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Circular Hessenberg pairs and the tridiagonal relations

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Pith's one-line read Every circular Hessenberg pair of linear maps satisfies the two tridiagonal relations conjectured by Lee.

desk verdict Clean proof of Lee’s 2022 conjecture on circular Hessenberg pairs; the non-elementary case analysis for d≥4 is original and holds up. read the letter →

arxiv 2607.05688 v1 pith:YIKGOVOH submitted 2026-07-06 math.CO math.RA

Kazumasa Nomura , Paul Terwilliger This is my paper · ORCID
classification math.COmath.RA MSC 05E3015A0415A21
keywords circularHessenbergpairtridiagonalrelationssystemLeonardparameterarrayβ-recurrentsequence
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

A circular Hessenberg pair consists of two diagonalizable linear maps on a finite-dimensional vector space such that each acts on an eigenbasis of the other in a circular Hessenberg fashion: the representing matrix is zero below the subdiagonal and above the superdiagonal except for a nonzero corner entry. Lee conjectured that every such pair obeys a pair of cubic commutation identities known as the tridiagonal relations. The paper proves the conjecture for every dimension. The argument proceeds by reducing the relations to simultaneous recurrence conditions on three sequences of eigenvalues and split parameters, verifying the low-dimensional cases by direct matrix calculation, and obtaining a contradiction for higher dimensions by assuming no common recurrence coefficient exists and then examining the resulting linear dependence among certain polynomials in the pair. The result places circular Hessenberg pairs on the same algebraic footing as the classical Leonard pairs that arise from Q-polynomial association schemes and terminating orthogonal polynomials.

What carries the argument

The reduction of the tridiagonal relations to the simultaneous β-recurrence of the three sequences {θ_i}, {θ*_i}, {ϑ_i} (Proposition 5.5), together with the directed-graph path-weight analysis and the 5 imes5 minors of the auxiliary matrix T that force a contradiction when no such β exists.

What would settle it

Exhibit a concrete circular Hessenberg pair (or its parameter array) of dimension at least 5 for which the three sequences {θ_i}, {θ*_i}, {ϑ_i} fail to be simultaneously β-recurrent for every scalar β; the resulting non-vanishing of the commutators would refute the claim.

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Extended reading notes

Core claim

Every circular Hessenberg pair A, A* on a nonzero finite-dimensional vector space satisfies the tridiagonal relations: there exist scalars β, γ, γ*, ρ, ρ* such that the commutator [A, A²A* - β AA*A + A*A² - γ(AA* + A*A) - ρ A*] vanishes and the dual commutator with A and A* interchanged also vanishes.

Load-bearing premise

The higher-dimensional proof assumes that no common recurrence coefficient β can make the three eigenvalue sequences recurrent at once, then derives a contradiction from that global non-existence; if such a β existed without forcing the minors of T to vanish, the argument would fail.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper proves Jae-ho Lee's 2022 conjecture that every circular Hessenberg pair A, A* on a nonzero finite-dimensional vector space V of dimension d+1 satisfies the tridiagonal relations: there exist scalars β, γ, γ*, ρ, ρ* such that [A, A^{2}A* − β AA*A + A*A^{2} − γ(AA* + A*A) − ρ A*] = 0 and the dual relation with A and A* interchanged. After recalling Hessenberg pairs/systems and their circular specializations (Definitions 3.3, 3.7, 4.3, 4.4), the authors reduce the claim, for d ≥ 3, to the existence of a single β making the eigenvalue sequences {θ_i}, {θ*_i} and the auxiliary sequence {ϑ_i} simultaneously β-recurrent (Proposition 5.5). The cases d = 2 and d = 3 are settled by direct matrix computation (Lemmas 5.2, 6.1). For d ≥ 4 the argument proceeds by contradiction: the non-existence of such a β produces a linear dependence among five commutators (Lemma 7.5), a four-term recurrence on the dual eigenvalues (Lemma 7.27), closed-form expressions for those eigenvalues (Lemma 7.36), and the vanishing of certain 5 × 5 minors of an auxiliary matrix T; the resulting identities force β = β* = −1 and then contradict ξ ≠ 0 (Lemma 7.48).

Significance. The result completes the classification of circular Hessenberg systems begun by Lee and places them on the same algebraic footing as Leonard pairs, which are known to satisfy the same tridiagonal relations. The proof is self-contained, uses only linear algebra and the theory of linear recurrences, and supplies explicit closed forms for the dual eigenvalues under the contradictory hypothesis. The introduction of the directed graph D, walk weights, and winding numbers (Definitions 7.6–7.23) is a clean technical device that organizes the lengthy case analysis for d ≥ 4. The paper therefore settles a concrete open conjecture in the literature on Hessenberg pairs and supplies a reusable toolkit for related problems involving circular or almost-tridiagonal actions.

minor comments (5)
  1. In the abstract and the final sentence of the introduction the authors state that the proof is “not elementary.” A brief parenthetical remark indicating what is meant (e.g., reliance on characteristic polynomials of linear recurrences and non-vanishing of 5 × 5 minors) would help the reader set expectations.
  2. Lemma 7.36 lists five cases according to the characteristic of F and the value of β. The verification that the closed forms satisfy the four-term recurrence of Lemma 7.27 is left to the reader; a one-line check for the generic case (i) would make the argument easier to follow.
  3. The auxiliary sequence ϑ_i is defined in (5) and used heavily thereafter, yet it is never given a name or short descriptive phrase. Calling it the “adjusted split sequence” (or similar) would improve readability.
  4. In Definition 7.37 the matrix T is displayed with a final column that already incorporates β. It would be clearer to write the five columns first without β and then state that the rightmost column is a linear combination of the first four (as proved in Lemma 7.38).
  5. A few typographical slips: “classfied” (p. 1), “ford=2” (p. 7), and the missing space before “where” in several displayed equations of Section 7.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the tridiagonal relations are derived from the circular Hessenberg definition by direct matrix algebra (d=2,3) and proof-by-contradiction on eigenvalue recurrences (d≥4), without assuming the target relations or load-bearing self-citation of the conjecture itself.

full rationale

The paper proves Lee's Conjecture 5.1 that every circular Hessenberg system satisfies the tridiagonal relations (3)-(4). For d=2 the relations are verified by explicit matrix multiplication on the bidiagonal forms (Lemma 5.2). For d=3 the circular conditions E_i A* E_j =0 (or eq0) produce the three 0-recurrent sequences via direct entry computations (7)-(12), yielding eta=0 and the explicit eta,eta*, ho, ho* (Lemma 6.1). For d≥4 the argument assumes the negation of Proposition 5.5(iv) (no eta makes { heta_i},{ heta*_i},{ϑ_i} simultaneously eta-recurrent) and derives a contradiction: Lemmas 7.2-7.5 produce a linear dependence among five commutators, which forces the dual eigenvalues to satisfy the four-term recurrence (31); closed forms (Lemma 7.36) then make the 5 imes5 minors of T and T* evaluate to nonzero expressions involving eta and eq0 eq0 (Lemmas 7.45-7.46), forcing eta=eta*=-1 and contradicting eq0 (Lemma 7.48). All steps begin from the circular definition (nonzero corner entries, zeros elsewhere above the superdiagonal) and the parameter-array representation; the only self-citations are to independent prior definitions/classifications (Godjali, Lee) and standard recurrence facts from Terwilliger's Leonard-pair papers, none of which presuppose the conjecture. No parameter is fitted, no uniqueness theorem is imported to force the result, and the contradiction fully discharges the global non-existence hypothesis. The derivation is therefore self-contained.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

Pure algebraic proof over an arbitrary field. No free parameters are fitted. All background notions (multiplicity-free maps, primitive idempotents, parameter arrays of Hessenberg systems, β-recurrence) are taken from the cited literature of Terwilliger, Godjali and Lee; the only new objects are the auxiliary scalars ϑ_i and the directed graph D used in the contradiction argument.

assumptions (4)
  • domain assumption A Hessenberg pair is multiplicity-free (Lemma 3.5, citing Godjali).
    Used throughout to guarantee the existence of ordered primitive idempotents and the parameter array.
  • domain assumption The parameter array of a Hessenberg system exists and is unique up to isomorphism (Proposition 3.9, citing Godjali).
    Supplies the scalars θ_i, θ*_i, ϕ_i that appear in every subsequent calculation.
  • standard math Standard facts about β-recurrent sequences (Terwilliger, Leonard pairs paper).
    Used in Lemmas 5.3–5.5 to convert the tridiagonal relations into recurrence conditions on the eigenvalues.
  • standard math The field F is arbitrary (possibly of characteristic 2); algebraic closure is taken when needed for closed-form solutions of recurrences.
    Explicitly handled in the five cases of Lemma 7.36.
invented entities (2)
  • auxiliary sequence ϑ_i = ϕ_i − (θ*_i − θ*_0)(θ_{d−i+1} − θ_0)
    purpose: Converts the circular condition into a third sequence that must be β-recurrent for the tridiagonal relations to hold.
    Defined ad hoc in equation (5); no independent existence outside the proof.
  • directed graph D on vertices {0, ho,d} with arcs given by nonzero E*_j A E*_i
    purpose: Encodes the support of walk-products that appear when expanding powers of A and A*; used to extract the linear recurrence on θ*_i.
    Introduced in Definition 7.6 solely for the contradiction argument of Section 7.

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Pith. "Pith review of Circular Hessenberg pairs and the tridiagonal relations." pith.science (2026). https://pith.science/paper/YIKGOVOH

@misc{pith2026260705688,
  author       = {Pith},
  title        = {Pith review of: Circular Hessenberg pairs and the tridiagonal relations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YIKGOVOH}},
  note         = {Machine review of arXiv:2607.05688}
}
read the original abstract

A square matrix is said to be Hessenberg whenever each entry below the subdiagonal is zero, and each entry on the subdiagonal is nonzero. A Hessenberg matrix is called circular whenever the top-right corner entry is nonzero, and every other entry above the superdiagonal is zero. A circular Hessenberg pair consists of two diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a circular Hessenberg fashion. In 2022, Jae-ho Lee conjectured that a circular Hessenberg pair satisfies two relations called the tridiagonal relations. In the present paper, we prove Lee's conjecture. Our proof is not elementary.

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Forward citations

Cited by 1 Pith paper

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

  1. Variations on a circular Hessenberg pair

    math.CO 2026-07 conditional novelty 6.0 of 10

    Quasi-circular Hessenberg systems and systems satisfying the tridiagonal relations are the same family; the tridiagonal-relations family splits exactly into the circular and tridiagonal-Hessenberg cases.

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Works this paper leans on

20 extracted references · 20 canonical work pages · cited by 1 Pith paper

  1. [1]

    Bannai, Et

    E. Bannai, Et. Bannai, T. Ito, R. Tanaka.Algebraic Combinatorics.De Gruyter Series in Discrete Math and Applications 5. De Gruyter, 2021. https://doi.org/10.1515/9783110630251

    work page doi:10.1515/9783110630251 2021

  2. [2]

    Bannai and T

    E. Bannai and T. Ito.Algebraic Combinatorics, I. Association schemes.Ben- jamin/Cummings, Menlo Park, CA, 1984

    work page 1984

  3. [3]

    A. Godjali. Thin Hessenberg pairs.Linear Algebra Appl.432 (2010) 3231–3249; arXiv:0911.4118

    work page Pith review arXiv 2010

  4. [4]

    Thin Hessenberg Pairs and Double Vandermonde Matrices

    A. Godjali, Thin Hessenberg pairs and double Vandermonde matrices.Linear Algebra Appl.436 (2012) 3018–3060;arXiv:1107.5369

    work page Pith review arXiv 2012

  5. [5]

    J. H. Lee. Circular Hessenberg pairs.Linear Algebra Appl.655 (2022) 202–235; arXiv:2209.02194

    work page Pith review arXiv 2022

  6. [6]

    D. A. Leonard. Orthogonal polynomials, duality and association schemes.SIAM J. Math. Anal.13 (1982) 656–663

    work page 1982

  7. [7]

    Krawtchouk polynomials, the Lie algebra $\mathfrak{sl}_2$, and Leonard pairs

    K. Nomura and P. Terwilliger. Krawtchouk polynomials, the Lie algebrasl 2, and Leonard pairs.Linear Algebra Appl.437 (2012) 345–375;arXiv:1201.1645

    work page Pith review arXiv 2012

  8. [8]

    Leonard pairs, spin models, and distance-regular graphs

    K. Nomura and P. Terwilliger. Leonard pairs, spin models, and distance-regular graphs. J. Combin. Theory Ser. A177 (2021), Paper No. 105312, 59 pp.;arXiv:1907.03900. 26

    work page Pith review arXiv 2021

Show all 20 references
  1. [9]

    Terwilliger

    P. Terwilliger. The subconstituent algebra of an association scheme I.J. Algebraic Combin.1 (1992) 363–388

    1992

  2. [10]

    Terwilliger

    P. Terwilliger. The subconstituent algebra of an association scheme II.J. Algebraic Combin.2 (1993) 73–103

    1993

  3. [11]

    Terwilliger

    P. Terwilliger. The subconstituent algebra of an association scheme III.J. Algebraic Combin.2 (1993) 177–210

    1993

  4. [12]

    Terwilliger

    P. Terwilliger. Two linear transformations each tridiagonal with respect to an eigenbasis of the other.Linear Algebra Appl.330 (2001) 149–203;arXiv:math.RA/0406555

    2001

  5. [13]

    Terwilliger

    P. Terwilliger. Two relations that generalize theq-Serre relations and the Dolan-Grady relations. InPhysics and Combinatorics 1999 (Nagoya), 377–398, World Scientific Pub- lishing, River Edge, NJ, 2001;arXiv:math.QA/0307016

    1999

  6. [14]

    Terwilliger

    P. Terwilliger. Leonard pairs from 24 points of view.Rocky Mountain J. Math.32 (2002) 827–888;arXiv:math/0406577

    2002 arXiv

  7. [15]

    Terwilliger

    P. Terwilliger. Introduction to Leonard pairs.J. Comput. Appl. Math.153 (2003) 463–475

    2003

  8. [16]

    Terwilliger

    P. Terwilliger. Leonard pairs and theq-Racah polynomials.Linear Algebra Appl.387 (2004) 235–276.arXiv:math.QA/0306301

    2004

  9. [17]

    Terwilliger

    P. Terwilliger. An algebraic approach to the Askey scheme of orthogonal polynomials. Orthogonal polynomials and special functions, 255–330, Lecture Notes in Math., 1883, Springer, Berlin, 2006;arXiv:math/0408390

    2006 arXiv

  10. [18]

    Terwilliger

    P. Terwilliger. Notes on the Leonard system classification.Graphs Combin.37 (2021) 1687–1748;arXiv:2003.09668

    2021 arXiv

  11. [19]

    Terwilliger

    P. Terwilliger. Distance-regular graphs, the subconstituent algebra, and theQ- polynomial property. London Math. Soc. Lecture Note Ser., 487 Cambridge University Press, London, 2024, 430–491;arXiv:2207.07747

    2024 arXiv

  12. [20]

    Terwilliger and R

    P. Terwilliger and R. Vidunas. Leonard pairs and the Askey-Wilson relations.J. Algebra Appl.3 (2004) 411–426;arXiv:math/0305356. Kazumasa Nomura Institute of Science Tokyo Kohnodai Ichikawa 272-0827 Japan email:knomura@pop11.odn.ne.jp Paul Terwilliger Department of Mathematics...

    2004 arXiv

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