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Variations on a circular Hessenberg pair

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Pith's one-line read A Hessenberg system satisfies the tridiagonal relations if and only if its associated matrices are quasi-circular Hessenberg, a family that splits into circular and tridiagonal types.

desk verdict New families and a clean logical map for Hessenberg systems, but the key CHS⇒TD implication comes from an unverified companion preprint and the paper leans on unshown computations. read the letter →

arxiv 2607.16964 v1 pith:NFCNKLSY submitted 2026-07-18 math.CO math.RA

Kazumasa Nomura , Paul Terwilliger This is my paper · ORCID
classification math.COmath.RA MSC 17B3715A21
keywords HessenbergpairsystemLeonardtridiagonalrelationsquasi-circularcircularparameterarray
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

This paper organizes the known families of Hessenberg systems into a single logical diagram. Its central claim is that the systems satisfying the tridiagonal relations (TD) are exactly the quasi-circular Hessenberg systems (QCHS), and that these split without overlap into the circular ones (CHS) and the tridiagonal ones (THS). It also proves the chain Leonard ⇒ tridiagonal ⇒ TD ⇒ recurrent ⇒ Hessenberg. If correct, this reduces the study of TD systems to a matrix-shape condition and gives explicit parameter-array recurrences for each family.

What carries the argument

The parameter array of a Hessenberg system — its eigenvalue sequence, dual eigenvalue sequence, and split sequence — together with the adjusted split sequence ϑ_i = ϕ_i − (θ*_i − θ*_0)(θ_{d−i+1} − θ_0). The classification is carried by recurrence conditions: a system is TD iff all three sequences are β-recurrent for a common β, and it is THS iff additionally the first and last adjusted split values coincide (ϑ_1 = ϑ_d). The proofs use vanishing of triple products E_i A* E_j and the explicit (r, r−2) matrix entries computed in Lemmas 12.5–12.11.

What would settle it

Construct a Hessenberg system with d≥3 that satisfies the tridiagonal relations (11)–(12) but has some entry E_i A* E_j nonzero for 1 < j−i < d (i.e., not quasi-circular), or construct a quasi-circular Hessenberg system that does not satisfy (11)–(12).

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Extended reading notes

Core claim

The paper proves QCHS⇔TD: a Hessenberg system satisfies the tridiagonal relations (11)–(12) if and only if, in the eigenbasis of each map, the other map is represented by a matrix whose only possibly nonzero entries on the far side are the subdiagonal, diagonal, superdiagonal, and the far-corner entry (the (d,0) position). Within this family, the far-corner entry is zero exactly for tridiagonal Hessenberg systems and nonzero exactly for circular Hessenberg systems; hence CHS is the complement of THS in TD. For d≥3, each of these conditions is characterized by the β-recurrence of the eigenvalue sequence, the dual eigenvalue sequence, and the adjusted split sequence {ϑ_i} defined in Definition

Load-bearing premise

The paper relies on a companion result that circular Hessenberg systems satisfy the tridiagonal relations; if that implication fails, the equivalence QCHS⇔TD and the complement claim are unsupported.

Editorial extensions

If this is right

  • The tridiagonal relations hold for a Hessenberg system exactly when its associated matrices are quasi-circular; no other Hessenberg shape can satisfy them.
  • Every TD system has all three of its defining sequences (eigenvalues, dual eigenvalues, adjusted split sequence) governed by a single three-term recurrence parameter β.
  • A TD system is a Leonard system precisely when it is irreducible tridiagonal, and these sit inside the THS family; the circular systems are the remaining TD systems.
  • The parameter-array characterizations give explicit existence and uniqueness statements for TD and THS systems, extending the known Leonard-system classification.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Since the implication CHS⇒TD is invoked from a companion preprint rather than proved here, the central equivalence stands or falls with that companion argument; checking it independently would settle the paper's main claim.
  • One testable extension is to read the (r, r−2) entry formulas as a direct computational criterion: given a Hessenberg system, the vanishing of these entries in one basis forces the full triple-product pattern.
  • The splitting of TD into CHS and THS by a single scalar condition (ϑ_1 = ϑ_d) suggests that q-Racah and other orthogonal-polynomial families from the Askey scheme may be classified by this one parameter.
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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

2 major / 4 minor

Summary. The paper studies Hessenberg systems and the relations among six families: circular Hessenberg systems (CHS), quasi-circular Hessenberg systems (QCHS), tridiagonal Hessenberg systems (THS), Leonard systems (LS), systems satisfying the tridiagonal relations (TD), and recurrent systems (REC). The central claim is the logical diagram of Section 1: QCHS is equivalent to TD, and the chain LS ⇒ THS ⇒ TD ⇒ REC ⇒ HS holds; moreover, within TD, CHS is the complement of THS. The paper gives parameter-array characterizations: LS by Lemma 8.4, THS by Propositions 12.2 and 12.8, TD by Propositions 11.7 and 11.8, REC by Definition 10.3, and HS by Lemma 5.11. The proofs use the split sequence and the adjusted split sequence {ϑ_i}, plus detailed computations of triple-product matrix entries. The main in-house contributions are the proofs of TD⇒REC, THS⇒TD, the disjoint-union statement (16), and the QCHS⇒TD direction of (17), the last of these contingent on the implication CHS⇒TD obtained in the authors' companion preprint [17].

Significance. If the main diagram is correct, the paper gives a clean structural organization of several previously studied families: the TD family coincides exactly with the quasi-circular Hessenberg systems, and the two ``extremal'' subfamilies—circular and tridiagonal—account for all of them. The parameter-array descriptions are concrete and useful, especially Propositions 11.7, 12.2, and 12.8, which characterize TD and THS by simple recurrence conditions involving {ϑ_i}. The paper also handles the low-dimensional case d=2 separately, which is necessary because several recurrence definitions degenerate there. A notable strength is that much of the argument is self-contained: the reader can trace the logical chain from the definitions through the matrix computations to the main equivalences, with the important exception of (13). The result should be of interest to researchers working on Leonard pairs, tridiagonal algebras, and the classification of Hessenberg-type matrix families.

major comments (2)
  1. [Sections 10 and 13, especially Eq. (13) and proof of (17)] The central equivalence QCHS⇔TD and the statement that CHS is the complement of THS in TD both depend on the implication CHS⇒TD, labelled (13), which is not proved in this paper. The proof of (17) in Section 13 uses (13) explicitly: after (16) splits QCHS into CHS and THS, the CHS case is closed by (13) and the THS case by (15). Section 6 also uses the companion preprint [17] to remove the additional conditions in the classification of CHS from [10]. Since [17] is cited as an arXiv preprint and is not included in this manuscript, the main theorem is conditional on an external result. This is a disclosed and addressable gap, but it is load-bearing. I ask the authors either to include a proof of (13) in this paper or an appendix, or to cite a published version of [17] and spell out precisely which statement is being used. Without this, the claimed diagram (4)–(5) is not fully established w
  2. [Section 12, Lemmas 12.5–12.11] The proofs of the THS characterization and hence of (15) and (16) rest on explicit formulas for the (r,r−2)-entry of (E_{d−r}A^*E_{d−r+2})♭, obtained in Lemmas 12.5, 12.6, 12.7, 12.9, 12.10, and 12.11. These are asserted as ``routine verification'' with no derivation shown. The formulas are complicated and they are used to identify exactly when triple products vanish, so an error here would propagate to Propositions 12.2, 12.12, and 12.13. I do not claim the formulas are wrong—the structure is plausible and consistent with the later recurrence reformulation—but because they underpin a central claim, the authors should provide a derivation, at least for one representative case, or make the computations available in a verifiable supplementary form. This is a secondary but genuine gap in the write-up.
minor comments (4)
  1. [Abstract and Section 1] The abstract contains the typo ``eignvalues'' for ``eigenvalues''. In Section 1, ``compliment'' should be ``complement''; the same typo appears in Section 10.
  2. [Definition 5.8] ``paramater array'' should be ``parameter array''.
  3. [Section 10] The list of statements to prove includes (17), but (17) depends on (13) from [17]. The text should indicate this dependence more prominently, perhaps by listing (13) as an external input in the same display, so the reader is not misled into thinking all of (14)–(17) are proved here.
  4. [Section 12, notation] The notation E♭_r and E∗♭_r in Lemma 12.4 is clear, but the indexing in Lemma 12.5 is slightly compressed; stating explicitly that the scalar multiplier is independent of i,j would improve readability.

Circularity Check

1 steps flagged · score 4.0 of 10

QCHS⇔TD depends on the self-cited companion implication CHS⇒TD ([17]); other directions are proved in-house.

  1. self citation load bearing [Section 10, equation (13); used in Section 13, proof of (17)]
    "In [17] we obtained the implication CHS⇒TD (13). ... Next assume that Φ is QCHS. We show that Φ is TD. By (16), Φ is CHS or THS. By this and (13), (15) the Hessenberg system Φ is TD."

    The QCHS⇒TD half of the central equivalence (17) is not derived in the present paper; the proof reduces it to (13), quoted verbatim from the authors' own companion preprint [17]. The paper supplies no proof of (13) inside this manuscript and no independent check. Since (13) is exactly the missing link that turns the definitional partition (16) into the claimed equality QCHS=TD, the advertised first-principles diagram (4) is, for one direction, an appeal to the same authors' prior claim. This is a disclosed question-begging dependency rather than a definitional identity; it does not affect the many in-house directions such as TD⇒QCHS or THS⇒TD, so the circularity is partial.

full rationale

The paper contains no data fitting, no fitted parameters renamed as predictions, and no equation-level circularity: the definitions of QCHS, THS, CHS, and TD are independent, and most implications (LS⇒THS, THS⇒TD, TD⇒REC, TD⇒QCHS) are proved from the definitions and from external (non-self) cited results such as [20] and [10]. The only load-bearing self-citation is the companion preprint [17] providing CHS⇒TD, which is used twice: in the proof of (17) to close the QCHS⇒TD direction, and in Section 6 to complete the CHS classification. This makes the central claim QCHS⇔TD dependent on an unproved-in-this-paper statement from the same authors. That is a verification gap and a partial circular-support structure, but not a reduction by construction, so the score is 4 rather than higher. If [17] is independently confirmed, the diagram (4) would be supported; if not, the complement claim 'CHS is the complement of THS in TD' would be unsupported.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants and no invented entities: the new families QCHS and THS are definitions, not explanatory postulates. All load-bearing premises are results inherited from the literature. The Godjali classification (Lemma 5.11) makes the parameter array the right coordinate system; Lee's Propositions 11.7 and 12.1 provide the TD/recurrence bridge; Terwilliger's [20] supplies the Leonard machinery and the TD relations; and the authors' own companion preprint [17] supplies CHS⇒TD. The only non-standard dependency is the last one — an unpublished, same-author, load-bearing citation.

assumptions (6)
  • domain assumption Classification of Hessenberg systems by their parameter array: a triple ({θ_i}, {θ*_i}, {ϕ_i}) with mutually distinct θ, θ* and nonzero ϕ_i exists and is unique up to isomorphism (Lemma 5.11).
    Cited from Godjali [8, Lemma 6.1/Theorem 6.3]; the whole parameter-array program of the paper takes this as input.
  • domain assumption CHS ⇒ TD: every circular Hessenberg system satisfies the tridiagonal relations (implication (13)).
    Not proved in this paper; cited from the authors' own companion preprint [17, arXiv:2607.05688]. Load-bearing for QCHS⇒TD in the proof of (17), Section 13.
  • domain assumption For d≥3, TD ⇔ existence of β such that {θ_i}, {θ*_i}, {ϑ_i} are all β-recurrent (Proposition 11.7).
    Attributed to Lee [10, Lemma 2.16]; used in the proofs of (14), (15), and (17).
  • domain assumption Leonard-system classification by parameter array (Lemma 8.4) and existence/uniqueness of TD-relation scalars for d≥3 (Lemma 10.1).
    Imported from Terwilliger [20, Theorems 1.9/1.12]; foundational, peer-reviewed, widely cited.
  • domain assumption Vanishing of certain triple products forces existence of β,γ,ϱ satisfying (11) ([20, Lemma 12.2]); Lemmas 11.5/11.6 convert (11),(12) into recurrence conditions.
    Used in the proof of Proposition 12.2 to bootstrap the THS characterization; from Terwilliger [20, Lemmas 12.2/12.5].
  • domain assumption Lemma 12.1: under β-recurrence of θ,θ*,ϑ, triple products vanish for 1<j−i<d, and E_0 A* E_d = 0 iff ϑ_1 = ϑ_d.
    Imported from Lee [10, Lemma 4.4]; central structural input to Propositions 12.2 and 12.13.

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Pith. "Pith review of Variations on a circular Hessenberg pair." pith.science (2026). https://pith.science/paper/NFCNKLSY

@misc{pith2026260716964,
  author       = {Pith},
  title        = {Pith review of: Variations on a circular Hessenberg pair},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NFCNKLSY}},
  note         = {Machine review of arXiv:2607.16964}
}
abstract

A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. A Hessenberg pair is an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a Hessenberg fashion. A Hessenberg system $\Phi$ an `oriented' version of a Hessenberg pair. It is known that $\Phi$ is determined up to isomorphism by its parameter array; this consists of the eigenvalue sequence of $\Phi$, the dual eigenvalue sequence of $\Phi$, and a sequence of nonzero scalars $\{\phi_i\}_{i=1}^d$ called the split sequence of $\Phi$. We are interested in some types of Hessenberg matrices, said to be circular, quasi-circular, tridiagonal, and irreducible tridiagonal. We are interested in the families of Hessenberg systems for which the associated Hessenberg matrices have one of the above types. A Hessenberg system of irreducible tridiagonal type is often called a Leonard system. In this case the associated Hessenberg pair satisfies two relations, called the tridiagonal relations. We are interested in the family of Hessenberg systems for which the associated Hessenberg pair satisfies the tridiagonal relations. We are also interested in the family of Hessenberg systems for which the eigenvalue sequence and dual eigenvalue sequence satisfy a linear three-term recurrence. In the present paper we have two main goals. First, we show how the above families of Hessenberg systems are related to each other. Second, we describe each family in terms of the parameter array.

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