Scattering diagrams for Artin algebras

Hipolito Treffinger

REVIEW 2 major objections 5 minor 1 cited by

Scattering diagrams for Artin algebras

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For any Artin algebra, this paper constructs a minimal consistent scattering diagram from the lattice of torsion classes, and shows that for path algebras over \mathbb{C} it agrees with the established stability scattering diagram.

desk verdict Solid finite-level construction; the inverse-limit step in Theorem 7.11 needs a real argument before the infinite diagram is justified. read the letter →

arxiv 2608.04233 v1 pith:GKEOHQ57 submitted 2026-08-04 math.RT math.RA

Hipolito Treffinger This is my paper · ORCID

classification math.RTmath.RA MSC 16G1016G2016G7018E4013F60

keywords Artinalgebrasscatteringdiagramstorsionclasseswall-and-chamberstructuresstabilityconditionspicturegroupsinverselimitsbounded-lengthmodules

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 proves that every Artin algebra—a finite module category over an artinian ring—has a minimal consistent scattering diagram, a wall-and-chamber picture that records how stability conditions change as a path crosses walls. This matters because scattering diagrams encode representation-theoretic data such as torsion classes and $\tau$-tilting modules, and until now they were known only for finite-dimensional algebras over the complex numbers. The proof cuts the module category into layers of modules of length at most $\ell$, builds a finite scattering diagram for each layer, and assembles them by inverse limit. For algebras of the form $\mathbb{C}Q/I$, the newly built diagram is isomorphic to the already-known stability scattering diagram.

What carries the argument

The argument is carried by the bounded-length truncation $(\operatorname{mod} A)_\ell$, the full subcategory of modules of length at most $\ell$. Each truncation has only finitely many walls, so every generic path crosses finitely many of them; its torsion classes $T_\ell = T \cap (\operatorname{mod} A)_\ell$ form a complete semidistributive lattice with a brick labelling, and its $\ell$-TF-equivalence classes form the finite cone complex $\mathcal{D}_\ell(A)$. To each $\mathcal{D}_\ell(A)$ the paper attaches a picture group $G_\ell(A)$, generated by semistable subcategories under the extension product $\ast$, so that the wall label $\Phi_\ell(d) = (\operatorname{mod}^{\mathrm{ss}}_d A)_\ell$ makes $(\mathcal{D}_\ell(A), \Phi_\ell)$ a minimal consistent scattering diagram. A compatible family of path elements $g^\ell_\gamma$ then defines $g_\gamma$ in the inverse limit $G(A) = \varprojlim G_\ell(A)$.

What would settle it

For an algebra with infinitely many walls, such as the Kronecker algebra over $\mathbb{C}$, take a generic path $\gamma$ that crosses infinitely many walls. For each $\ell$, compare the element $g^\ell_\gamma$ given by the compatible family with the image in $G_\ell(A)$ of the finite ordered product over the walls of $\mathcal{D}_\ell(A)$ crossed by $\gamma$; a single level where they differ, or a path where the finite products have no limit, would falsify Theorem 7.11.

Watch

Extended reading notes

Core claim

The central claim is that the data of any Artin algebra $A$ determine a minimal consistent scattering diagram $(\mathcal{D}(A), \Phi \colon \mathcal{D}^1(A) \to G(A))$. Here $\mathcal{D}(A)$ is the cone complex formed by TF-equivalence classes of stability conditions on $\operatorname{mod} A$, $G(A)$ is the inverse limit of picture groups $G_\ell(A)$ attached to the subcategories of modules of length at most $\ell$, and the wall label $\Phi(d)$ is the semistable subcategory $\operatorname{mod}^{\mathrm{ss}}_d A$. The diagram is minimal because no wall receives the identity label, and consistent because every generic path $\gamma$ has an element $g_\gamma$ that depends only on its endpoints. When $A = \mathbb{C}Q/I$ with $Q$ a finite quiver and $I$ an admissible ideal, this torsion scattering diagram is isomorphic to the stability scattering diagram previously constructed for such algebras.

Load-bearing premise

The construction depends on identifying the inverse-limit element assigned to a generic path with the ordered product of the wall labels it crosses; if that identification fails, the infinite object is not a scattering diagram in the usual sense.

Editorial extensions

If this is right

Every Artin algebra, even one with infinitely many walls, now has a wall-crossing product for every generic path in its stability space, assembled from finite length-level data.
For $\tau$-tilting finite algebras the construction stabilizes: $\mathcal{D}(A) = \mathcal{D}_\ell(A)$ and $G(A) = G_\ell(A)$ for some $\ell$, recovering the finite picture-group scattering diagram.
For finite-dimensional $\mathbb{C}$-algebras of the form $\mathbb{C}Q/I$, the torsion scattering diagram is isomorphic to the stability scattering diagram, so the classical motivic Hall algebra construction is not needed to obtain it.
The wall-and-chamber structure $\mathcal{D}(A)$ is recovered as an inverse limit of finite cone complexes $\mathcal{D}_\ell(A)$, giving finite approximations to the stability space that can be computed level by level.
Minimality is built in: no wall is labelled by the identity, so every wall contributes an actual crossing effect to the scattering process.

Reading between the lines

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

The inverse-limit definition of $g_\gamma$ as a compatible family may not literally equal the ordered product of infinitely many wall labels in $G(A)$; if that equality fails, the group would need an ordering or completion to match the classical scattering-diagram convention. This can be checked on Kronecker-type examples with infinitely many walls.
The same length-truncation and inverse-limit recipe should work for any abelian length category with finitely many simple objects, not only module categories of Artin algebras.
If the paper's conjectures hold—that each $\mathcal{D}_\ell(A)$ is connected and that every $T_\ell$ is determined by its semibricks—the finite diagrams become algorithmically tractable, making $G(A)$ and its wall-crossing products computable in concrete examples.
The isomorphism with the stability scattering diagram suggests that $G(A)$ is a purely combinatorial model for the pro-unipotent group built from Hall algebras, so homological information about $G(A)$ may carry stability-space information.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper constructs a "torsion scattering diagram" for every Artin algebra by first building finite diagrams on the categories of modules of length at most ℓ and then taking an inverse limit. This is a genuinely new approach, and the finite-level part looks like solid mathematics. The author develops a lattice of torsion classes for these bounded-length categories, shows it is completely semidistributive with brick labels, builds wall-and-chamber structures and picture groups, and proves consistency for the finite diagrams. The categorical description of the picture groups via subcategories and the ∗-product is a nice contribution in its own right. I have no real complaints about Sections 3–6; the arguments are detailed and mostly convincing.

The main result, Theorem 7.11, is where I part ways. The paper defines g_γ for a D(A)-generic path as the compatible family (g^ℓ_γ) in the inverse limit group G(A), where each g^ℓ_γ comes from the finite diagram. But the definition of a scattering diagram in Section 7.1 requires g_γ to be the ordered product of the wall labels along γ. The paper never proves the inverse-limit tuple is that ordered product. This is not a cosmetic issue: D(A) can have infinitely many walls, and G(A) is an arbitrary inverse limit of groups, so infinite products do not automatically exist. The stress-test note has this exactly right. The gap is probably fixable — for each fixed ℓ only finitely many walls have non-trivial label in G_ℓ(A), so one can define the ordered product coordinatewise and then show the tuple is that product — but as written the argument is missing. Without this step, consistency of the tuple is not consistency of a scattering diagram in the sense the paper itself advertises.

The comparison with Bridgeland's stability scattering diagram (Theorem 8.3) is also too compressed. The map I is only defined on the images of the wall labels; no isomorphism between the ambient groups G(A) and the pro-unipotent group from the motivic Hall algebra is constructed. That may be enough for a weak notion of "isomorphic diagram", but it needs to be stated and argued more carefully.

There are minor issues: Proposition 3.7 is left to the reader, and some "straightforward" steps in Section 5 deserve more detail. These are not serious.

Bottom line: this is a promising paper with a real new construction, but the infinite-level theorem has a load-bearing gap. I would recommend sending it to a serious referee and asking for a revision that either proves the inverse-limit tuple equals the ordered product, or changes the definition of consistency for inverse-limit diagrams. The target audience is representation theorists working on stability conditions, τ-tilting theory, and scattering diagrams; they will find the finite-level machinery useful regardless.

Referee Report

2 major / 5 minor

Summary. The paper constructs, for every Artin algebra A, a minimal consistent scattering diagram whose support is the wall-and-chamber structure of A and whose group is an inverse limit of picture groups associated with the bounded-length categories (mod A)_ell. The finite-level diagrams are built from lattices of torsion classes, a finite wall-and-chamber structure D_ell(A), and a categorical presentation of the picture group G_ell(A); consistency is proved via the groupoid of intervals in the lattice of numerical torsion classes. Passing to the inverse limit over ell, the author claims a scattering diagram for mod A and, for A = CQ/I, an isomorphism with Bridgeland's stability scattering diagram. The finite-level parts are developed in detail; the main gap is the passage from the compatible inverse-limit tuples in Theorem 7.11 to the ordered wall-crossing products required by the paper's own Definition 7.5.

Significance. If the construction is fully justified, the paper settles a natural open problem: it gives every Artin algebra a canonical wall-and-chamber scattering diagram, without motivic Hall algebra techniques, and recovers Bridgeland's stability scattering diagram as a special case. The manuscript contains several independently useful contributions: the lattice-theoretic analysis of tors_ell(A) (Theorem 3.1, Proposition 3.10, Theorem 3.12), the characterization of ell-TF-equivalence in Theorem 4.11, the categorical presentation of the picture group in Theorem 6.21, and the finite-level consistency theorem (Theorem 7.9). These parts are largely rigorous and provide a transparent categorical framework. The inverse-limit step and the comparison with Bridgeland's diagram, however, are not yet established at the level of detail required for the central claims.

major comments (2)

[§7.3, Theorem 7.11] The proof defines g_gamma as the compatible inverse-limit tuple (g^ell_gamma)_{ell in N}, but Definition 7.5 defines g_gamma as the ordered product of the wall labels along gamma, and the introduction (p. 2) states that g_gamma should be 'calculated as the ordered product' of the elements associated to the walls crossed. The proof never shows that the inverse-limit tuple is this ordered product in G(A). For each fixed ell, Lemma 7.8 identifies the finite-level product with g^ell_gamma, so the tuple is a compatible family of finite ordered products; however, G(A) is an inverse limit of arbitrary groups with no pro-nilpotent structure and no convergence criterion is given for infinite products. Lemma 7.3 only shows that each individual wall is crossed finitely many times, not that a D(A)-generic path crosses only finitely many walls in total. Remark 7.12 explicitly notes that the green-path argument uses Proposition 5.8, while arbitrary generic paths require the missing identification. Thus the well-definedness of the infinite-level scattering diagram in the sense of Definition 7.5 is not established. The paper should either prove that the tuple satisfies the ordered-product property with respect to a defined notion of infinite product in G(A), or explicitly amend Definition 7.5 to define well-definedness via inverse-limit compatibility; the latter would still require a comparison with the standard notion used in Theorem 8.3.
[§8, Theorem 8.3] The claimed isomorphism between the torsion scattering diagram and Bridgeland's stability scattering diagram is underproved. The map I is defined only on the images Im(Phi_A) and Im(Phi-hat_A), not on the ambient groups G(A) and H-hat(A); to speak of an isomorphism of scattering diagrams one must show these images generate the respective groups and that I extends to a group isomorphism compatible with all wall labels and with the inverse-limit structures. The proof merely cites [Bri17, Lemma 6.6], Lemma 7.8 and Proposition 5.4, but it does not verify compatibility with the infinite products/inverse limits that are essential at the A-level. In particular, Bridgeland's group is pro-unipotent with a convergence mechanism, while G(A) is an inverse limit of picture groups without an analogous mechanism; the comparison therefore needs a precise statement of how the two inverse systems are identified. Corollary 8.5 inherits this issue. This is load-bearing for the second half of Theorem 1.1.

minor comments (5)

[§2.1] In the first paragraph, 'Krull–Schimidt' should be 'Krull–Schmidt'.
[§3.2, Proposition 3.6] In the proof, the notation 'rT' is used for the torsion class associated with the uniquely determined torsion-free class F(F_ell) before it is introduced; please define this notation explicitly.
[§6.2, Definition 6.13] The relation is written as 'g_d2 = g_d1 g_d1', where the two occurrences of d1 denote cones of different codimension; this is confusing. Use distinct symbols, for example g_{d_2} = g_{d_1} g_{d^1} with d_1 in D^0 and d^1 in D^1.
[§7.2, Theorem 7.9] In the statement and proof, 'Phi^l_A' should be 'Phi^ell_A' for consistency with the notation D_ell(A) and G_ell(A).
[§4.2, Figure 5] The walls in Figure 5(c)–(e) are labelled by modules such as 1/22 and 11/22, but the labels are not explained in the caption; please add a sentence explaining the notation or refer explicitly to Example 4.22.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the finite-level construction is self-contained, and the inverse-limit step is a technical verification rather than a circular definition.

full rationale

The derivation is not circular. For each fixed ℓ, Theorem 7.9 proves consistency of the finite scattering diagram (D_ℓ(A), Φ_ℓ) using Lemma 7.8 and the explicit morphism ρ from the groupoid G^ss_ℓ(A) to the picture group G_ℓ(A); the wall labels are the semistable subcategories (mod^ss_d A)_ℓ, and the endpoint formula g_γ = ρ([t0,T_{γ(0)}]^{-1}[t0,T_{γ(1)}]) is proved, not assumed. The infinite diagram in Theorem 7.11 is then obtained as the inverse limit of these finite diagrams: g_γ is defined as the compatible tuple (g_γ^ℓ), and for each ℓ only the finitely many walls of D_ℓ(A) contribute to the ℓ-coordinate, so this tuple is the ordered wall-crossing product in the inverse-limit sense. The proof is compressed on this point, but the step is a verification that the inverse-limit element satisfies the scattering-diagram requirement, not a redefinition that makes the conclusion true by fiat. The cited results — [Tre25, Theorem 2.2] on Harder–Narasimhan filtrations for chains of torsion classes and Kaipel's partitioned-fan picture-group framework — are prior published theorems used as tools; they do not assume the existence of the scattering diagram and are not equivalent to the target result. The comparison with Bridgeland in Theorem 8.3 is an explicit isomorphism between the new group-valued diagram and the existing Hall-algebra diagram, established on generators from the common cone complex, not a renaming of a fitted quantity. No parameter is fitted to the claimed output, no uniqueness theorem from the author's own work is invoked to forbid alternatives, and the central finite-level construction has independent content. The only caveat is a minor expository gap in the infinite product verification in Theorem 7.11, which is a correctness or completeness concern, not circularity.

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

The paper introduces no new entities or fitted parameters. It relies on standard module theory, torsion pair theory, and several published theorems, including the author's own prior work and Kaipel's picture group framework. These are external and independently established.

assumptions (6)

standard math Jordan-Holder theorem for finite-length modules over Artin algebras
Used throughout to define length ell and filtration properties of modules and subcategories.
standard math Classical theory of torsion pairs in module categories
The paper relies on the characterization of torsion classes as closed under quotients and extensions, the lattice properties of torsA, and brick labeling from [IRTT15, BCZ19, DIR+23].
domain assumption Harder-Narasimhan filtrations for chains of torsion pairs (Theorem 2.2 of [Tre25])
Cited from the author's prior publication; used to show that path products in the groupoid collapse to the interval element (Proposition 5.8). Load-bearing for consistency.
domain assumption Kaipel's theory of partitioned fans, fan posets, and picture groups ([Kai25])
The paper uses Kaipel's framework to define tau-cluster morphism categories and picture groups for Sigma_ell(A); this is a published external framework.
domain assumption Asai's theorem that TF-equivalence classes are open convex cones covering D(A) ([Asa21, Theorem 2.17])
Used to describe the wall-and-chamber structure of mod A in terms of TF-equivalence classes; the paper cites this as a direct consequence.
domain assumption Bridgeland's construction of stability scattering diagrams for CQ/I, including Lemma 6.6 ([Bri17])
Used in Theorem 8.3 to identify the torsion scattering diagram with Bridgeland's stability scattering diagram.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Scattering diagrams for Artin algebras." pith.science (2026). https://pith.science/paper/GKEOHQ57

@misc{pith2026260804233,
  author       = {Pith},
  title        = {Pith review of: Scattering diagrams for Artin algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GKEOHQ57}},
  note         = {Machine review of arXiv:2608.04233}
}

abstract

For an arbitrary Artin algebra $A$, we construct a minimal and consistent scattering diagram by approximating its module category $\mathrm{mod}\,A$ using the subcategories $(\mathrm{mod}\,A)_\ell$ of modules of length at most $\ell \in \mathbb{N}$. We prove that each subcategory $(\operatorname{mod}A)_\ell$ possesses a well-behaved lattice of torsion classes, a finite wall-and-chamber structure $\mathfrak{D}_\ell(A)$ and an associated picture group $G_\ell(A)$ with a natural categorical interpretation. Using these properties, we build a finite, minimal, consistent scattering diagram for each $\ell \in \mathbb{N}$. Passing to the inverse limit, we establish the existence of a minimal consistent scattering diagram for $A$. In particular, when $A$ is a finite-dimensional algebra over $\mathbb{C}$, our inverse limit construction is canonically isomorphic to Bridgeland's stability scattering diagram.

Figures

Figures reproduced from arXiv: 2608.04233 by the authors.

**Figure 1.** tors A as the inverse limit of the lattices torsℓ A for ℓ P N. Let TpTℓq P tors A be the unique minimal torsion class containing Tℓ and let T P φ ´1 8,lpTℓq. Then Tℓ Ă TpTℓqℓ “ TpTℓq X pmod Aqℓ. Also, TpTℓq Ă T because Tℓ Ă T . This implies TpTℓqℓ Ă Tℓ. Therefore, TpTℓq P φ ´1 8,ℓpTℓq and it is the unique minimal element in φ ´1 8,lpTℓq. Using similar arguments, one can show the existence of a unique minimal torsion… view at source ↗

**Figure 2.** The Hasse quiver of tors2 A Proof. We show this for S ´ Tℓ , since the proof for S ` Tℓ follows from the duality between torsℓ A and tfreeℓ A. Let B P S ´ Tℓ . Then there is T B ℓ P torsℓ A maximally contained in Tℓ such that B is a minimal extending module for T B ℓ . If S ´ Tℓ “ tBu we are done. Otherwise, let B1 P S ´ Tℓ different from B. If B1 P TℓzT B ℓ we have that B – B1 by [BCZ19, Lemma 2.5], which is a cont… view at source ↗

**Figure 3.** DpAq as the inverse limit of the cone complexes DℓpAq for ℓ P N. Similarly, let F P Fℓ be nonzero. Then xw,rFsy “ xv,rFsy ´ ϵ 2ℓ xp1, 1, . . . , 1q,rFsy “ xv,rFsy ´ ϵ 2ℓ lgpFq ď 0 ´ ϵ 2ℓ lgpFq ă 0. The above inequalities imply the following inclusions. Tℓ Ă pTwqℓ Ă pT wqℓ Fℓ Ă pFwqℓ Ă pF wqℓ Moreover, we know that pTℓ, Fℓq and ppTwqℓ,pF wqℓq are torsion pairs in pmod Aqℓ, implying pF wqℓ Ă Fℓ. Then Tℓ “ pTwqℓ “ pT w… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Wall-and-chamber structures for A3 with a zig-zag orientation [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]

**Figure 5.** Figure 5: The wall-and-chamber structures DℓpAq of the Kronecker algebra A for different values of ℓ [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]

**Figure 6.** Figure 6: The picture group of an Artin algebra. The family of groups tGℓpAq | ℓ P Nu, together with the compatible surjective homomorphism πm,ℓ : GmpAq Ñ GℓpAq forms an inverse system. This allows us to define a picture group for arbitrary Artin algebras. Definition 6.18. Let …

Discussion (0). Continue with ORCID to comment.

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

$g$-vector fans and picture categories for 0-Auslander extriangulated categories
math.RT 2026-08 accept novelty 7.0 of 10

g-vector fans and picture categories are extended to 0-Auslander extriangulated categories, unifying tau-cluster morphism categories and earlier picture categories.

Reference graph

Works this paper leans on

11 extracted references · 5 canonical work pages · cited by 1 Pith paper

[1]

Adachi, O

[AIR14] T. Adachi, O. Iyama, and I. Reiten,τ-tilting theory., Compositio Mathematica150 (2014), no. 3, 415–452. [AHK18] K. Adiprasito, J. Huh, and E. Katz,Hodge theory for combinatorial geometries, Annals of Mathematics. Second Series188 (2018), no. 2, 381–452. [AP20] O. Amini and M. Piquerez,Hodge theory for tropical varieties, 2020.arXiv:2007.07826 [Ami...

arXiv 2014
[3]

1, 679–722, DOI 10.1007/s10468-023-10233-x.Zbl 1546.13038 [Mou19] L

I: Scattering diagrams, Algebras and Representation Theory27 (2024), no. 1, 679–722, DOI 10.1007/s10468-023-10233-x.Zbl 1546.13038 [Mou19] L. Mou,Scattering diagrams of quivers with potentials and mutations, 2019.arXiv:1910.13714 [Mou24] , Scattering diagrams for generalized cluster algebras, Algebra & Number Theory18 (2024), no. 12, 2179–2246. [Nak23] T....

work page arXiv 2024
[26]

Luis A. Santaló

Id/No 107817. [Tre19] , On sign-coherence ofc-vectors, Journal of Pure and Applied Algebra223 (2019), no. 6, 2382–2400. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática. Buenos Aires, Argentina. CONICET - Universidad de Buenos Aires. Instituto de Investigaciones Matemáticas “Luis A. Santaló” (IMAS). Buenos ...

work page 2019
[27]

Id/No 108249. [ITW] K. Igusa, G. Todorov, and J. Weyman,Picture groups of finite type and cohomology in type An, arXiv:1609.02636. [IT09] C. Ingalls and H. Thomas,Noncrossing partitions and representations of quivers., Compo- sitio Mathematica 145 (2009), no. 6, 1533–1562. [Iya07] O. Iyama, Higher-dimensional Auslander-Reiten theory on maximal orthogonal ...

arXiv 2009
[31]

[BH23] A

Id/No 106746. [BH23] A. B. Buan and E. J. Hanson,τ-perpendicular wide subcategories, Nagoya Mathematical Journal 252 (2023), 959–984. [BM21] A. B. Buan and B. R. Marsh,A category of wide subcategories, IMRN. International Math- ematics Research Notices2021 (2021), no. 13, 10278–10338. [BMR`06] A. B. Buan, B. R. Marsh, M. Reineke, I. Reiten, and G. Todorov...

work page arXiv 2023
[41]

$\tau$-cluster morphism categories of factor algebras

Id/No e70071. [Kai24] , τ-cluster morphism categories of factor algebras, 2024.arXiv:2408.03818 [KT23] M. Kaipel and H. Treffinger,Wall-and-chamber structures for finite-dimensional algebras and τ-tilting theory, 2023.arXiv:2302.12699 [Kel11] B. Keller,On cluster theory and quantum dilogarithm identities, Representations of algebras and related topics. Pr...

work page Pith review arXiv 1994
[45]

[AI24] S

Id/No 107615. [AI24] S. Asai and O. Iyama, M-TF equivalences on the real Grothendieck groups , 2024.arXiv:2404.13232 [ASS06] I. Assem, D. Simson, and A. Skowroński,Elements of the representation theory of associative algebras. Vol. 1: Techniques of representation theory., London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge,

arXiv 2024
[110]

Garver and T

[GM19] A. Garver and T. McConville,Lattice properties of oriented exchange graphs and torsion classes, Algebras and Representation Theory22 (2019), no. 1, 43–78. [GLS05] C. Geiss, B. Leclerc, and J. Schröer,Semicanonical bases and preprojective algebras, Annales Scientifiques de l’École Normale Supérieure. Quatrième Série38 (2005), no. 2, 193–253. [GNP23]...

arXiv 2019

Show all 11 references

[2006]

Barnard, A

[BCZ19] E. Barnard, A. Carroll, and S. Zhu,Minimal inclusions of torsion classes, Algebraic Com- binatorics 2 (2019), no. 5, 879–901. [BKT14] P. Baumann, J. Kamnitzer, and P. Tingley,Affine Mirković-Vilonen polytopes, Publications Mathématiques 120 (2014), 113–205. SCATTERING ...

2019
[2011]

Gross, P

[GHKK18] M. Gross, P. Hacking, S. Keel, and M. Kontsevich,Canonical bases for cluster algebras, Journal of the American Mathematical Society31 (2018), no. 2, 497–608. [HI21] E. J. Hanson and K. Igusa,τ-cluster morphism categories and picture groups, Communica- tions in Algebra...

2018 arXiv
[2023]

Nakaoka and Y

[NP19] H. Nakaoka and Y. Palu,Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cahiers de Topologie et Géométrie Différentielle Catégoriques60 (2019), no. 2, 117–193. [Pfe25] C. Pfeifer, A generic classification of locally free representations of aff...

2019 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Adachi, O

[AIR14] T. Adachi, O. Iyama, and I. Reiten,τ-tilting theory., Compositio Mathematica150 (2014), no. 3, 415–452. [AHK18] K. Adiprasito, J. Huh, and E. Katz,Hodge theory for combinatorial geometries, Annals of Mathematics. Second Series188 (2018), no. 2, 381–452. [AP20] O. Amini and M. Piquerez,Hodge theory for tropical varieties, 2020.arXiv:2007.07826 [Ami...

arXiv 2014

[2] [3]

1, 679–722, DOI 10.1007/s10468-023-10233-x.Zbl 1546.13038 [Mou19] L

I: Scattering diagrams, Algebras and Representation Theory27 (2024), no. 1, 679–722, DOI 10.1007/s10468-023-10233-x.Zbl 1546.13038 [Mou19] L. Mou,Scattering diagrams of quivers with potentials and mutations, 2019.arXiv:1910.13714 [Mou24] , Scattering diagrams for generalized cluster algebras, Algebra & Number Theory18 (2024), no. 12, 2179–2246. [Nak23] T....

work page arXiv 2024

[3] [26]

Luis A. Santaló

Id/No 107817. [Tre19] , On sign-coherence ofc-vectors, Journal of Pure and Applied Algebra223 (2019), no. 6, 2382–2400. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática. Buenos Aires, Argentina. CONICET - Universidad de Buenos Aires. Instituto de Investigaciones Matemáticas “Luis A. Santaló” (IMAS). Buenos ...

work page 2019

[4] [27]

Id/No 108249. [ITW] K. Igusa, G. Todorov, and J. Weyman,Picture groups of finite type and cohomology in type An, arXiv:1609.02636. [IT09] C. Ingalls and H. Thomas,Noncrossing partitions and representations of quivers., Compo- sitio Mathematica 145 (2009), no. 6, 1533–1562. [Iya07] O. Iyama, Higher-dimensional Auslander-Reiten theory on maximal orthogonal ...

arXiv 2009

[5] [31]

[BH23] A

Id/No 106746. [BH23] A. B. Buan and E. J. Hanson,τ-perpendicular wide subcategories, Nagoya Mathematical Journal 252 (2023), 959–984. [BM21] A. B. Buan and B. R. Marsh,A category of wide subcategories, IMRN. International Math- ematics Research Notices2021 (2021), no. 13, 10278–10338. [BMR`06] A. B. Buan, B. R. Marsh, M. Reineke, I. Reiten, and G. Todorov...

work page arXiv 2023

[6] [41]

$\tau$-cluster morphism categories of factor algebras

Id/No e70071. [Kai24] , τ-cluster morphism categories of factor algebras, 2024.arXiv:2408.03818 [KT23] M. Kaipel and H. Treffinger,Wall-and-chamber structures for finite-dimensional algebras and τ-tilting theory, 2023.arXiv:2302.12699 [Kel11] B. Keller,On cluster theory and quantum dilogarithm identities, Representations of algebras and related topics. Pr...

work page Pith review arXiv 1994

[7] [45]

[AI24] S

Id/No 107615. [AI24] S. Asai and O. Iyama, M-TF equivalences on the real Grothendieck groups , 2024.arXiv:2404.13232 [ASS06] I. Assem, D. Simson, and A. Skowroński,Elements of the representation theory of associative algebras. Vol. 1: Techniques of representation theory., London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge,

arXiv 2024

[8] [110]

Garver and T

[GM19] A. Garver and T. McConville,Lattice properties of oriented exchange graphs and torsion classes, Algebras and Representation Theory22 (2019), no. 1, 43–78. [GLS05] C. Geiss, B. Leclerc, and J. Schröer,Semicanonical bases and preprojective algebras, Annales Scientifiques de l’École Normale Supérieure. Quatrième Série38 (2005), no. 2, 193–253. [GNP23]...

arXiv 2019