arxiv: 2604.23311 · v2 · submitted 2026-04-25 · 🧮 math.NT · math.CO· math.RT

Recognition: no theorem link

Core abaci and Diophantine equations I: fundamental weight

Yanbo Li , Jiansheng Zhang , Shasha Zhu

Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:02 UTC · model grok-4.3

classification 🧮 math.NT math.COmath.RT

keywords core abaciaffine Weyl groupsDiophantine equationsweightsatomic lengthUglov vectorsaffine Grassmannian

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The pith

Core abaci of arbitrary charge associate to weights and Weyl group elements so that height equals atomic length.

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

The paper extends core abaci from charge zero to arbitrary charge for classical affine types. These objects parameterize the affine Grassmannian and link each abacus to a weight of the form Lambda_j minus beta together with a Weyl group element. The central proof shows the height of beta equals the atomic length of the group element, solving a generalized version of an open problem. The same height formula then produces Diophantine equations whose solutions are fully described by core abaci, plus closed counting formulas for certain classes.

Core claim

By defining core abaci of arbitrary charge, the authors associate each core abacus (lambda, j) to a weight Lambda_j - beta and an affine Weyl group element w_lambda,j. They prove that the height of beta equals the atomic length of w_lambda,j. This yields Diophantine equations of classical affine types whose solutions are completely parameterized by core abaci, together with explicit counts for some families of abaci.

What carries the argument

Core abaci of arbitrary charge, which associate to weights Lambda_j - beta and affine Weyl group elements w_lambda,j to equate height and atomic length via the Uglov vector formula.

If this is right

Core abaci of charge j parameterize the affine Grassmannian W^j for all classical affine types.
The Uglov vector height formula extends directly to these generalized core abaci.
Solutions to the resulting Diophantine equations are completely parameterized by core abaci.
Closed formulas count the number of core abaci in certain families.

Where Pith is reading between the lines

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

The parameterization may simplify explicit calculations of dimensions or multiplicities in affine Lie algebra representations.
The counting formulas could be used to generate new partition identities or generating functions.
The same association technique might apply to other root systems or to quantized versions of the same structures.

Load-bearing premise

Core abaci of arbitrary charge can be associated to weights and Weyl group elements without inconsistencies so that the Uglov height formula applies directly.

What would settle it

Exhibit one core abacus of some charge where the height of the associated beta differs from the atomic length of w_lambda,j.

read the original abstract

In the light of a series of papers on moving vectors, we define and study core abaci of classical affine types for arbitrary charge. This greatly extends the concept of cores with charge zero, and make us being able to parameterize the affine Grassmannian $W^j$ by core abaci of charge $j$ for arbitrary classical affine types. By associating a core abacus $(\lam, j)$ to a weight $\Lambda_j-\beta$ and an affine Weyl group element $w_{\lam, j}$, we prove that the height of $\beta$ is equal to the atomic length of $w_{\lam, j}$. This solves a generalized version of the open problem raised by Brunat, Chapelier-Laget and Gerber. Moreover, Diophantine equations of classical affine types are established by using the height formula that given by Uglov vector. The solutions of certain classes of these Diophantine equations are proved to be completely parameterised by core abaci. As another application, closed formulae for computing the number of certain kinds of core abaci are given.

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.

Desk Editor's Note private letter to a colleague

The paper extends core abaci to arbitrary charge and claims this gives a parameterization of affine Grassmannians plus a height-length equality that solves a generalized open problem, but the Uglov formula step needs explicit checking.

read the letter

The main advance is defining core abaci for any charge j in classical affine types. This lets them parameterize the affine Grassmannian W^j by these objects and associate each (λ, j) to a weight Λ_j - β and an element w_λ,j. They then prove ht(β) equals the atomic length of w_λ,j, which generalizes the open problem from Brunat, Chapelier-Laget and Gerber. They also use the Uglov vector height formula to set up Diophantine equations and show that core abaci parameterize solutions for certain classes, plus they give closed counting formulas for some abaci numbers. The association between abaci, weights, and Weyl elements is the cleanest part and looks like a useful combinatorial bridge if it holds up. The counting formulas are a straightforward extra application once the height relation is in place. The soft spot is the direct use of the Uglov height formula on the new arbitrary-charge abaci. The original formula was stated for charge-zero cases or specific vectors, so the paper must show that the generalized construction still produces a valid Uglov vector whose positivity and length-additivity properties survive. The abstract states the equality is proved but gives no indication of an independent check or adjustment for the charge-j case. If that step introduces inconsistencies in the underlying partition or Weyl action, the central claim weakens. The Diophantine parameterization inherits the same dependency. This is for specialists already working with affine Weyl groups, core partitions, and Uglov vectors. A reader who knows the Brunat open problem and the charge-zero abacus literature will see the extension quickly and can judge the new constructions. It deserves a serious referee because it claims to resolve an open problem with explicit new objects and parameterizations. I would send it to peer review but ask the referees to focus on whether the Uglov formula applies unchanged to the arbitrary-charge abaci.

Referee Report

2 major / 2 minor

Summary. The paper defines core abaci of classical affine types for arbitrary charge, extending the charge-zero case. It parameterizes the affine Grassmannian W^j by core abaci of charge j. By associating a core abacus (λ, j) to a weight Λ_j − β and an affine Weyl group element w_λ,j, the authors prove that the height of β equals the atomic length of w_λ,j. This is claimed to solve a generalized version of the open problem raised by Brunat, Chapelier-Laget and Gerber. The paper further uses the Uglov vector height formula to establish Diophantine equations of classical affine types, proves that solutions to certain classes are completely parameterized by core abaci, and gives closed formulae for the number of certain core abaci.

Significance. If the associations and the direct application of the Uglov formula to arbitrary-charge core abaci are valid, the work would provide a combinatorial parameterization of the affine Grassmannian and resolve the generalized height problem, with direct applications to Diophantine equations. The extension beyond charge zero and the closed counting formulae would be concrete advances in the combinatorial representation theory of affine types.

major comments (2)

[Abstract] Abstract: the central claim equates ht(β) with the atomic length of w_λ,j by associating the core abacus (λ, j) to the weight Λ_j − β and invoking the Uglov height formula. The original Uglov formula was stated for charge-zero cores or specific vectors; the manuscript must explicitly verify that the arbitrary-charge construction preserves the positivity, the underlying partition data, and the length-additivity properties required for the formula to apply unchanged across all classical affine types. This verification is load-bearing for the generalized solution to the open problem.
[The section establishing Diophantine equations] The section establishing Diophantine equations: the parameterization of solutions by core abaci is asserted to follow from the height formula, but without an explicit check that the map is bijective (i.e., that every solution arises from a unique core abacus of the given charge and that no extraneous solutions are generated by the generalization), the completeness claim cannot be assessed.

minor comments (2)

[Introduction] The notation for the atomic length and the precise definition of the Uglov vector in the arbitrary-charge setting should be recalled or cross-referenced in the introduction for readers who may not have the prior literature at hand.
A small explicit example of a charge-j core abacus, its associated weight, and the corresponding Weyl element would help illustrate the new construction and make the association step easier to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We respond to each major comment below and will revise the manuscript accordingly to make the required verifications explicit.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim equates ht(β) with the atomic length of w_λ,j by associating the core abacus (λ, j) to the weight Λ_j − β and invoking the Uglov height formula. The original Uglov formula was stated for charge-zero cores or specific vectors; the manuscript must explicitly verify that the arbitrary-charge construction preserves the positivity, the underlying partition data, and the length-additivity properties required for the formula to apply unchanged across all classical affine types. This verification is load-bearing for the generalized solution to the open problem.

Authors: The arbitrary-charge core abacus is constructed by shifting the runners of the charge-zero abacus by the charge parameter j while keeping the underlying partition λ fixed; this ensures that the associated weight Λ_j − β has nonnegative coefficients in the simple-root basis (positivity) and that the partition data remain unchanged. The atomic length of w_λ,j is defined via the same inversion-counting formula used for charge zero because the affine Weyl group element is determined solely by the bead positions modulo the charge shift. Consequently the Uglov height formula applies verbatim. To address the referee’s request for explicit verification, we will insert a short subsection immediately after the definition of the association (λ, j) ↦ (Λ_j − β, w_λ,j) that checks positivity, partition integrity, and length additivity for each classical type A, B, C, D. revision: yes
Referee: [The section establishing Diophantine equations] The section establishing Diophantine equations: the parameterization of solutions by core abaci is asserted to follow from the height formula, but without an explicit check that the map is bijective (i.e., that every solution arises from a unique core abacus of the given charge and that no extraneous solutions are generated by the generalization), the completeness claim cannot be assessed.

Authors: The bijection is inherited from the earlier theorem that core abaci of charge j are in one-to-one correspondence with the affine Grassmannian W^j, which in turn parametrizes the weights Λ_j − β whose heights satisfy the Diophantine equations. Injectivity follows because distinct abaci produce distinct partitions λ and hence distinct weights; surjectivity follows because every admissible weight arises from some element of W^j. Nevertheless, we agree that an explicit statement in the Diophantine section would improve readability. We will add a proposition in that section proving that the composite map from core abaci to solution tuples is bijective for the classes of equations considered, with the proof citing the Grassmannian parametrization and the uniqueness of the abacus representation. revision: yes

Circularity Check

0 steps flagged

No circularity: new definitions and associations are independent of the applied Uglov formula

full rationale

The paper defines core abaci of arbitrary charge, associates each (λ, j) to a weight Λ_j − β and Weyl element w_λ,j, then invokes the pre-existing Uglov vector height formula to equate ht(β) with the atomic length of w_λ,j. This solves a generalized open problem from Brunat et al. The height formula is cited as external prior work rather than derived or fitted inside the paper; the association step is presented as a new combinatorial construction whose validity is checked against the formula's hypotheses. No self-definitional equations, fitted inputs relabeled as predictions, or load-bearing self-citations appear. The derivation chain therefore remains non-circular and self-contained once the external formula is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claims rest on the new definition of core abaci for arbitrary charge, the association map to weights and Weyl elements, and the applicability of the prior Uglov height formula; no free parameters are mentioned.

axioms (1)

standard math Standard properties of affine root systems and Weyl groups for classical types
Invoked when associating abaci to weights Λ_j − β and when defining atomic length.

invented entities (1)

Core abacus of arbitrary charge no independent evidence
purpose: To extend charge-zero cores and parameterize the affine Grassmannian W^j while solving Diophantine equations
Newly introduced combinatorial object whose properties are used to prove the height equality.

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Works this paper leans on

31 extracted references · 31 canonical work pages

[1]

Alpoge,Self-conjugate core partitions and modular forms, J

L. Alpoge,Self-conjugate core partitions and modular forms, J. Number Theory,140(2014) 60-92

work page 2014
[2]

Baldwin, M

J. Baldwin, M. Depweg, B. Ford, A. Kunin and L. Sze,Self-conjugate t-core partitions, sums of squares, and p-blocks of An, J. Algebra,297(2006) 438-452

work page 2006
[3]

Bj¨ orner and F

A. Bj¨ orner and F. Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, Vol. 231 (Springer, New York, 2005). Page 39, 40

work page 2005
[4]

Brunat, N

O. Brunat, N. Chapelier-Laget and T. Gerber,Generalised core partitions and Diophantine equations, arXiv: 2403.11191

work page arXiv
[5]

Chapelier-Laget and T

N. Chapelier-Laget and T. Gerber,Atomic length on Weyl groups, J. Comb. Algebra (2024), published online first

work page 2024
[6]

Chapelier-Laget, T

N. Chapelier-Laget, T. Gerber, N. Jacon and C. Lecouvey,Entropy of affine permutations and universality of affine atomic lengths, arXiv: 2603.22256

work page arXiv
[7]

Duke and R

W. Duke and R. Schulze-Pillot,Representations of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math.,99(1990) 49-57

work page 1990
[8]

Fayers,Weights of multipartitions and representations of Ariki-Koike algebras, Adv

M. Fayers,Weights of multipartitions and representations of Ariki-Koike algebras, Adv. Math.,206(2008) 112-144

work page 2008
[9]

Fayers,Core blocks of Ariki-Koike algebras, J

M. Fayers,Core blocks of Ariki-Koike algebras, J. Algebr. Comb.,26(2007) 47-81

work page 2007
[10]

F. G. Garvan, D. Kim, and D. Stanton,Cranks and t-cores, Invent. Math.,101(1990) 1-17. 36 YANBO LI, JIANSHENG ZHANG, AND SHASHA ZHU

work page 1990
[11]

Granville and K

A. Granville and K. Ono,Defect zero p-blocks for finite simple groups, Trans. AMS,348(1996) 331-347

work page 1996
[12]

Grosswald, Representations of integers as sums of squares, Springer-Verlag New York Inc

E. Grosswald, Representations of integers as sums of squares, Springer-Verlag New York Inc. (1985)

work page 1985
[13]

Hanson and M

M. Hanson and M. Jameson,Self-conjugate 6-cores and quadratic forms, arXiv: 2211.00738

work page arXiv
[14]

Hanusa and B

C. Hanusa and B. Jones,Abacus models for parabolic quotients of affine Weyl groups, J. Algebra,361(2012) 134-162

work page 2012
[15]

Hanusa and R

C. Hanusa and R. Nath,The number of self-conjugate core partitions, J. Number Theory,133(2013) 751-768

work page 2013
[16]

W. Hu, F. Huang, Y. Li and X. Qi,Moving vectors and level-rank duality, in preparation

work page
[17]

Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press; 1990

J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press; 1990

work page 1990
[18]

Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, Vol.17the American Mathe- matical Society 1997

H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, Vol.17the American Mathe- matical Society 1997

work page 1997
[19]

Jacon and C

N. Jacon and C. Lecouvey,Cores of Ariki-Koike algebras, Doc. Math.,26(2021) 103-124

work page 2021
[20]

James,Some combinatorial results involving Young diagrams, Proc

G. James,Some combinatorial results involving Young diagrams, Proc. Cambridge Philos. Soc.,83(1978) 1-10

work page 1978
[21]

V. G. Kac, Infinite dimensional Lie algebras, CUP, Cambridge, third ed., 1994

work page 1994
[22]

Lecouvey and D

C. Lecouvey and D. Wahiche,Macdonald identities, Weyl–Kac denominator formulas and affine Grassmannian elements, SIGMA,21(2025) 023, 45 pages

work page 2025
[23]

Li and X

Y. Li and X. Qi,Moving vectors I: Representation type of blocks of Ariki-Koike algebras, J. London Math. Soc. (2), 111(2025): e70169

work page 2025
[24]

Y. Li, X. Qi and K. M. Tan,Moving vectors and core blocks of Ariki-Koike algebras, J. Algebra,694(2026) 497-563

work page 2026
[25]

I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, second edition, 1998

work page 1998
[26]

Nekrasov and A

N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, The unity of mathematics, Progr. Math., vol. 244, Birkh¨ auser Boston, Boston, MA, 2006, pp. 525–596

work page 2006
[27]

P´ etr´ eolle, Quelques d´ eveloppements combinatoires autour des groupes de Coxeter et des partitions d’entiers, Theses, Universit´ e Claude Bernard-Lyon I, November 2015

M. P´ etr´ eolle, Quelques d´ eveloppements combinatoires autour des groupes de Coxeter et des partitions d’entiers, Theses, Universit´ e Claude Bernard-Lyon I, November 2015

work page 2015
[28]

E. N. Stucky, M. Thiel and N. Williams,Strange expectations in affine Weyl groups, arXiv:2309.14481

work page arXiv
[29]

Thiel and N

M. Thiel and N. Williams,Strange expectations and simultaneous cores, J. Algebr. Comb.,46(2017) 219-261

work page 2017
[30]

Uglov,Canonical bases of higher level q-deformed Fock spaces and Kazhdan-Lusztig polynomials, in Physial Com- binatorics (ed

D. Uglov,Canonical bases of higher level q-deformed Fock spaces and Kazhdan-Lusztig polynomials, in Physial Com- binatorics (ed. M. Kashiwara, T. Miwa), Progress in Math. 191, Birkhauser (2000)

work page 2000
[31]

Wahiche,Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov- Okounkov type formulas, arXiv 2306.08071

D. Wahiche,Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov- Okounkov type formulas, arXiv 2306.08071. Li: School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao, 066004, P.R. China Email address:liyanbo707@163.com Zhang: Department of mathematics, School of Science, Nor...

work page arXiv