pith. sign in

arxiv: 2306.08071 · v2 · pith:WVZ7GRL7new · submitted 2023-06-13 · 🧮 math.CO · math-ph· math.MP· math.RT

Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov--Okounkov type formulas

David Wahiche This is my paper

Pith reviewed 2026-05-24 08:07 UTC · model grok-4.3

classification 🧮 math.CO math-phmath.MPmath.RT
keywords Macdonald identitiesaffine root systemsSchur functionsNekrasov-Okounkov formulashook lengthsinteger partitionsbi-infinite wordscombinatorial interpretations
0
0 comments X

The pith

Viewing partitions as bi-infinite words supplies combinatorial interpretations of the Macdonald identities for all seven infinite families of affine root systems via Schur functions.

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

The paper connects vectors of integers to integer partitions interpreted as bi-infinite words. This produces enumerations that relate products of hook lengths to these vectors. The connection supplies combinatorial interpretations of the Macdonald identities for affine root systems of the seven infinite families, expressed using Schur functions, symplectic Schur functions, and special orthogonal Schur functions. These interpretations enable derivation of q-Nekrasov--Okounkov formulas for each type, whose limits as q tends to one recover the Nekrasov--Okounkov formulas matching Macdonald's specializations.

Core claim

By viewing partitions as bi-infinite words and relating them to vectors of integers, enumerations are obtained that connect products of hook lengths with these vectors. This supplies a combinatorial interpretation of the Macdonald identities for the seven infinite families of affine root systems in terms of Schur functions and their symplectic and special orthogonal counterparts. From these, q-Nekrasov--Okounkov formulas associated to each type are derived, and their limits as q approaches one yield the Nekrasov--Okounkov type formulas corresponding to all the specializations given by Macdonald.

What carries the argument

The representation of partitions as bi-infinite words connected to vectors of integers, which produces uniform enumerations and interpretations for the seven families.

Load-bearing premise

The methodology of viewing partitions as bi-infinite words and connecting them to vectors of integers produces the required enumerations and interpretations uniformly across all seven infinite families of affine root systems.

What would settle it

Direct computation for a small vector of integers in one family, such as type A, showing that the product of hook lengths does not equal the enumeration predicted by the bi-infinite word model would falsify the claim.

Figures

Figures reproduced from arXiv: 2306.08071 by David Wahiche.

Figure 1
Figure 1. Figure 1: Ferrers diagram and some partition statistics. For each box v in the Ferrers diagram of a partition λ (for short we will say for each box v in λ), one defines the arm-length (respectively leg-length) as the number of boxes in the same row (respectively in the same column) as v strictly to the right of (respec￾tively strictly below) the box v. The hook length of v, denoted by hv(λ) or hv , is the number of … view at source ↗
Figure 2
Figure 2. Figure 2: A self-conjugate partition, a doubled distinct partition and its conjugate filled with ε. Let n be a positive integer and let x = (x1, . . . , xn) be a set of any independent variables. Recall the Weyl denominator formula for a reduced root system R (see for instance p.185 of [Bou68]) (1.6) X σ∈W sgn(σ)e σ(ρ)−ρ = Y α∈R+ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: ∂λ and its binary correspondence for λ = (5, 5, 3, 2) with a hook. Lemma 2.1. The application ψ maps bijectively a box s of hook length hs of the Ferrers diagram of λ to a pair of indices (is, js) ∈ Z 2 of the word ψ(λ) such that (1) is < js, (2) cis = 1, cjs = 0, (3) js − is = hs, (4) s is a box above the main diagonal in the Ferrers diagram of λ if and only if the number of letters “1” with negative inde… view at source ↗
Figure 4
Figure 4. Figure 4: ω = (11, 6, 4, 2, 2, 1, 1, 1, 1, 1) ∈ DD(6) and its binary correspon￾dence. Here g = 2t + 2 = 6 and v1 is defined in Section 4.2. The subset of self-conjugate 2t-cores arises in the Macdonald identity for C˜∨ t . A restriction of this previous results to SC(2t) yields some additional conditions on the associated vector of integers. Let ω ∈ SC(2t) and φ(ω) = n ∈ Z 2t by the bijection of Theorem 4.1. Equival… view at source ↗
Figure 5
Figure 5. Figure 5: ω ∈ P(6) Lemma 4.6. Let t be a positive integer. Set ω ∈ P(t) and (v1, . . . , vt) ∈ Z t its associated Vt,t-coding. Then the largest part of ω corresponds to the collection of boxes of indices in I t,+ vi,v1−t . The proof is straightforward from the fact that the boxes in ω1 are those whose index of the letter “0” is v1 − g and the definition of g-intervals. Lemma 4.5 is necessary to prove Theorem 1.2, bu… view at source ↗
read the original abstract

We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us on the one hand to obtain enumerations connecting products of hook lengths and vectors of integers. This yields on the other hand a combinatorial interpretation of the Macdonald identities for affine root systems of the $7$ infinite families in terms of Schur functions, symplectic and special orthogonal Schur functions. From these results, we are able to derive $q$-Nekrasov--Okounkov formulas associated to each type. The latter for limit cases of $q$ yield Nekrasov--Okounkov type formulas corresponding to all the specializations given by Macdonald.

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

0 major / 0 minor

Summary. The paper introduces a methodology connecting vectors of integers to integer partitions represented as bi-infinite words. This yields enumerations of products of hook lengths in terms of such vectors and combinatorial interpretations of the Macdonald identities for the seven infinite families of affine root systems, expressed using Schur functions, symplectic Schur functions, and special orthogonal Schur functions. From these interpretations, q-Nekrasov--Okounkov formulas are derived for each type; their q→1 limits recover the Nekrasov--Okounkov formulas corresponding to Macdonald's specializations.

Significance. If the central derivations hold, the work supplies a uniform combinatorial framework for the Macdonald identities across all seven infinite affine families and produces the associated q-Nekrasov--Okounkov formulas. The bi-infinite-word construction appears to generate the required enumerations and interpretations without type-specific adjustments, which would constitute a genuine advance in algebraic combinatorics.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report contains no major comments requiring a point-by-point reply.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on a direct combinatorial construction: partitions viewed as bi-infinite words are connected to vectors of integers to enumerate hook-length products and interpret Macdonald identities uniformly for the seven affine families via Schur, symplectic, and orthogonal Schur functions, from which the q-Nekrasov-Okounkov formulas follow. No quoted step reduces a claimed prediction or identity to a fitted parameter, self-citation, or definitional renaming; the central methodology is presented as producing the enumerations independently. This is the normal case of a self-contained combinatorial argument.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard combinatorial facts about partitions and hook lengths together with the algebraic definitions of Schur functions and their variants; no new free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard properties of integer partitions, hook lengths, and bi-infinite words
    Invoked to obtain enumerations connecting products of hook lengths to vectors of integers.
  • domain assumption Existence and basic properties of Schur functions, symplectic Schur functions, and special orthogonal Schur functions for the relevant root systems
    Used as the target objects for the combinatorial interpretations of the Macdonald identities.

pith-pipeline@v0.9.0 · 5645 in / 1358 out tokens · 62977 ms · 2026-05-24T08:07:11.453347+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 2 Pith papers

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

  1. Core abaci and Diophantine equations I: fundamental weight

    math.NT 2026-04 unverdicted novelty 7.0

    Core abaci of arbitrary charge parameterize affine Grassmannians and equate the height of β to the atomic length of the corresponding Weyl group element w_λ,j, solving a generalized open problem and yielding closed fo...

  2. Core abaci and Diophantine equations I: fundamental weight

    math.NT 2026-04 unverdicted novelty 7.0

    Core abaci of arbitrary charge parameterize the affine Grassmannian and equate the height of weights to atomic lengths of Weyl elements, solving a generalized open problem and parameterizing solutions to certain Dioph...

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · cited by 1 Pith paper

  1. [1]

    Factorization of classical charact ers twisted by roots of unity

    [AK22] A. Ayyer and N. Kumari. “Factorization of classical charact ers twisted by roots of unity”. J. Algebra 609 (2022), pp. 437–483. [Alb22] S. Albion. Universal characters twisted by roots of unity

    work page 2022

  2. [2]

    Modified Hall-Littlewood polynomials and charac ters of affine Lie algebras

    arXiv: 2212.07343. [Bar14] N. Bartlett. “Modified Hall-Littlewood polynomials and charac ters of affine Lie algebras” (2014). [Bet+18] D. Betea et al. “The free boundary Schur process and ap plications I”. Ann. Henri Poincar´ e19.12 (2018), pp. 3663–3742. [BN19] O. Brunat and R. Nath. Cores and quotients of partitions through the Frobenius symbol

    work page arXiv 2014

  3. [3]

    [Bou68] N

    arXiv: 1911.12098. [Bou68] N. Bourbaki. ´El´ ements de math´ ematique. Fasc. XXXIV. Groupes et alg` ebres de Lie. Chapitre IV: Groupes de Coxeter et syst` emes de Tits. Ch apitre V: Groupes engendr´ es par des r´ eflexions. Chapitre VI: syst` emes de racines. Actualit´ es Sci- entifiques et Industrielles [Current Scientific and Industrial Topics ], No

    work page arXiv 1911

  4. [4]

    Vertex operators a nd character vari- eties

    Hermann, Paris, 1968, 288 pp. (loose errata). [CR V18] E. Carlsson and F. Rodriguez Villegas. “Vertex operators a nd character vari- eties”. Adv. Math. 330 (2018), pp. 38–60. [DH11] P.-O. Dehaye and G.-N. Han. “A multiset hook length formula an d some appli- cations”. Discrete Math. 311.23-24 (2011), pp. 2690–2702. [FH91] W. Fulton and J. Harris. Represen...

    work page 1968

  5. [5]

    The hook graphs of t he symmetric groups

    Graduate Texts in Mathematics. A first course, Readings in Mathematics. Springer-V erlag, New York, 1991, pp. xvi+551. [FRT54] J. Frame, G. Robinson, and R. Thrall. “The hook graphs of t he symmetric groups”. Canad. J. Math. 6 (1954), pp. 316–324. [GKS90] F. Garvan, D. Kim, and D. Stanton. “Cranks and t-cores”. Invent. Math. 101.1 (1990), pp. 1–17. REFEREN...

    work page 1991

  6. [6]

    Lattice points and simultaneous core partitio ns

    Encyclopedia of Mathematics and its Applications. With a fore word by P. M. Cohn, With an introduction by Gilbert de B. Robinson. Addison-W esley Publishing Co., Reading, Mass., 1981, pp. xxviii+510. [Joh18] P. Johnson. “Lattice points and simultaneous core partitio ns”. Electron. J. Com- bin. 25.3 (2018), Paper No. 3.47. [Kac90] V. Kac. Infinite-dimension...

    work page 1981

  7. [7]

    Advanced determinant calculus: a com plement

    IMA Vol. Math. Appl. Springer, New York, 1990, pp. 226–261. [Kra05] C. Krattenthaler. “Advanced determinant calculus: a com plement”. Linear Al- gebra Appl. 411 (2005), pp. 68–166. [Kra99] C. Krattenthaler. “Advanced determinant calculus”. Vol

    work page 1990

  8. [8]

    1999, Art

    The Andrews Festschrift (Maratea, 1998). 1999, Art. B42q,

    work page 1998

  9. [9]

    Young diagrammatic methods for th e restriction of rep- resentations of complex classical Lie groups to reductive subgrou ps of maximal rank

    [KT90] K. Koike and I. Terada. “Young diagrammatic methods for th e restriction of rep- resentations of complex classical Lie groups to reductive subgrou ps of maximal rank”. Adv. Math. 79.1 (1990), pp. 104–135. [Mac72] I. G. Macdonald. “Affine root systems and Dedekind’s η-function”. Invent. Math. 15 (1972), pp. 91–143. [Mac95] I. G. Macdonald. Symmetric f...

    work page 1990

  10. [10]

    A Nekrasov-Okounkov type formula for ˜C

    Progr. Math. Birkh¨ auser Boston, Boston, MA, 2006, pp. 525–596. [P´ e15a] M. P´ etr´ eolle. “A Nekrasov-Okounkov type formula for ˜C”. Proceedings of FP- SAC 2015 . Discrete Math. Theor. Comput. Sci. Proc. Assoc. Discrete Math . Theor. Comput. Sci., Nancy, 2015, pp. 535–546. [P´ e15b] M. P´ etr´ eolle. “Quelques d´ eveloppements combinatoires autour des ...

    work page 2006

  11. [11]

    Elliptic determinant evaluat ions and the Mac- donald identities for affine root systems

    54 REFERENCES [RS06] H. Rosengren and M. Schlosser. “Elliptic determinant evaluat ions and the Mac- donald identities for affine root systems”. Compos. Math. 142.4 (2006), pp. 937–

    work page 2006

  12. [12]

    A Nekrasov-Okounkov formula fo r Macdonald poly- nomials

    [R W18] E. Rains and S. Warnaar. “A Nekrasov-Okounkov formula fo r Macdonald poly- nomials”. J. Algebraic Combin. 48.1 (2018), pp. 1–30. [Sta89] D. Stanton. “An elementary approach to the Macdonald ide ntities”.q-series and partitions (Minneapolis, MN,

    work page 2018

  13. [13]

    Multiplication theorems for self-conjugat e partitions

    IMA Vol. Math. Appl. Springer, New York, 1989, pp. 139–149. [Wah22a] David Wahiche. “Multiplication theorems for self-conjugat e partitions”. Comb. Theory 2.2 (2022), Paper No

    work page 1989

  14. [14]

    Multiplication theorems for self-conjugat e partitions

    [Wah22b] David Wahiche. “Multiplication theorems for self-conjugat e partitions”. S´ em. Lothar. Combin. 86B, Art. 64 (2022), p

    work page 2022

  15. [15]

    Universal characters from the Macdonald identities

    [Wes06] B. Westbury. “Universal characters from the Macdonald identities”. Adv. Math. 202.1 (2006), pp. 50–63. [WW20] A. Walsh and S. Warnaar. “Modular Nekrasov-Okounkov for mulas”. S´ em. Lothar. Combin. 81, Art. B81c (2020), p

    work page 2006