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This paper proves two symmetric algebras are 3-representation-finite and still escape the orbit-algebra construction.

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load-bearing objection Strong, ambitious paper whose non-orbit obstruction is compelling, but whose 3-representation-finite claim rests on an imported unproved classification from Erdmann's preprint. the 4 major comments →

arxiv 2607.18497 v1 pith:AQ4JJ7UN submitted 2026-07-20 math.RT

Cyclic covers and non-orbit 3-representation-finite symmetric algebras

Tor Kringeland This is my paper
classification math.RT MSC 16G1016D5016G70
keywords higher Auslander-Reiten theory3-representation-finite algebrasself-injective algebrasorbit algebrasrepetitive categoriescluster-tilting modulescyclic coversweighted surface algebras
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that the orbit-algebra construction, long the principal source of d-representation-finite self-injective algebras, is not exhaustive. It exhibits a 36-dimensional symmetric algebra A, a characteristic-zero lift of a quaternion-type algebra known to be 3-representation-finite in characteristic 2, and an 84-dimensional 3-spherical algebra S, both with explicit 3-cluster-tilting modules. For each algebra, the paper proves no presentation as an orbit category of a repetitive category exists, regardless of the global dimension of the underlying algebra. The proof works by showing that any such presentation would force a half-dimensional square-zero grading, and that for these algebras every possible grading is killed by a trace obstruction. A curious reader should care because this settles an open question in higher Auslander-Reiten theory, showing the orbit construction misses genuine examples.

Core claim

The paper's central claim is that the algebra A, presented by an explicit quiver with relations, is 3-representation-finite with the explicit 3-cluster-tilting module M = A ⊕ S_2 ⊕ Ω^2 S_1 ⊕ Ω^2 S_3 ⊕ U, where U is a uniserial module of composition series (2,1,2); and that A is not isomorphic to any orbit algebra Λ-hat/〈φ〉 for any finite-dimensional algebra Λ and admissible automorphism φ. The same conclusion holds for the second algebra S. The non-orbit proof is independent of the cluster-tilting construction and of any global-dimension assumption on Λ. The key mechanism: if an orbit presentation existed, then A or a connected cyclic cover of A would admit a Z-grading supported in degrees 0

What carries the argument

The load-bearing device is a pair of invariants. First, a balanced square-zero extension: a Z-grading with support {0,1} in which the degree-1 half has dimension equal to half the algebra. Second, the group of arrow characters H_t(C), built from two integer matrices (a character-relation matrix B and a vertex-gauge matrix P), which classifies the possible degree vectors of connected cyclic covers. For A, H_t(A) is nonzero only for t=2 and leaves exactly three covers, each ruled out by a trace-zero theorem for idempotent-annihilating derivations. For S, the identity ker B_S = im P shows every connected cyclic cover collapses, so only the derivation obstruction remains.

Load-bearing premise

The argument leans on a cited classification, not reproduced in the paper, stating that every module in the 2-orthogonal of the candidate module is a uniserial subquotient of two specific syzygy modules; if that classification misses some summand, the exhibited module might not actually be 3-cluster-tilting.

What would settle it

Exhibit an indecomposable module X over the algebra A (or S) with Ext^i(M,X)=0 for i=1,2 but X not in add M; or produce an idempotent derivation E on A (or on one of the three double covers) with trace equal to half the dimension, which would supply the forbidden grading.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the paper is correct, the orbit construction is not exhaustive for d-representation-finite self-injective algebras in infinite representation type.
  • The non-orbit obstruction holds for any finite-dimensional Λ, regardless of global dimension, strengthening what the open question originally asked.
  • The trace criterion gives a practical method to test whether a connected symmetric algebra is an orbit algebra: check for existence of a half-dimensional square-zero grading.
  • The explicit cluster-tilting module shows that a single well-chosen summand U suffices to complete a previously known rigid candidate to a full 3-cluster-tilting module.
  • The second example S shows the phenomenon is not isolated: the same obstruction applies to a different weighted surface algebra.

Where Pith is reading between the lines

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

  • The trace-grading obstruction may be convertible into a general cohomological criterion for orbitness of connected symmetric algebras, beyond the two examples treated here.
  • The method suggests that other weighted surface algebras in infinite representation type could be tested in the same way; computing H_t and the derivation spaces for the whole family is a natural next step.
  • The fact that adjoining a single summand U, rather than all rigid uniserial candidates, succeeds hints that earlier combinatorial rigidity conditions for cluster-tilting were too coarse.
  • Since the non-orbit proof is independent of the cluster-tilting claim, the two results stand or fall separately: even if the cluster-tilting module relied on a classification flaw, the non-orbit conclusion would be untouched.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 9 minor

Summary. The paper exhibits two symmetric algebras over an algebraically closed field of characteristic zero, A (36-dimensional) and S (84-dimensional), and claims they are 3-representation-finite but not isomorphic to any orbit algebra bΛ/⟨φ⟩ of a repetitive category, for any finite-dimensional Λ and admissible φ. For A, the proof constructs an explicit 3-cluster-tilting module M = M0 ⊕ U (Proposition 3), and then develops a general obstruction theory: any orbit presentation of a connected symmetric algebra forces either the algebra or a connected cyclic cover to admit a half-dimensional square-zero grading, detected by idempotent derivations (Lemmas 8–13). A finite group of arrow characters H_t(A) leaves only three double covers at t=2, each excluded by a trace criterion. For S, the character lattice satisfies ker B_S = im P, eliminating all nontrivial connected cyclic covers, and the derivation obstruction excludes the t=1 case. The paper concludes that both algebras are counterexamples to Darpö–Iyama's Question 1, even without assuming finite global dimension of Λ.

Significance. If correct, the paper settles a natural open question in higher Auslander–Reiten theory: the orbit construction of Darpö–Iyama is not exhaustive for self-injective d-representation-finite algebras, even in the symmetric case. The non-orbit part of the argument is notably stronger than the question asks, since it does not require finite global dimension of Λ. The paper contains a parameter-free obstruction theory, explicit modules and bases, and states that all computations were double-checked in exact arithmetic over Q — a strength that allows independent verification. The proposed counterexamples are concrete and the proof strategy (grading obstructions via idempotent derivations, cyclic covers controlled by character lattices, and Mostow conjugacy to diagonalize gradings) is likely to be reusable. However, the 3-representation-finite claim rests substantially on an imported, unproved classification result of Erdmann [5, Proposition 5.4], which the paper itself acknowledges requires a correction in Lemma 5.3; this is the main risk to the central claim.

major comments (4)
  1. [§3, Proposition 3 and Remark 4] The 3-representation-finite claim for A depends entirely on [5, Proposition 5.4], which classifies indecomposable modules in the 2-orthogonal of M0 as uniserial subquotients of Ω²S1 or Ω²S3 with top and socle S2. This proposition is neither proved nor reproduced; it is imported from a preprint that the paper itself admits contains a flaw in Lemma 5.3 (Remark 4). Remark 4 patches the statement of Lemma 5.3 by passing to the quotient by e(soc Λ)e, but the interaction of this correction with the subsequent string classification in [5] is not addressed in the present paper. If Proposition 5.4 fails for A, additional indecomposable summands could lie in the 2-orthogonal of M0, and M would not be 3-cluster-tilting. The non-orbit part of Theorem 2 would still hold, but A would cease to be a counterexample to Question 1. This is load-bearing for the paper's central claim.
  2. [§5, Proposition 15] The same unproved [5, Proposition 5.4] is invoked for S, with the same Remark 4 caveat, and the proof of Proposition 15 explicitly relies on it twice: first to identify all possible indecomposable nonprojective modules in the 2-orthogonal of M0, and second through Lemma 5.5 of [5] for orthogonality of the uniserial candidates. Since S is the second counterexample and the paper's abstract names it prominently, the cluster-tilting half of the S result inherits the same risk. A failure of the classification would leave N not 3-cluster-tilting, again removing S as a counterexample to Question 1 even though the non-orbit theorem would remain true.
  3. [§5, Remark 16] Remark 16 states that Proposition 15 'corrects' the quoted conclusion of [5, Corollary 5.6 discussion] that M cannot be extended to a 3-cluster-tilting module for n-spherical algebras when n≥3. This is a substantial mathematical claim about the prior literature. The present paper does not reproduce the argument from [5] that rules out adjoining all candidates, and does not fully explain why the displayed extension 0→B2→S_{a2}⊕Ω²S_{d3}→B1→0 invalidates it. If the reading of [5] is inaccurate, the novelty attribution in Remark 16 would need adjustment. As written, this is more than a bibliographic remark and should be either substantiated or softened.
  4. [§4, Lemma 12] Lemma 12 asserts that if χ(D)=χ(G), then G = N⋊D and uses Mostow's theorem [10, Theorem 7.1] to conjugate any finite-order element of G into D. The proof observes that N is unipotent and connected, and that both F_D and ⟨σ⟩ are maximal fully reducible subgroups of H. This is a delicate algebraic-group argument. The statement that 'N has no nontrivial element of finite order' uses the logarithm, which is valid because N is unipotent; however, the proof then claims that both F_D and ⟨σ⟩ are fully reducible as subgroups of H, which requires that they act semisimply on C. This is asserted but not demonstrated. The reader must verify that the finite group generated by σ is semisimple on the whole algebra; this follows from σ having finite order and char 0, but the proof would benefit from an explicit sentence. Similarly, the use of Mostow's theorem requires that the ambient group G is an alge
minor comments (9)
  1. [Abstract and Introduction] The abstract says '3-representation-finite' but the introduction defines 'd-representation-finite' with a hyphen; please unify notation. Also 'Darpö and Iyama's orbit construction is not exhaustive' is a clear summary in the reader's wording but the abstract should be careful to state that the counterexamples are symmetric and the non-orbit statement holds without a global-dimension hypothesis, which it does.
  2. [§1, equation (1)] The relations are written 'byb=bdnybdn, yby=dnybdny, ndn=nybdnyb, dnd=ybdnybd, bybd=0, ndny=0'. It may help the reader to explicitly note that the first four are binomial relations and the last two are monomial, and that 'paths composed left to right' means the quiver path convention is opposite to the standard composition of functions. This is already stated briefly but could be highlighted for accessibility.
  3. [§3, proof of Proposition 3] The paragraph 'We first present the two second syzygies' says that the images of e2, d, dn, dny, dnyb, dnybd, dnybdn form a basis of P2/yA. Since the basis of A in Appendix A lists B2 with 16 elements, the reader must check that yA has basis of size 9. This is done in the preceding sentence but the reasoning could be condensed. The notation U(2,3,2,1,2,3,2) is used without reciting the definition of composition series order (top to socle), which is defined earlier in the section; still, a direct pointer would help.
  4. [§4, Lemma 8] The proof of Lemma 8 is somewhat dense: the step 'bνx = ¯x for every x' follows from C being symmetric, because the Nakayama functor of C permutes the indecomposable projectives trivially. This should be stated explicitly. The notation tx is then used for the exponent, and it is shown to be constant using local boundedness; the argument is correct but the reader must supply several standard facts about repetitive categories. Adding a reference to [3, §2] for the level function and local boundedness would improve readability.
  5. [§4, Lemma 10] The proof says 'Conjugating the grading by u therefore makes the given ei homogeneous of degree 0'. It may be helpful to note that the unit u is constructed from the isomorphism of projective modules, and that after conjugation one obtains an honest Z-grading with the same support. This is standard but a one-line justification is advisable.
  6. [§4, proof of Theorem 7] When t=1, the paper invokes 'Proposition 18(1) and Lemma 13' to show every idempotent-annihilating derivation has trace zero. The condition 'dime vC = dimCe v' is satisfied because A is symmetric; this is stated. However, the reader must also check that the quiver of A has no loops and no parallel arrows; this is stated in Section 1 but not repeated. A cross-reference to Section 1 would be convenient.
  7. [§5, proof of Theorem 17] In the t≥2 case, the proof says 'The identity morphisms of the quotient category are homogeneous of degree0' and then 'the unit-conjugation argument in the second half of the proof of Lemma 10 moves the idempotents'. This is somewhat implicit. The model of [2] for the quotient category is used without discussing how the grading on the quotient category is obtained from the free action; a reference to [2, Theorem 3.8] is given but the exposition could be expanded slightly, especially because the categorical smash product is then replaced by an algebra isomorphism.
  8. [Appendix A] The basis verification for A says 'one checks that the six relations of (1) act as zero' but the details are not shown. Since the paper emphasizes computer verification, it would be reassuring to include the explicit multiplication table or at least state that the check is symbolic over Q. The path bases for S in (5) are asserted to span and then their cardinalities are matched to the projective dimensions from [7]; this is a reasonable shortcut, but the reader must trust the cited dimensions.
  9. [References] Reference [4] is a self-citation (Darpö–Kringeland) that is cited in a side remark in the introduction and again in Remark 14. It is not used in the proofs, which is fine, but it might be worth noting that the classification in [4] is for finite representation type and distinct from the infinite-type case studied here, to avoid confusion.

Circularity Check

0 steps flagged

No significant circularity: the central non-orbit obstruction is a parameter-free contradiction proof, and the cluster-tilting construction rests on external classification results, not on the paper's own conclusion.

full rationale

The derivation chain is not circular. A is fixed by a quiver and explicit relations; the claimed 3-representation-finite property is proved by explicitly exhibiting M = M0 ⊕ U and then checking the two 2-orthogonals. The classification of indecomposable modules in the 2-orthogonal of M0 is imported from Erdmann [5, Proposition 5.4 and Lemma 5.5], and Remark 4 records a correction to [5, Lemma 5.3]. That is an external mathematical dependency and a real correctness risk, but it is not a reduction of the paper's conclusion to its own inputs: the paper does not define M, the 2-orthogonal, or the non-orbit property in terms of the target claims. The rigidity and extension checks for U, V, W are direct computations. The non-orbit half is self-contained: assuming an orbit presentation, Lemmas 8–13 force a balanced square-zero grading and hence an idempotent derivation in V(A); Proposition 18 and Lemma 13 then compute that every idempotent-annihilating derivation of A or of the three double covers has trace zero, contradicting V nonemptiness. No parameter is fitted to a predicted quantity, and no notion is defined in terms of 'orbit algebra' or '3-representation-finite'. The only self-citation, [4] (Darpö–Kringeland), appears in the introduction as a classification statement and is not used in the proofs. Thus no circular step can be exhibited; the acknowledged gap concerning Erdmann's Proposition 5.4 is an external-dependency concern, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central claims do not involve fitted parameters or invented objects. The algebras A and S are concrete, fully specified quivers with relations. The non-orbit argument is a parameter-free obstruction; the 3-representation-finite part imports structural classification results, including one whose proof is corrected in Remark 4. Those imports are the main external dependencies.

axioms (6)
  • domain assumption k is an algebraically closed field of characteristic 0.
    Used throughout; needed for Lemma 10 (2 invertible, algebraic-group semisimplicity) and Mostow's theorem.
  • domain assumption Darpö–Iyama orbit-category framework [3]: admissibility, Serre duality, Nakayama automorphism, and ν_d-finiteness.
    Defines the target class of orbit algebras; Lemma 8 and the orbit-category model rely on [3].
  • domain assumption Weighted-surface-algebra theory [6,7,8]: A and S are general weighted surface algebras, hence symmetric, periodic of period 4, with Ext-symmetry (3) and stated dimension formulas.
    Imported classification results used in Proposition 3 and Proposition 15 before the explicit module checks.
  • domain assumption Erdmann [5, Prop 5.4 and Lemma 5.5] classification of the 2-orthogonal of M0 applies to A and S after the correction in Remark 4.
    Load-bearing for the 3-cluster-tilting proof; the paper does not reproduce this classification fully.
  • standard math Mostow's conjugacy theorem for maximal fully reducible subgroups in characteristic 0 [10] and standard algebraic-group facts [12].
    Used in Lemma 12 to make finite-order automorphisms diagonal on arrows.
  • standard math Galois covering and smash-product equivalence of Cibils–Marcos [2] and Green [9].
    Used to pass from orbit categories to graded algebras and to identify covers with smash products.

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read the original abstract

Over an algebraically closed field of characteristic zero, we exhibit two symmetric algebras that are 3-representation-finite but are not orbit algebras of repetitive categories. The first is a 36-dimensional characteristic-zero lift $A$ of $Q(3A)^2_2$, the quaternion-type algebra that B\"ohmler and Marczinzik proved 3-representation-finite in characteristic 2; we construct an explicit 3-cluster-tilting module. We show that any orbit presentation of a connected symmetric algebra forces the algebra, or a connected cyclic cover of it, to admit a half-dimensional square-zero grading. Such gradings are detected by idempotent derivations, and a finite group of arrow characters constrains the possible covers. For $A$, only three double-cover candidates remain and the derivation obstruction excludes all of them. Thus $A$ is not an orbit algebra of any finite-dimensional algebra, regardless of global dimension, answering a question of Darp\"o and Iyama. The 84-dimensional 3-spherical weighted surface algebra is likewise 3-representation-finite and not an orbit algebra.

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Reference graph

Works this paper leans on

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