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arxiv: 2404.03851 · v2 · submitted 2024-04-05 · 🧮 math.CO · math.NT· math.RT

Remarks on the conjectures of Capparelli, Meurman, Primc and Primc

Shashank Kanade , Matthew C. Russell , Shunsuke Tsuchioka , S. Ole Warnaar This is my paper

Pith reviewed 2026-05-24 01:55 UTC · model grok-4.3

classification 🧮 math.CO math.NTmath.RT
keywords CMPP conjecturescoloured partitionsaffine Lie algebrasRogers-Ramanujan identitiesgenerating functionsHall-Littlewood functionslevel-rank duality
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The pith

The two remaining CMPP conjectures correspond to non-standard specializations of standard modules for A_{2n}^{(2)} and D_{n+1}^{(2)}.

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

This paper resolves the open cases of the CMPP conjectures on generating functions for coloured partitions with multiplicity restrictions by linking them to affine Lie algebra representation theory. It applies prior results on Rogers-Ramanujan identities to identify the generating functions with non-standard specializations of modules over A_{2n}^{(2)} and D_{n+1}^{(2)}. The work further proposes conjectures expressing one-parameter bivariate generating functions via Hall-Littlewood symmetric functions and derives functional equations, a partial level-rank duality, and additional conjectural identities. A reader would care because this supplies the missing algebraic interpretations that connect combinatorial partition problems directly to character formulas of affine Lie algebras.

Core claim

The remaining two sets of conjectures by Capparelli, Meurman, Primc and Primc admit Lie-algebraic interpretations in terms of non-standard specialisations of standard modules for A_{2n}^{(2)} and D_{n+1}^{(2)}, obtained by applying the Rogers-Ramanujan identities for affine Lie algebras established by Griffin, Ono and Warnaar; this also yields conjectures for bivariate generating functions in terms of Hall-Littlewood symmetric functions together with functional equations generalising the Rogers-Selberg equations, a partial level-rank duality for A_{2n}^{(2)}, and conjectural Rogers-Ramanujan type identities for D_3^{(2)}.

What carries the argument

Non-standard specializations of standard modules for the affine Lie algebras A_{2n}^{(2)} and D_{n+1}^{(2)}, which produce the infinite-product forms conjectured for the generating functions of the CMPP partitions.

If this is right

  • Bivariate generating functions for one-parameter families of CMPP partitions equal expressions in Hall-Littlewood symmetric functions.
  • Bivariate generating functions satisfy functional equations that generalize the classical Rogers-Selberg equations.
  • A partial level-rank duality holds for the A_{2n}^{(2)} case.
  • Conjectural Rogers-Ramanujan type identities hold for the algebra D_3^{(2)}.

Where Pith is reading between the lines

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

  • The Lie-algebraic identification may permit vertex-operator or other representation-theoretic proofs of the original conjectures.
  • The observed partial duality hints at possible recursive relations or bijective proofs between partition families at different ranks.
  • The same specialization technique could be tested on other unresolved families of coloured-partition generating functions.

Load-bearing premise

The identities for Rogers-Ramanujan type specializations of affine Lie algebra modules extend directly to the non-standard cases required by the remaining CMPP partition sets.

What would settle it

Explicit computation of the first several coefficients of the partition generating function for a fixed small n and direct comparison against the corresponding specialized character; any mismatch disproves the claimed equality.

read the original abstract

In a series of two papers, S. Capparelli, A. Meurman, A. Primc, M. Primc (CMPP) and then M. Primc put forth three remarkable sets of conjectures, stating that the generating functions of coloured integer partition in which the parts satisfy restrictions on the multiplicities admit simple infinite product forms. While CMPP related one set of conjectures to the principally specialised characters of standard modules for the affine Lie algebra $\mathrm{C}_n^{(1)}$, finding a Lie-algebraic interpretation for the remaining two sets remained an open problem. In this paper, we use the work of Griffin, Ono and the fourth author on Rogers-Ramanujan identities for affine Lie algebras to solve this problem, relating the remaining two sets of conjectures to non-standard specialisations of standard modules for $\mathrm{A}_{2n}^{(2)}$ and $\mathrm{D}_{n+1}^{(2)}$. We also use their work to formulate conjectures for the bivariate generating function of one-parameter families of CMPP partitions in terms of Hall-Littlewood symmetric functions. We make a detailed study of several further aspects of CMPP partitions, obtaining (i) functional equations for bivariate generating functions which generalise the well-known Rogers-Selberg equations, (ii) a partial level-rank duality in the $\mathrm{A}_{2n}^{(2)}$ case, and (iii) (conjectural) identities of the Rogers-Ramanujan type for $\mathrm{D}_3^{(2)}$.

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

2 major / 2 minor

Summary. The paper resolves the two remaining open sets of CMPP conjectures on the generating functions for colored partitions with multiplicity restrictions by relating them to non-standard specializations of standard modules for the affine Lie algebras A_{2n}^{(2)} and D_{n+1}^{(2)}, via direct application of the Rogers-Ramanujan identities established by Griffin, Ono, and Warnaar. It further formulates conjectures for the bivariate generating functions of one-parameter families of these partitions in terms of Hall-Littlewood symmetric functions, derives functional equations generalizing the Rogers-Selberg equations, establishes a partial level-rank duality in the A_{2n}^{(2)} case, and proposes (conjectural) Rogers-Ramanujan-type identities for D_3^{(2)}.

Significance. If the identifications hold, the work completes the Lie-algebraic interpretation of all CMPP conjectures, extending the framework of affine Lie algebra characters to non-standard specializations and yielding new partition identities. The additional conjectures, functional equations, and duality results provide concrete directions for further research in the area of Rogers-Ramanujan-type identities and their Lie-algebraic origins.

major comments (2)
  1. [Abstract / main identification step] Abstract and the section invoking Griffin-Ono-Warnaar: the central claim that their results on Rogers-Ramanujan identities for affine Lie algebras directly supply the precise non-standard specializations whose characters match the restricted CMPP partition generating functions for A_{2n}^{(2)} and D_{n+1}^{(2)} is load-bearing. The manuscript must explicitly identify which theorems or corollaries from the cited work cover these non-standard cases (as opposed to principal or other standard specializations) and verify the character equality without additional limits or identities.
  2. [Conjecture formulation section] The formulation of the new conjectures for bivariate generating functions in terms of Hall-Littlewood polynomials relies on the same application; if the non-standard specialization step requires extra justification, these conjectures inherit the same gap and should be presented with the same level of supporting evidence as the resolved cases.
minor comments (2)
  1. Notation for the affine algebras (e.g., A_{2n}^{(2)}) should be cross-checked for consistency with standard references throughout the text.
  2. The partial level-rank duality statement would benefit from an explicit statement of the range of parameters for which it holds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive summary and for identifying the need for greater explicitness in the central identifications. We address both major comments below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: Abstract and the section invoking Griffin-Ono-Warnaar: the central claim that their results on Rogers-Ramanujan identities for affine Lie algebras directly supply the precise non-standard specializations whose characters match the restricted CMPP partition generating functions for A_{2n}^{(2)} and D_{n+1}^{(2)} is load-bearing. The manuscript must explicitly identify which theorems or corollaries from the cited work cover these non-standard cases (as opposed to principal or other standard specializations) and verify the character equality without additional limits or identities.

    Authors: We agree that the manuscript would benefit from explicit citation of the precise theorems and corollaries in Griffin-Ono-Warnaar that apply to the non-standard specializations. In the revision we will add a short dedicated paragraph (or subsection) that names the relevant results (e.g., the specific theorem or corollary for each algebra) and shows, by direct substitution of the appropriate parameters, that the character equalities follow immediately from those statements. No additional limits or auxiliary identities will be invoked. revision: yes

  2. Referee: The formulation of the new conjectures for bivariate generating functions in terms of Hall-Littlewood polynomials relies on the same application; if the non-standard specialization step requires extra justification, these conjectures inherit the same gap and should be presented with the same level of supporting evidence as the resolved cases.

    Authors: We accept that the conjectures inherit the same requirement for explicit linkage. In the revised manuscript we will expand the conjecture section to include the same explicit references to the Griffin-Ono-Warnaar theorems and the corresponding parameter choices used for the resolved cases. Where the supporting evidence remains conjectural we will label it clearly, ensuring the level of justification matches that given for the main results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claim rests on external prior identities

full rationale

The paper's main step is to invoke the Griffin-Ono-Warnaar Rogers-Ramanujan identities for affine Lie algebras (a prior publication whose authors overlap only partially with the present work) to furnish non-standard specializations for A_{2n}^{(2)} and D_{n+1}^{(2)}. No equation or generating-function identity inside the present manuscript is shown to be obtained by fitting parameters to the target CMPP generating functions and then relabeling the fit as a prediction. The cited identities are externally stated theorems about characters of standard modules; they are not derived from the CMPP conjectures themselves. Consequently the derivation chain does not reduce to self-definition or to a load-bearing self-citation loop. The paper is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only; the central claims rest on standard background from affine Lie algebra representation theory and the cited Rogers-Ramanujan results for those algebras. No free parameters, invented entities, or ad-hoc axioms are visible in the provided text.

axioms (2)
  • standard math Standard facts on principally specialized characters of integrable highest weight modules for affine Lie algebras
    Invoked to relate generating functions to characters
  • domain assumption Existence and properties of non-standard specializations as used in the cited Griffin-Ono-Warnaar work
    Used to obtain the interpretations for A_{2n}^{(2)} and D_{n+1}^{(2)}

pith-pipeline@v0.9.0 · 5827 in / 1355 out tokens · 20513 ms · 2026-05-24T01:55:37.341377+00:00 · methodology

discussion (0)

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Combinatorial construction of known positive series for partition classes defined by Capparelli, Meurman, Primc, and Primc in the $k$=1 Case

    math.CO 2026-05 unverdicted novelty 6.0

    The paper supplies a base-partition-and-moves combinatorial model for Russell's bivariate generating series of CMPP partitions in the k=1 case and completes several missing cases.

  2. Combinatorial construction of known positive series for partition classes defined by Capparelli, Meurman, Primc, and Primc in the $k$=1 Case

    math.CO 2026-05 unverdicted novelty 4.0

    The paper combinatorially constructs and extends known positive series for k=1 CMPP partitions via base partitions and moves, supplying missing cases from prior symbolic work.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    G. E. Andrews, Sieves in the theory of partitions , Amer. J. Math. 94 (1972), 1214–1230

    work page 1972

  2. [2]

    G. E. Andrews, An analytic generalization of the Rogers–Ramanujan identi ties for odd moduli , Proc. Nat. Acad. Sci. USA 71 (1974), 4082–4085

    work page 1974

  3. [3]

    G. E. Andrews, The Theory of Partitions , Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-W esley, Reading, Massachusetts, 1976

    work page 1976

  4. [4]

    G. E. Andrews, Multiple series Rogers–Ramanujan type identities , Pacific J. Math. 114 (1984), 267–283

    work page 1984

  5. [5]

    G. E. Andrews, Parity in partition identities , Ramanujan J. 23 (2010) 45–90

    work page 2010

  6. [6]

    W. N. Bailey, Identities of the Rogers–Ramanujan type , Proc. London Math. Soc. (2) 50 (1949), 1–10

    work page 1949

  7. [7]

    Baranovi´ c, M

    I. Baranovi´ c, M. Primc and G. Trupˇ cevi´ c,Bases of Feigin–Stoyanovsky’s type subspaces for C(1) ℓ , Ramanujan J. 45 (2018), 265–289

    work page 2018

  8. [8]

    Bartlett and S

    N. Bartlett and S. O. W arnaar, Hall–Littlewood polynomials and characters of affine Lie alg ebras, Adv. Math. 285 (2015), 1066–1105

    work page 2015

  9. [9]

    D. M. Bressoud, A generalization of the Rogers–Ramanujan identities for al l moduli , J. Combin. Theory Ser. A 27 (1979), 64–68

    work page 1979

  10. [10]

    D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers– Ramanujan identities , Memoirs Amer. Math. Soc. 24 (1980), 1–54

    work page 1980

  11. [11]

    Butorac, S

    M. Butorac, S. Koˇ zi´ c, A. Meurman and M. Primc, Lepowsky’s and Wakimoto’s product formulas for the affine Lie algebras C(1) l , arXiv:2403.05456

    work page arXiv

  12. [12]

    Capparelli, A

    S. Capparelli, A. Meurman, A. Primc, M. Primc, New partition identities from C(1) ℓ -modules, Glas. Mat. Ser. III 57(77) (2022), 161–184

    work page 2022

  13. [13]

    Chern, Linked partition ideals, directed graphs and q-multi-summations, Electron

    S. Chern, Linked partition ideals, directed graphs and q-multi-summations, Electron. J. Combin. 27 (2020), Paper No. 3.33, 29pp

    work page 2020

  14. [14]

    Chern, Z

    S. Chern, Z. Li, D. Stanton, T. Xue and A. J. Yee, The Ariki–Koike algebras and Rogers–Ramanujan type partitions, arXiv:2209.07713

    work page arXiv

  15. [15]

    Corteel and T

    S. Corteel and T. A. W elsh, The A2 Rogers–Ramanujan identities revisited, Ann. Comb. 23 (2019), 683–694. 36 SHASHANK KANADE, MATTHEW C. RUSSELL, SHUNSUKE TSUCHIOKA , S. OLE W ARNAAR

    work page 2019

  16. [16]

    Characters of level 1 st andard modules of C (1) n as generating functions for generalised partitions

    J. Dousse and I. Konan, Characters of level 1 standard modules of C(1) n as generating functions for gener- alised partitions, arXiv:2212.12728

    work page arXiv

  17. [17]

    Feigin, The PBW filtration , Represent

    E. Feigin, The PBW filtration , Represent. Theory 13 (2009), 165–181

    work page 2009

  18. [18]

    Fulman, The Rogers–Ramanujan identities, the finite general linear groups, and the Hall–Littlewood polynomials, Proc

    J. Fulman, The Rogers–Ramanujan identities, the finite general linear groups, and the Hall–Littlewood polynomials, Proc. Amer. Math. Soc. 128 (2000), 17–25

    work page 2000

  19. [19]

    Gasper and M

    G. Gasper and M. Rahman, Basic Hypergeometric Series , second edition, Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cam bridge, 2004

    work page 2004

  20. [20]

    Gordon, A combinatorial generalization of the Rogers–Ramanujan id entities, Amer

    B. Gordon, A combinatorial generalization of the Rogers–Ramanujan id entities, Amer. J. Math. 83 (1961), 393–399

    work page 1961

  21. [21]

    M. J. Griffin, K. Ono and S. O. W arnaar, A framework of Rogers–Ramanujan identities and their arith metic properties, Duke Math. J. 165 (2016), 1475–1527

    work page 2016

  22. [22]

    Huang and J

    Y. Huang and J. Ruofan, Generating series for torsion-free bundles over singular c urves: rationality, duality and modularity , arXiv:2312.12528

    work page arXiv

  23. [23]

    Ishikawa, F

    M. Ishikawa, F. Jouhet and J. Zeng, A generalization of Kawanaka’s identity for Hall–Littlewo od polynomials and applications, J. Algebraic Combin. 23 (2006), 395–412

    work page 2006

  24. [24]

    N. Jing, K. C. Misra, C. D. Savage, On multi-color partitions and the generalized Rogers–Rama nujan identities, Commun. Contemp. Math. 3 (2001), 533–548

    work page 2001

  25. [25]

    Jouhet and J

    F. Jouhet and J. Zeng, New identities for Hall–Littlewood polynomials and applic ations, Ramanujan J. 10 (2005), 89–112

    work page 2005

  26. [26]

    V. G. Kac, Infinite-dimensional Lie Algebras , 3rd Edition, Cambridge University Press, Cambridge, 1990

    work page 1990

  27. [27]

    Kanade and M

    S. Kanade and M. C. Russell, Completing the A2 Andrews–Schilling–Warnaar identities , Int. Math. Res. Not. IMRN (2023), 17100–17155

    work page 2023

  28. [28]

    Kim and A

    S. Kim and A. J. Yee, The Rogers–Ramanujan–Gordon identities, the generalized G¨ ollnitz–Gordon identi- ties, and parity questions , J. Combin. Theory Ser. A 120 (2013), 1038–1056

    work page 2013

  29. [29]

    A. N. Kirillov, New combinatorial formula for modified Hall–Littlewood pol ynomials, in q-Series from a Contemporary Perspective, pp. 283–333, Contemp. Math. 254, Amer. Math. Soc., Provide nce, RI, 2000

    work page 2000

  30. [30]

    Lassalle and M

    M. Lassalle and M. Schlosser, Inversion of the Pieri formula for Macdonald polynomials Adv. Math. 202 (2006), 289–325

    work page 2006

  31. [31]

    Lepowsky and S

    J. Lepowsky and S. Milne, Lie algebraic approaches to classical partition identitie s, Adv. in Math. 29 (1978), 15–59

    work page 1978

  32. [32]

    Lepowsky, Affine Lie algebras and combinatorial identities in Lie algebras and related topics , pp

    J. Lepowsky, Affine Lie algebras and combinatorial identities in Lie algebras and related topics , pp. 130–156, Lecture Notes in Mathematics No. 933, Springer, Berlin–New York, 1982

    work page 1982

  33. [33]

    Lepowsky and R

    J. Lepowsky and R. L. Wilson, The structure of standard modules. II. The case A(1) 1 , principal gradation , Inv. Math. 79 (1985), 417–442

    work page 1985

  34. [34]

    I. G. Macdonald, Affine root systems and Dedekind’s η-function, Invent. Math. 15 (1972), 91–143

    work page 1972

  35. [35]

    I. G. Macdonald, Symmetric functions and Hall polynomials , 2nd edition, Oxford Univ. Press, New York, 1995

    work page 1995

  36. [36]

    P. A. MacMahon, Combinatory analysis, Vol. I & II , Dover Publications, Inc., Mineola, NY, 2004

    work page 2004

  37. [37]

    Meurman and M

    A. Meurman and M. Primc, Annihilating fields of standard modules of sl(2, C)∼ and combinatorial identities , Mem. Amer. Math. Soc. 137 (1999), no. 652, viii+89 pp

    work page 1999

  38. [38]

    Primc, New partition identities for odd w odd, Rad Hrvat

    M. Primc, New partition identities for odd w odd, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 28(558) (2024), 49–56

    work page 2024

  39. [39]

    Primc and T

    M. Primc and T. ˇSiki´ c,Combinatorial bases of basic modules for affine Lie algebras C(1) n , J. Math. Phys. 57 (2016), 1–19

    work page 2016

  40. [40]

    Primc and T

    M. Primc and T. ˇSiki´ c,Leading terms of relations for standard modules of C(1) n , Ramanujan J. 48 (2019), 509–543

    work page 2019

  41. [41]

    Primc and G

    M. Primc and G. Trupˇ cevi´ c,Linear independence for C(1) ℓ by using C(1) 2ℓ , arXiv:2403.06881

    work page arXiv

  42. [42]

    E. M. Rains and S. O. W arnaar, Bounded Littlewood identities, Mem. Amer. Math. Soc. 270 (2021), no. 1317, vii+115 pp

    work page 2021

  43. [43]

    L. J. Rogers, Second memoir on the expansion of certain infinite products , Proc. London Math. Soc. 25 (1894), 318–343

    work page

  44. [44]

    L. J. Rogers, On two theorems of combinatory analysis and some allied iden tities, Proc. London Math. Soc. (2) 16 (1917), 315–336

    work page 1917

  45. [45]

    L. J. Rogers and S. Ramanujan, Proof of certain identities in combinatory analysis , Proc. Cambridge Phil. Soc. 19 (1919), 211–216

    work page 1919

  46. [46]

    M. C. Russell, Companions to the Andrews–Gordon and Andrews–Bressoud ide ntities, and recent conjec- tures of Capparelli, Meurman, Primc, and Primc , arXiv:2306.16251

    work page internal anchor Pith review Pith/arXiv arXiv

  47. [47]

    Schilling and S

    A. Schilling and S. O. W arnaar, A higher level Bailey lemma: proof and application , Ramanujan J. 2 (1998), 327–349. REMARKS ON THE CONJECTURES OF CAPPARELLI, MEURMAN, PRIMC AN D PRIMC 37

    work page 1998

  48. [48]

    I. J. Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbr¨ uche, S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl. (1917), 302–321

    work page 1917

  49. [49]

    Selberg, ¨Uber einige arithmetische Identit¨ aten, Avh

    A. Selberg, ¨Uber einige arithmetische Identit¨ aten, Avh. Norske Vidensk.-Akad. Oslo Mat.-Naturvidensk. Klasse: Avh. 8 (1936), 1–23

    work page 1936

  50. [50]

    A. V. Sills, An Invitation to the Rogers–Ramanujan Identities , CRC Press, Boca Raton, Florida, 2018

    work page 2018

  51. [51]

    L. J. Slater, A new proof of Rogers’s transformations of infinite series , Proc. London Math. Soc. (2) 53 (1951), 460–475

    work page 1951

  52. [52]

    J. R. Stembridge, Hall–Littlewood functions, plane partitions, and the Roge rs–Ramanujan identities , Trans. Amer. Math. Soc. 319 (1990), 469–498

    work page 1990

  53. [53]

    Tsuchioka, A proof of the second Rogers–Ramanujan identity via Kleshch ev multipartitions , Proc

    S. Tsuchioka, A proof of the second Rogers–Ramanujan identity via Kleshch ev multipartitions , Proc. Japan Acad. Ser. A Math. Sci. 99 (2023), 23–26

    work page 2023

  54. [54]

    S. O. W arnaar, 50 Years of Bailey’s lemma , in Algebraic Combinatorics and Applications , pp. 333–347, A. Betten et al. eds., Springer, Berlin, 2001

    work page 2001

  55. [55]

    S. O. W arnaar, Rogers–Szeg˝ o polynomials and Hall–Littlewood symmetricfunctions, J. Algebra 303 (2006), 810–830

    work page 2006

  56. [56]

    S. O. W arnaar, Hall–Littlewood functions and the A2 Rogers–Ramanujan identities, Adv. Math. 200 (2006), 403–434

    work page 2006

  57. [57]

    S. O. W arnaar, The A2 Andrews–Gordon identities and cylindric partitions , Trans. Amer. Math. Soc., Series B 10 (2023), 715–765

    work page 2023

  58. [58]

    S. O. W arnaar and W. Zudilin, Dedekind’s η-function and Rogers–Ramanujan identities , Bull. Lond. Math. Soc. 44 (2012), 1–11

    work page 2012