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Fans and polytopes in tilting theory III: Classification of convex $g$-fans of rank 3

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Pith's one-line read This paper proves that the rank-3 convex g-fans are exactly the 61 fans listed in Table 5.

desk verdict Real progress on the rank-3 classification, but the completeness count hangs on a computer enumeration that the paper doesn't ship. read the letter →

arxiv 2508.18678 v1 pith:64GCNVZ6 submitted 2025-08-26 math.RT math.CO

Toshitaka Aoki , Akihiro Higashitani , Osamu Iyama , Ryoichi Kase , Yuya Mizuno This is my paper
classification math.RTmath.CO MSC 16G1016E3552B20
keywords tiltingtheoryg-fansg-polytopesreflexivepolytopessign-coherentfanssiltingmutationrank3classificationminimalnumberofgenerators
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

Tilting theory attaches to every finite-dimensional algebra a fan, the g-fan, in its real Grothendieck group; when the union of its simplices is convex, the resulting g-polytope is reflexive, so in each dimension only finitely many such fans can occur. This paper settles the dimension-3 case: it proves that there are exactly 61 convex g-fans of rank 3 up to isomorphism of sign-coherent fans, and Table 5 gives a complete set of representatives. The proof shows that for a g-convex rank-3 algebra the whole fan is determined by a small numerical invariant recording minimal numbers of left/right generators and a radical-squared condition for the six bimodules between primitive idempotents. It also explains why five plausible sign-coherent fans are not realizable, a phenomenon absent in rank 2.

What carries the argument

The load-bearing object is the datum d(A,e) = (d_ij) for ordered pairs of primitive idempotents of a rank-3 algebra, with d_ij = (l_ij, r_ij, h_ij), where l_ij and r_ij are the minimal numbers of generators of e_i A e_j as a left e_iAe_i-module and as a right e_jAe_j-module, and h_ij records whether e_iAe_j lies outside rad² A. Theorem 1.6 shows that in the (+−+) orthant the entire subfan is determined by the pair (d_12, d_32) together with h_13 in one exceptional case, and lists the 14 possible fans. The proof mechanism combines sign-decomposition, reduction at a ray to rank-2 convex g-fans (whose classification is already known), maximal-path analysis of Hasse quivers of mutation, and an H

What would settle it

Independently re-run the enumeration of Section 5.2: generate all pairs for (d_12, d_32), form all data d in the allowed set, impose the Proposition 5.1 constraints for every g in the symmetry group, and count the orbits. If the count is not 66, and after excluding the Proposition 5.2 cases is not 61, the classification is wrong; comparing the orbit representatives with Tables 4 and 5 gives a concrete computational check.

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Extended reading notes

Core claim

Theorem 1.5(2) is the central claim: up to isomorphism of sign-coherent fans, the number of convex g-fans of rank 3 is 61, and Table 5 lists one representative of each class. The companion Theorem 1.5(1) states that if A is g-convex of rank 3, then the datum d(A,e) completely determines the fan Σ(A,e). The strategy is to decompose the fan into the 2^3 orthants; using the symmetry group S3 × {±1}, each orthant is isomorphic to the standard (+−+) orthant, and Theorem 1.6 classifies the possible subfans there via the 14 data patterns d(0),…,d(13). A computer-assisted enumeration yields 66 convex sign-coherent fans satisfying these local constraints; 61 are realized by explicit algebras in Table

Load-bearing premise

The load-bearing premise is that the computer enumeration in step (i) of the proof is complete: it must produce exactly the 66 convex sign-coherent fans satisfying the local orthant constraints. The paper supplies neither the program nor its raw output, so the final count of 61 inherits the correctness of that enumeration.

Editorial extensions

If this is right

  • For every g-convex rank-3 algebra, the g-polytope is one of 61 reflexive polytopes, and Table 5 records a realizing algebra and its number of 2-term silting complexes for each.
  • The classification is complete: any convex g-fan in R³ is isomorphic to a sign-coherent fan in Table 5 under coordinate permutation and sign reversal.
  • The datum d(A,e) is a complete invariant of the fan for g-convex rank-3 algebras; two g-convex algebras with the same d have isomorphic g-fans.
  • Five sign-coherent fans that satisfy the local orthant conditions cannot be realized, showing that in rank 3 the gluing of orthant data is more constrained than in rank 2, where quadrants are independent.
  • Since g-finiteness is equivalent to completeness of the g-fan, each of the 61 fans is complete and comes from a g-finite algebra.

Reading between the lines

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

  • An independent re-implementation of the step-(i) enumeration would convert the count 66, and hence 61, into a checkable computational certificate rather than a stated program output.
  • The same orthant-decomposition strategy could in principle be attempted in rank 4, but the number of possible d-data and local fans grows quickly; a new structural restriction would likely be needed rather than a direct extrapolation.
  • Because the g-polytopes are reflexive, the 61 fans give 61 lattice polytopes; comparing this list with classifications of three-dimensional reflexive polytopes could reveal which reflexive polytopes arise from tilting theory, a direction the paper does not pursue.
  • The five excluded fans suggest that convexity plus sign-coherence is close to, but not identical with, realizability; testing those same five under a weakened convexity condition might isolate exactly which convexity hypothesis is responsible.
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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 / 2 minor

Summary. The paper classifies convex g-fans of rank 3. Theorem 1.5(1) states that for a g-convex finite dimensional algebra of rank 3, the fan Σ(A,e) is completely determined by the numerical datum d(A,e) of minimal generator numbers of the bimodules eiAej. Theorem 1.5(2) states that there are precisely 61 convex g-fans of rank 3 up to isomorphism of sign-coherent fans, with representatives in Table 5. The proof strategy is sign-decomposition of the fan into the eight orthants, a detailed analysis in Theorem 1.6 of the possible (+−+)-subfans, constraints from Proposition 5.1, a computer enumeration reducing the possible data to 66 G-orbits, realization of 61 of these by explicit algebras, and exclusion of the remaining 5 via ring-theoretic arguments in Proposition 5.2.

Significance. If correct, the paper gives the first complete classification of convex g-fans in rank 3, extending the rank-2 result of the authors' previous work and providing strong evidence for the general finiteness philosophy of g-convex algebras. The structural statement Theorem 1.5(1) is valuable and appears to be proved non-circularly from the datum d(A,e). The case analysis in Sections 4 and 5 is detailed, and the non-realizability proofs in Section 5.4 are genuine ring-theoretic arguments, not merely appeals to computation. The main weakness is that the exact count in Theorem 1.5(2) depends on an undocumented computer enumeration; the manuscript ships no code, pseudocode, or machine-readable output, so the central completeness claim is not independently verifiable from the paper as written.

major comments (2)
  1. [§5.2, proof of Theorem 1.5(2)] The statement 'Indeed, we may use a computer to obtain this list' is load-bearing. The proof reduces the space of possible data to 66 G-orbits by applying Proposition 5.1 to the 7^6 possible data, but neither the program, the algorithm, nor the list of 66 representatives is provided. Since the exact count 61 is the headline result and the analytic results only narrow the search space, an error or omission in this enumeration would invalidate Theorem 1.5(2). The authors should supply the code or an independently checkable certificate, including the 66 representatives and the 5 excluded data, and specify the algorithm used for the orbit count and completeness check.
  2. [§5.2 / Table 5] The lower bound 'at least 61' also depends on the undocumented enumeration. The proof asserts that the list of candidates is 'up to isomorphism,' and Theorem 1.5(1) only gives uniqueness of the fan for each datum, not injectivity of the correspondence d ↦ Σ(A,e). Thus, without a repairable enumeration certificate, the pairwise non-isomorphism of the 61 fans in Table 5 is as unverified as the upper bound. If the enumeration is supplied, this distinctness point should be stated explicitly.
minor comments (2)
  1. [§1, Table 4/Table 5] The table captions in the arXiv text are confusing: the caption 'Table 4' appears to precede a long table that looks like the 61-row Table 5, while the five excluded fans in (5.1) are not displayed as a separate table. Please correct the labelling and make the layout of Table 4 and Table 5 unambiguous.
  2. [§4.10] In 'Staring at the positive cone σ+', the word should be 'Starting'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the classification is anchored in the independently defined datum d(A,e); the unverified computer enumeration is a reproducibility gap, not a circular step.

full rationale

The central invariant d(A,e) is defined directly from the bimodule generator numbers (l_{ij}, r_{ij}, h_{ij}) and is independent of the g-fan. Theorem 1.5(1) is proved by reducing each orthant subfan to the (+-+) case via the group action and then invoking Theorem 1.6, which is proved by explicit path and polytope computations. The rank-2 classification from the authors' prior work is used as a lower-dimensional, parameter-free external result; it is load-bearing but not circular, since it does not assume the rank-3 classification. The only conspicuous gap is the statement in the proof of Theorem 1.5(2): 'Indeed, we may use a computer to obtain this list' — no code or raw data is supplied, so the completeness of the 66-element enumeration is not reproducible from the paper alone. That is a rigor/reproducibility concern, not a circular reduction: the enumeration is an exhaustive search of the finite set S of d-data satisfying the explicit constraints of Proposition 5.1, and the final 61/5 split is then justified by explicit algebra constructions (Table 5) and independent ring-theoretic exclusions (Proposition 5.2). No step assumes Theorem 1.5(2) to prove itself, and no fitted parameter is renamed as a prediction. Hence the derivation chain is not circular.

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

The paper introduces no new postulated entities or fitted constants. Its central claim rests on background tilting theory, the prior rank-2 classification, and an unverified computational enumeration; all are external or asserted rather than derived in this paper.

assumptions (4)
  • domain assumption The rank-2 classification of convex g-fans (7 fans) from [AHIKM2] is correct and applies.
    Used in Lemma 3.2, Propositions 3.6, 3.9, and throughout the path analysis; it is an external result by the same authors and is treated as a black box.
  • ad hoc to paper The computer enumeration of 66 convex sign-coherent fans is complete and correct.
    Stated in Strategy (i) and proof of Theorem 1.5(2); no code or detailed search algorithm is provided, so the completeness of the list is an unverified computational assumption.
  • domain assumption Foundational silting and tilting theory results (e.g., sign-coherent property, reduction theorem, order properties of the 2-silt poset) hold as cited.
    Used pervasively (Sections 2.1, 2.2, 2.4) from [DIJ], [AiI], [AIR], [AHIKM1]; these background facts are not proved in this paper.
  • domain assumption g-convexity is preserved under idempotent truncation and reduction (Propositions 2.12, 2.13).
    These are central tools for restricting to rank 2 and for path arguments; they are taken from [AHIKM1] and used without reproof.

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Pith. "Pith review of Fans and polytopes in tilting theory III: Classification of convex $g$-fans of rank 3." pith.science (2026). https://pith.science/paper/64GCNVZ6

@misc{pith2026250818678,
  author       = {Pith},
  title        = {Pith review of: Fans and polytopes in tilting theory III: Classification of convex $g$-fans of rank 3},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/64GCNVZ6}},
  note         = {Machine review of arXiv:2508.18678}
}
abstract

The $g$-fan $\Sigma(A)$ of a finite dimensional algebra $A$ is a non-singular fan in its real Grothendieck group, defined by tilting theory. If the union ${\rm P}(A)$ of the simplices associated with the cones of $\Sigma(A)$ is convex, we call $A$ $g$-convex. In this case, the $g$-polytope ${\rm P}(A)$ of $A$ is a reflexive polytope. Thus, in each dimension, there are only finitely many isomorphism classes of fans that can be realized as $g$-fans of $g$-convex algebras. An important problem is to classify such fans for a fixed dimension $d$. In this paper, we give a complete answer for the case $d=3$: we prove that there are precisely 61 convex $g$-fans of dimension 3 up to isomorphism. Our method is based on the decomposition of fans into the $2^3$ orthants in the real Grothendieck group of $A$, together with a detailed analysis of possible sequences of $g$-vectors arising from iterated mutations.

Figures

Figures reproduced from arXiv: 2508.18678 by the authors.

Figure 1
Figure 1. 4.3. Case d(6). We assume that d+−+(A, e) = d(6) = ((2, 1, 1),(2, 1, 0)). By Proposition 3.5, we have σ+−+,max = h 1 0 0 [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗
Figure 2
Figure 2. 4.4. Case d(7). Assume that d+−+(A, e) = d(7) = ((1, 1, 1),(1, 1, 1)). By Proposition 3.5, we have σ+−+,max = h 1 0 0 [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗
Figure 3
Figure 3. We first show that the above cone τ lies in Σ+−+(A, e). Consider an arrow σ+−+,max = h 1 0 0 [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: If this is (a), then there exists an arrow τ = h 0 −1 1 [PITH_FULL_IMAGE:figures/full_fig_p031_4.png]
Figure 5
Figure 5. Figure 5: P+−+(A, e) = P+−+(X) by Proposition 2.17. Therefore, there are 3 possibility of Σ+−+(A, e) described in [PITH_FULL_IMAGE:figures/full_fig_p032_5.png]
Figure 6
Figure 6. Figure 6: Since Σ(A, e) is ordered, there is an arrow τ :=  1 −2 0 [PITH_FULL_IMAGE:figures/full_fig_p032_6.png]
Figure 7
Figure 7. Figure 7: To prove the assertion, it suffices to show that τ ∈ Σ+−+(A, e), where τ = h 1 −2 0 [PITH_FULL_IMAGE:figures/full_fig_p034_7.png]
Figure 8
Figure 8. Figure 8: From now on, we show that κ, φ lies in Σ+−+(A, e), where κ, φ are cones in (4.9). Firstly, we study maximal cones of Σ+−+(A, e) adjacent to the maximum and minimum. We consider an arrow τ :=  1 −1 0 [PITH_FULL_IMAGE:figures/full_fig_p035_8.png]
Figure 9
Figure 9. Figure 9: If this is (a), then Σ(A, e) contains the following maximal cones. γ := h 2 −1 0 [PITH_FULL_IMAGE:figures/full_fig_p037_9.png]

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