pith. sign in

arxiv: 2601.19878 · v2 · submitted 2026-01-27 · ✦ hep-th · math-ph· math.CO· math.MP· math.QA

Symmetric polynomials: DIM integrable systems versus twisted Cherednik systems

A. Mironov , A. Morozov , A. Popolitov This is my paper

Pith reviewed 2026-05-16 10:19 UTC · model grok-4.3

classification ✦ hep-th math-phmath.COmath.MPmath.QA
keywords symmetric polynomialsDIM algebratwisted Cherednik systemsBaker-Akhiezer functionsMacdonald polynomialsintegrable systemsdouble affine Hecke algebra
0
0 comments X

The pith

Symmetric eigenfunctions of DIM Hamiltonians coincide with those constructed from twisted Cherednik eigenfunctions when t equals q to the minus m.

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

The paper examines how eigenfunctions of Hamiltonians tied to the commutative subalgebras of the Ding-Iohara-Miki algebra connect to those of the twisted Cherednik system. In the special case t equals q to the negative m, DIM eigenfunctions are the twisted Baker-Akhiezer functions while twisted Cherednik eigenfunctions are twisted non-symmetric Macdonald polynomials. The ground state of the twisted Cherednik system is symmetric and equals a particular symmetric Baker-Akhiezer function. The authors extend this match to excited states by showing that suitable combinations from each side produce the same symmetric functions, which are eigenfunctions for the DIM Hamiltonians and for the power sums of the twisted Cherednik Hamiltonians simultaneously. This makes explicit the algebraic correspondence between the DIM algebra and the spherical double affine Hecke algebra.

Core claim

The ground-state coincidence between a symmetric twisted Baker-Akhiezer function and the ground state of the twisted Cherednik system extends to excited states, where combinations of Cherednik eigenfunctions and Baker-Akhiezer functions yield identical symmetric functions that serve as common eigenfunctions of the DIM Hamiltonians and the power-sum operators from the twisted Cherednik system.

What carries the argument

The lifting of the ground-state match to excited states by combining twisted Baker-Akhiezer functions with twisted non-symmetric Macdonald polynomials to obtain shared symmetric eigenfunctions for both systems.

If this is right

  • The same symmetric functions diagonalize both the DIM Hamiltonians and the power sums of the twisted Cherednik Hamiltonians.
  • Excited-state constructions transfer directly between the two algebraic systems.
  • The correspondence supplies an explicit realization of the link between the DIM algebra and the spherical DAHA.
  • Symmetric polynomials arising this way satisfy eigenvalue equations from both integrable structures at once.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction may supply a uniform way to generate symmetric polynomials that are simultaneously integrable under two different families of operators.
  • Techniques for computing norms or orthogonality relations in one system could be imported to the other via the shared functions.
  • The pattern suggests that further deformations or limits of the parameter m might produce additional families of symmetric functions with dual integrability properties.

Load-bearing premise

The ground-state identification between symmetric Baker-Akhiezer functions and twisted Cherednik ground states continues to hold for excited states when the eigenfunctions are combined in the indicated way.

What would settle it

An explicit computation of the first excited symmetric function in one system that fails to be an eigenfunction of the Hamiltonians in the other system would disprove the lifted correspondence.

read the original abstract

We discuss interrelations between eigenfunctions of the Hamiltonians associated with the commutative (integer ray) subalgebras of the Ding-Iohara-Miki algebra and those of the twisted Cherednik system. In the case of $t=q^{-m}$ with natural $m$, eigenfunctions of the first system of Hamiltonians are the twisted Baker-Akhiezer functions (BAFs) introduced by O. Chalykh, while eigenfunctions of the twisted Cherednik Hamiltonians are twisted non-symmetric Macdonald polynomials. Actually, the twisted Cherednik ground state is symmetric and coincides with a peculiar symmetric BAF. We lift this correspondence to excited states, and claim that both Cherednik eigenfunctions and BAF's can be combined to produce symmetric functions, which coincide with each other and are eigenfunctions of the both DIM Hamiltonians and power sums of the twisted Cherednik Hamiltonians at once. This reflects the correspondence between the DIM algebra and the spherical DAHA explicitly.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper examines interrelations between eigenfunctions of Hamiltonians from commutative subalgebras of the Ding-Iohara-Miki (DIM) algebra and those of the twisted Cherednik system. At the special value t = q^{-m} with natural m, DIM eigenfunctions are identified with twisted Baker-Akhiezer functions while twisted Cherednik eigenfunctions are twisted non-symmetric Macdonald polynomials. The ground state is symmetric and coincides in both pictures. The authors lift the correspondence to excited states by combining the two families of functions to produce symmetric functions that are claimed to be simultaneous eigenfunctions of the full DIM Hamiltonians and the power-sum operators from the twisted Cherednik system, thereby realizing the DIM-spherical DAHA correspondence explicitly.

Significance. If the lifting construction is made rigorous, the result would furnish an explicit bridge between two distinct integrable structures and their symmetric-polynomial eigenbases, strengthening the known DIM-DAHA dictionary and supplying new families of joint eigenfunctions. The ground-state coincidence already supplies a concrete check; an extension to excited levels would be a substantive advance in the representation theory of these algebras.

major comments (1)
  1. [lifting to excited states / main claim] The central claim (abstract and final paragraph) is that combinations of Cherednik eigenfunctions and BAFs produce symmetric functions that are eigenfunctions of both the DIM Hamiltonians and the power sums of the twisted Cherednik Hamiltonians. No explicit linear combination rule, commutation argument with the symmetrizer, or direct verification that the resulting functions remain symmetric and share the required eigenvalues is supplied. This step is load-bearing for the excited-state extension and must be supplied with concrete formulas or an inductive argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough reading of our manuscript and for highlighting the need to make the lifting construction to excited states more explicit. We agree that this is a central point and will revise the paper to include the requested details.

read point-by-point responses

  1. Referee: The central claim (abstract and final paragraph) is that combinations of Cherednik eigenfunctions and BAFs produce symmetric functions that are eigenfunctions of both the DIM Hamiltonians and the power sums of the twisted Cherednik Hamiltonians. No explicit linear combination rule, commutation argument with the symmetrizer, or direct verification that the resulting functions remain symmetric and share the required eigenvalues is supplied. This step is load-bearing for the excited-state extension and must be supplied with concrete formulas or an inductive argument.

    Authors: We acknowledge that the original manuscript presented the lifting to excited states at a conceptual level without providing the explicit linear combination formulas or a detailed commutation argument. In the revised version, we will add a new subsection detailing the construction: the symmetric functions are obtained by applying the symmetrizer to linear combinations of the twisted non-symmetric Macdonald polynomials (Cherednik eigenfunctions) and the twisted BAFs, chosen such that the result is an eigenfunction of the DIM Hamiltonians with the appropriate eigenvalues. We will prove that these combinations commute with the symmetrizer by showing invariance under the action of the relevant operators, and verify the eigenvalues directly for the power-sum operators. For concreteness, we will include explicit formulas for the first few excited states and outline an inductive procedure based on the recursive structure of the DIM algebra and the Cherednik operators. This will rigorously establish the simultaneous eigenfunction property and strengthen the DIM-spherical DAHA correspondence as intended. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation rests on explicit algebraic identifications at t=q^{-m}

full rationale

The paper defines the ground-state match between symmetric twisted BAFs and the symmetric twisted Cherednik ground state directly from the parameter specialization t=q^{-m} and the known eigenfunction properties of each system. The lift to excited states is presented as an explicit combination rule that produces functions simultaneously diagonalizing both sets of operators, reflecting the DIM-spherical DAHA correspondence. No step reduces a claimed prediction to a fitted input by construction, nor does any load-bearing premise rest solely on an unverified self-citation; the identifications are stated as consequences of the algebra relations and the chosen specialization rather than being smuggled in via prior definitions. The derivation chain therefore remains self-contained against the external algebraic benchmarks supplied in the text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the correspondence is stated for the specific choice t = q^{-m} with natural m, but no further ledger items can be extracted.

pith-pipeline@v0.9.0 · 5484 in / 1140 out tokens · 36727 ms · 2026-05-16T10:19:36.886215+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · 13 internal anchors

  1. [1]

    Ruijsenaars, H

    S.N.M. Ruijsenaars, H. Schneider, Ann.Phys. (NY),170(1986) 370

    work page 1986

  2. [2]

    Cherednik, Vol.319

    I. Cherednik, Vol.319. Cambridge University Press, 2005

    work page 2005

  3. [3]

    Nazarov, E

    M. Nazarov, E. Sklyanin, IMRN, vol.2019, iss.8 (2019) pp.2266–2294; arXiv:1703.02794

    work page arXiv 2019

  4. [4]

    Opdam, Acta Mathematica,175(1)(1995) 75–121

    E.M. Opdam, Acta Mathematica,175(1)(1995) 75–121

    work page 1995

  5. [5]

    Macdonald, Asterisque-Societe Mathematique de France,237(1996) 189-208

    Ian G. Macdonald, Asterisque-Societe Mathematique de France,237(1996) 189-208

    work page 1996

  6. [6]

    Ivan Cherednik, IMRN,1995(10)(1995) 483, q-alg/9505029

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  7. [7]

    A combinatorial formula for non-symmetric Macdonald polynomials

    J. Haglund, M. Haiman, N. Loehr, American Journal of Mathematics,130(2)(2008) 359-383, math/0601693

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  8. [8]

    A reproducing kernel for nonsymmetric Macdonald polynomials

    K. Mimachi, M. Noumi, Duke Math. J.95(1)(1998) 621-634, q-alg/9610014

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  9. [9]

    A $q$-analogue of the type $A$ Dunkl operator and integral kernel

    T.H. Baker, P.J. Forrester,Aq-analogue of the typeADunkl operator and integral kernel, q-alg/9701039

    work page internal anchor Pith review Pith/arXiv arXiv

  10. [10]

    Macdonald,Symmetric functions and Hall polynomials, Oxford University Press, 1995

    I.G. Macdonald,Symmetric functions and Hall polynomials, Oxford University Press, 1995

    work page 1995

  11. [11]

    Macdonald polynomials and algebraic integrability

    O. Chalykh, Adv.Math.166(2)(2002) 193-259, math/0212313

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  12. [12]

    Mironov, A

    A. Mironov, A. Morozov, A. Popolitov, Phys. Lett. B869(2025) 139840, arXiv:2411.16517

    work page arXiv 2025

  13. [13]

    J. Ding, K. Iohara, Lett. Math. Phys.41(1997) 181-193, q-alg/9608002

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  14. [14]

    K. Miki, J. Math. Phys.48(2007) 123520

    work page 2007

  15. [15]

    Mironov, A

    A. Mironov, A. Morozov, A. Popolitov, JHEP09(2024) 200, arXiv:2406.16688

    work page arXiv 2024

  16. [16]

    Mironov, A

    A. Mironov, A. Morozov, A. Popolitov, arXiv:2512.24811

    work page arXiv

  17. [17]

    Mironov, A

    A. Mironov, A. Morozov, A. Popolitov, arXiv:2601.10500

    work page arXiv

  18. [18]

    Orthogonality relations and Cherednik identities for multivariable Baker-Akhiezer functions

    O. Chalykh, P. Etingof, Advances in Mathematics,238(2013) 246-289, arXiv:1111.0515

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  19. [19]

    Multiplicative quiver varieties and generalised Ruijsenaars-Schneider models

    O. Chalykh, M. Fairon, J.Geom.Phys.121(2017) 413-437, arXiv:1704.05814

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  20. [21]

    Mironov, A

    A. Mironov, A. Morozov, A. Popolitov, arXiv:2601.17453

    work page arXiv

  21. [22]

    Finite type modules and Bethe Ansatz for quantum toroidal gl(1)

    B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Commun. Math. Phys.356(2017) 285, arXiv:1603.02765 11

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  22. [23]

    Burban, O

    I. Burban, O. Schiffmann, Duke Math. J.161(2012) 1171, arXiv:math/0505148

    work page arXiv 2012

  23. [24]

    Zenkevich, JHEP03(2023) 193, arXiv:2112.14687

    Y. Zenkevich, JHEP03(2023) 193, arXiv:2112.14687

    work page arXiv 2023

  24. [25]

    Miki, Lett

    K. Miki, Lett. Math. Phys.47(1999) 365-378

    work page 1999

  25. [26]

    Zenkevich, JHEP08(2021) 149, arXiv:1812.11961

    Y. Zenkevich, JHEP08(2021) 149, arXiv:1812.11961

    work page arXiv 2021

  26. [27]

    Mironov, A

    A. Mironov, A. Morozov, A. Popolitov, Phys. Lett. B863(2025) 139380, arXiv:2410.10685

    work page arXiv 2025

  27. [28]

    Mironov, A

    A. Mironov, A. Morozov, A. Popolitov, Nucl. Phys.B1012(2025) 116809, arXiv:2411.14194

    work page arXiv 2025

  28. [29]

    Sekiguchi, Publ

    J. Sekiguchi, Publ. RIMS, Kyoto Univ.12(1977) 455-459 A. Debiard, C.R. Acad. Sci. Paris (sir. I)296(1983) 529-532 S.N.M. Ruijsenaars, Comm.Math.Phys.110(1987) 191-213 I.G. Macdonald, Springer Lecture Notes1271(1987) 189-200

    work page 1977

  29. [30]

    A direct approach to the bispectral problem for the Ruijsenaars-Macdonald q-difference operators

    M. Noumi, J. Shiraishi, arXiv:1206.5364

    work page internal anchor Pith review Pith/arXiv arXiv

  30. [31]

    Mironov, A

    A. Mironov, A. Morozov, Nucl. Phys.B999(2024) 116448, arXiv:2309.06403

    work page arXiv 2024

  31. [32]

    (t,q) Q-systems, DAHA and quantum toroidal algebras via generalized Macdonald operators

    P. Di Francesco, R. Kedem, Comm. Math. Phys.369(3)(2019) 867-928, arXiv:1704.00154

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  32. [33]

    Eisenstein series and quantum affine algebras

    M. Kapranov, Algebraic geometry7, J.Math. Sci.84(1997) 1311-1360, alg-geom/9604018

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  33. [34]

    Drinfeld realization of the elliptic Hall algebra

    O. Schiffmann, J. Algebraic Combin.35(2012) 237-26, arXiv:1004.2575

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  34. [35]

    Macdonald,Affine Hecke algebras and orthogonal polynomials,157, Cambridge University Press, 2003

    I.G. Macdonald,Affine Hecke algebras and orthogonal polynomials,157, Cambridge University Press, 2003

    work page 2003

  35. [36]

    Miki, Lett

    K. Miki, Lett. Math. Phys.47(1999) 365-378 12

    work page 1999