly state they do not know a deeper meaning and use them as a simplification trick; no falsifiable prediction or independent algebraic characterization is provided.

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Cite this review

Pith. "Pith review of Super-Hamiltonians for super-Macdonald polynomials." pith.science (2026). https://pith.science/paper/J76JPDFG

@misc{pith2026250114714,
  author       = {Pith},
  title        = {Pith review of: Super-Hamiltonians for super-Macdonald polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J76JPDFG}},
  note         = {Machine review of arXiv:2501.14714}
}

abstract

The Macdonald finite-difference Hamiltonian is lifted to a super-generalization. In addition to canonical bosonic time variables $p_k$ new Grassmann time variables $\theta_k$ are introduced, and the Hamiltonian is represented as a differential operator acting on a space of functions of both types of variables $p_k$ and $\theta_k$. Eigenfunctions for this Hamiltonian are a suitable generalization of Macdonald polynomials to super-Macdonald polynomials discussed earlier in the literature. Peculiarities of the construction in comparison to the canonical bosonic case are discussed.

Figures

Figures reproduced from arXiv: 2501.14714 by the authors.

**Figure 1.** Sets Add(⋆) and Rem(⋆). We denote by Add(λ) a subset of border places in diagram λ such that one can add a box to that place and the resulting diagram remains a Young diagram. Rem(λ) is the set of boxes in diagram λ that one can remove and the resulting diagram remains a Young diagram. An explanatory picture is presented on Fig.1. 2.2 Higher Hamiltonians By combining two properties – the sum over boxes in eigenvalue… view at source ↗

**Figure 2.** Sets Add(⋆) and Rem(⋆). where Cλ,λ± and Cλ,λ± are some constants. The definition of sets Add(λ) and Rem(λ) is clear from the Fig.2. 3.2 Higher Hamiltonians To construct higher Hamiltonians for the super-case we follow a procedure similar to the one presented in Sect.2. At the first step we define higher operators recursively: Eˆ k := h Wˆ , Eˆ k−1 i Eˆ k := h Wˆ , Eˆ k−1 i (53) Fˆ k := − h Wˆ , Fˆ k−1 i Fˆ k := − h … view at source ↗

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

I. G. Macdonald, Symmetric Functions and Hall Polynomials . Oxford Mathematical Monographs, 1998

work page 1998

[2] [2]

Macdonald po lynomials for super-partitions,

D. Galakhov, A. Morozov, and N. Tselousov, “Macdonald po lynomials for super-partitions,” Phys. Lett. B 856 (2024) 138911 , arXiv:2407.03301 [hep-th]

arXiv 2024

[3] [3]

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D. Galakhov, A. Morozov, and N. Tselousov, “Supersymmet ric polynomials and algebro-combinatorial duality,” SciPost Phys. 17 no. 4, (2024) 119 , arXiv:2407.04810 [hep-th]

arXiv 2024

[4] [4]

Macdonald polynomials in superspace: conjectural deﬁnition and positivity conje ctures,

O. Blondeau-Fournier, P. Desrosiers, L. Lapointe, and P. Mathieu, “Macdonald polynomials in superspace: conjectural deﬁnition and positivity conje ctures,” Lett. Math. Phys. 101 (2012) 27–47 , arXiv:1112.5188 [math-ph]

arXiv 2012

[5] [5]

Macdonald polynomials in superspace as eigenfunctions of commuting operators,

O. Blondeau-Fournier, P. Desrosiers, L. Lapointe, and P. Mathieu, “Macdonald polynomials in superspace as eigenfunctions of commuting operators,” J. Comb. 3 (2012) 495–561 , arXiv:1202.3922 [math-ph]

arXiv 2012

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D. Galakhov, A. Morozov, and N. Tselousov, “Super-Schur polynomials for Aﬃne Super Yangian Y( ˆgl1|1),” JHEP 08 (2023) 049 , arXiv:2307.03150 [hep-th]

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