Generalised tree modules: Hom-sets and indecomposability

Amit Kuber; Annoy Sengupta

REVIEW 3 major objections 4 minor 1 cited by

Generalised tree modules: Hom-sets and indecomposability

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read For ghost-free pairs of generalised tree modules, every Hom-space is spanned by finitely many generalised graph maps.

desk verdict A genuine extension of Crawley-Boevey's Hom basis to a broader class, but the central theorem currently depends on an unproved combinatorial lemma that needs to be either proved or made explicit before the result is established. read the letter →

arxiv 2504.18996 v4 pith:3ILMIZCY submitted 2025-04-26 math.RT

Annoy Sengupta , Amit Kuber This is my paper · ORCID

classification math.RT MSC 16G20

keywords zero-relationalgebrageneralisedtreemodulegraphmapHom-spaceindecomposabilityDynkinquivertypeD2-coveringnetwork

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

For a zero-relation algebra, presented as a quiver in which certain paths are declared zero, this paper studies modules built from a tree by pushing forward along a map that sends the tree into the quiver. The classical tree-module condition requires distinct arrows with a common source or target to be sent to distinct arrows; the paper drops this condition and shows that, as long as a certain ghost-free hypothesis holds, the space of homomorphisms between two such generalised tree modules is spanned by finitely many explicitly constructed combinatorial maps called generalised graph maps. These maps are signed subnetworks of a 2-covering pullback network, and they generalise the graph maps that form a basis in the classical tree-module case, where they may become linearly dependent. The paper applies this spanning result to give checkable sufficient conditions for indecomposability and decomposability of generalised tree modules, and it constructs explicit generalised tree modules realising every indecomposable module over a Dynkin quiver of type D.

What carries the argument

The load-bearing object is the 2-covering network $N^{[2]}$ associated with the pair of generalised tree modules: a two-sheeted cover of the pullback quiver of the two defining trees, in which each pullback vertex $(n,m)$ appears in two signed copies $(n,m,1)$ and $(n,m,-1)$, and each undirected edge that records a possible sign flip connects $(n,m,j)$ to $(n',m',-j)$. A subnetwork of $N^{[2]}$ is complete when it satisfies the existence conditions needed for the associated linear map to commute with the quiver action, and $R^{[2]}$-free when it satisfies the corresponding uniqueness conditions, expressed by forbidding certain length-two traversals inside triangles of the pullback network. A generalised graph map is a connected, involution-free, complete, $R^{[2]}$-free subnetwork; the machinery of the paper shows that the support of any homomorphism is complete, that completeness can be made $R^{[2]}$-free without changing the vertex set, and that under the ghost-free hypothesis a non-zero homomorphism always contains a generalised graph map, enabling an induction on support size that proves Theorem A.

What would settle it

Check the unproved Proposition 3.6 by exhaustive search over small trees: find a complete subnetwork of a 2-covering network that is $R^{[2]}$-free yet admits, at some vertex $(n,m,j)$, two distinct incoming arrows $(n',m',j)$ and $(n'',m'',j)$ both satisfying Condition (1a) of Definition 3.2. Such a configuration would falsify the uniqueness characterisation and break the proof of Proposition 3.9, hence of Theorem A.

Watch

Extended reading notes

Core claim

The central claim is Theorem A: for a ghost-free pair of generalised tree modules $M_1,M_2$, every homomorphism $H:M_1\to M_2$ is a finite linear combination of homomorphisms $H_G$ attached to generalised graph maps $G$. A generalised graph map is a non-empty, connected, involution-free subnetwork of the 2-covering network $N^{[2]}$ built from the pullback of the two defining trees, satisfying two conditions called completeness and $R^{[2]}$-freeness; the attached linear map sends each basis vector $v_n$ to the signed sum of basis vectors $w_m$ appearing as vertices $(n,m,j)$ with sign $j$ in the subnetwork. The proof takes the support of $H$, carves it into a complete $R^{[2]}$-free subnetwork with the same vertex set, extracts a generalised graph map whose support lies inside the support of $H$ under the ghost-free hypothesis, and then subtracts a scalar multiple to shrink the support, giving an induction. The ghost-free hypothesis rules out non-empty, connected, involution-invariant complete $R^{[2]}$-free subnetworks, which are algebraically invisible because they define the zero homomorphism. For ordinary tree modules the construction recovers the classical graph maps, and the generating set is in fact a basis.

Load-bearing premise

The load-bearing premise is that the pair of modules is ghost-free: the signed two-cover network contains no non-empty connected sign-invariant complete and $R^{[2]}$-free subnetwork, since the key lemma that extracts a generalised graph map from a homomorphism's support breaks down exactly when every connected component of the carved network is sign-invariant.

Editorial extensions

If this is right

Hom-spaces between ghost-free pairs of generalised tree modules have a finite, explicitly described generating set: the homomorphisms attached to the generalised graph maps, which can be read off directly from the 2-covering network.
A generalised tree module $M$ satisfying the ghost-free hypothesis is indecomposable whenever no generalised graph map from $M$ to itself contains a vertex $(n_1,n_2,j)$ with $n_1\neq n_2$ coming from a subtree of the form $n_1\leftarrow n\rightarrow n_2$ or $n_1\rightarrow n\leftarrow n_2$ with equal arrow images, and no two generalised graph maps contain $(n_1,n_2,j)$ and $(n_2,n_1,j')$ respecti
For trees shaped like a central vertex with several branches satisfying a uniqueness condition on arrows, the existence of a generalised graph map containing $(n_1,n_2)$ in its support forces $M$ to decompose, and the paper conjectures that this decomposability criterion holds for every generalised tree module.
Every indecomposable module over a Dynkin quiver of type $D$ is isomorphic to a generalised tree module; the paper constructs an explicit tree for each dimension vector and shows the correct tree choice is indecomposable while wrong choices give decomposable modules.

Reading between the lines

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

If the answer to the paper's Question 5.8 is affirmative, the ghost-free condition is an artefact of the support-carving induction rather than a genuine obstruction; a direct test is to take the ghost of Example 5.4 and check whether the zero homomorphism it defines can be written as a combination of generalised graph maps with strictly smaller supports.
Because the generalised graph maps are finite in number but not linearly independent in general, the dimension of the Hom-space can be computed by evaluating the finite set of homomorphisms $H_G$ on a basis and taking the rank, yielding a purely combinatorial dimension formula for ghost-free pairs.
The explicit construction for type $D$ suggests a pattern for other Dynkin quivers: choose the defining tree so that completeness conditions fail at exactly the vertices that would create off-diagonal support, then apply Theorem B; since every indecomposable over a Dynkin quiver is exceptional, such explicit trees should exist, and finding them would turn the known existence theorem for exceptiona
In characteristic 2 the 2-covering network collapses to the pullback network, so signs disappear and the distinction between graph maps and generalised graph maps vanishes; any statement that relies on the sign structure, including the ghost phenomenon, needs a separate formulation in characteristic 2.

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 is worth reading. It takes Crawley-Boevey's theorem on Hom-sets between tree modules and tries to extend it to generalized tree modules, where the tree condition is relaxed and you can get sign-flip homomorphisms and ghosts. The main new content is the network formalism—pullback networks, 2-covering networks, generalized graph maps—and the claim that these give a finite generating set for the Hom-space when the ghost-free condition holds. The indecomposability criterion in Theorem B is a natural application, and the Dynkin D section is honest: it gives explicit constructions and correctly notes that the classification statement already follows from Ringel. There is real work here, and the exposition is mostly careful. The soft spot is exactly where the reader put it. Proposition 3.6 is stated without proof and is load-bearing. It asserts that a complete subnetwork being R[2]-free is equivalent to a set of "exactly one of (1a)/(1b), at most one arrow, at most one edge" uniqueness conditions at every vertex. Proposition 3.9 uses that uniqueness in the coefficient-counting argument to show the associated linear map is a homomorphism. If Proposition 3.6 is false, or even just not proved, then the central construction may not produce homomorphisms, and Theorem A collapses. The stress-test note is right: Proposition 4.5 and Lemma 5.7 carve out complete R[2]-free subnetworks, but they do not bypass this lemma. The paper asks in Question 5.8 whether the ghost-free hypothesis can be dropped, and that is a real limitation, not a flaw—but Proposition 3.6 is a gap, not a limitation. This is a case where the overall idea is plausible and the paper is honest about its weak points, but the main theorem is not yet fully supported. The proof of Proposition 3.6 might be a routine verification; if so, the authors just forgot to include it. But as it stands, the referee needs to see it. The paper deserves a serious referee. It is written by people who know the literature, the problem is a natural one, and the combinatorial machinery is new even if the final payoff is still conditional. I would not desk-reject it. I would send it to a representation theorist and ask specifically: prove or disprove Proposition 3.6, and make the carving arguments in Section 4 precise. If Proposition 3.6 survives, this is likely a solid paper; if it fails, Theorem A may fail too. Either way, the authors have done enough to earn engagement.

Referee Report

3 major / 4 minor

Summary. The paper studies generalised tree modules over zero-relation algebras, obtained by dropping the injectivity condition in Crawley-Boevey's definition of tree module. The main result (Theorem A) claims that, when char(K) differs from 2 and the pair (M1,M2) is 'ghost-free', the space Hom(M1,M2) is spanned by homomorphisms attached to explicit finite combinatorial objects called generalised graph maps, defined as non-empty connected involution-free complete R[2]-free subnetworks of the 2-covering network N[2]. The proof proceeds by induction on the support of a homomorphism: Proposition 5.5 shows the support is complete, Proposition 4.5 carves out a complete R[2]-free subnetwork with the same vertices, Proposition 3.9 turns such subnetworks into homomorphisms, and Lemma 5.7 extracts a generalised graph map under the ghost-free hypothesis. Theorem B gives a sufficient condition for indecomposability of a generalised tree module, and Section 7 applies the results to construct indecomposable generalised tree modules for Dynkin quivers of type D, complementing Ringel's theorem that exceptional modules are generalised tree modules.

Significance. If the main theorem is correct, it is a meaningful extension of Crawley-Boevey's graph-map basis to a setting where basis elements may collide, giving a finite explicit generating set for Hom-spaces between generalised tree modules. The paper also provides a checkable indecomposability criterion and an explicit construction for type D Dynkin quivers, together with instructive examples of sign-flip homomorphisms and ghosts. The authors are transparent about the ghost-free scope and pose the natural open question whether it can be removed. However, the proof as written is not complete: Proposition 3.6 is stated without proof and is load-bearing for Proposition 3.9, while the carving arguments in Proposition 4.5 and Lemma 5.7 are underdetailed. The result is plausible, but the manuscript needs substantial revision before the central claim is established.

major comments (3)

[Section 3, Proposition 3.6] Proposition 3.6 is stated with the words 'we mention without proof', yet it is used essentially in the proof of Proposition 3.9. Specifically, it supplies the uniqueness of arrows and edges needed for the partition into I2 and J2, for the existence and uniqueness of the arrow in Condition (2a), and for the injectivity of the maps F and F' that yield the equality of the sums over I3 and J3. The asserted equivalence between R[2]-freeness and the local exactly-one conditions is not a formal consequence of the definitions as far as the text shows; one must prove, for example, that two arrows satisfying Condition (1a) for the same incoming arrow of T1 would produce a traversal in R[2], and conversely that absence of all R[2]-traversals forces the exactly-one conditions at every vertex. A complete proof, or a precise reference, is required before Theorem A can be considered established.
[Section 4, Proposition 4.5] The proof of Proposition 4.5 does not, as written, establish the claim. The construction partitions the vertices of each R[2]-system and chooses perfect matchings locally, but the text never verifies that the union of these choices is R[2]-free when hexagons overlap in antipodal vertices, a situation explicitly allowed by Remark 4.4. The completeness of the resulting subnetwork is also asserted rather than proved: the final sentence claims that for every removed link alpha there are links beta and gamma in M' with beta alpha and alpha gamma in R[2], but no construction or argument is supplied. Since Corollary 5.6 and hence the induction in Theorem A depend directly on this proposition, this is a load-bearing gap.
[Section 5, Lemma 5.7] The proof of Lemma 5.7 is underdetailed at the two points that matter for the induction in Theorem A. First, the modification of M to M' is described by cases, but the argument does not show that the modified links do not create new blocked traversals together with links from adjacent R[2]-systems. Second, the completeness of G is asserted by saying that any link between two vertices of G0 present in M is already present in G; this is not immediate because M' was explicitly allowed to delete links from M, and a deleted link between two vertices of G0 would not be present in G. A direct verification of the completeness conditions at each vertex of G for every arrow of T1 and T2 is needed. As the inductive step of Theorem A, this lemma is load-bearing.

minor comments (4)

[Definition 3.2(2b)] In Condition (2b), the displayed edge should be between (n,m,j) and (n,m'',-j); the printed version with (n'',m,-j) is inconsistent with the preceding vertex (n,m'',-j) and with the intended T2-sign-flip situation.
[Proposition 3.9 proof] In the definition of J2, the set is written with set-minus I1; it should be set-minus J1, since J1 was already defined as the set of indices with zero coefficient.
[Lemma 5.7] The line 'there exists (n0,m0,j0) in M'_0' refers to M' before M' has been defined; it should refer to the connected component M0 of the previously constructed subnetwork.
[Proposition 3.9 proof] In the sentence discussing the coefficient of w_m, the subscript 0 is dropped; the intended coefficient is that of w_{m0}.

Circularity Check

0 steps flagged · score 1.0 of 10

No substantive circularity; the only self-citation is motivational and the central spanning theorem is proved from definitions.

full rationale

The paper's central claim Theorem A is not circular: generalised graph maps are defined directly as non-empty connected involution-free complete R[2]-free subnetworks of N[2] (Definition 3.10), and Proposition 3.9 independently proves that the associated linear maps are homomorphisms from the completeness and uniqueness conditions. The proof of Theorem A then shows by induction that every homomorphism is a finite linear combination of such maps; it uses Proposition 5.5 (support is complete), Proposition 4.5 (carving to a complete R[2]-free subnetwork), and Lemma 5.7 (extracting a generalised graph map from the support), none of which assumes the conclusion. No fitted parameters or definitions of the form 'X is defined as the object that makes the claimed conclusion true' occur. The only self-citation is [SK24] in Section 2, which is used solely as motivation for defining traversals; Definition 2.2 is self-contained and no proof relies on [SK24]. The ghost-free condition is an explicit hypothesis whose necessity is left open in Question 5.8, so it is a scope limitation rather than a circular assumption. I do flag Proposition 3.6 as stated without proof and load-bearing for Proposition 3.9 and hence for Theorem A; this is an omitted proof, not circularity, and therefore does not raise the circularity score. The Dynkin type D application is checked against explicit constructions and the quoted Ringel theorem, not derived from itself.

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

The paper introduces new combinatorial definitions such as generalised tree modules, pullback networks, 2-covering networks, generalised graph maps, and ghosts. These are mathematical definitions, not postulated entities with independent empirical evidence, so the invented-entities list is empty. The load-bearing assumptions are the characteristic-2 exclusion, the ghost-free hypothesis, and the cited background theorems.

assumptions (6)

domain assumption The field K has characteristic not equal to 2.
Assumed in the abstract and used in Definition 2.11 and Remark 2.12 so that the signs +1 and -1 remain distinct; over characteristic 2 the 2-covering network collapses to N[1].
standard math Gabriel's theorem: tree modules over a zero-relation algebra are indecomposable.
Cited as Theorem 1.2 and used as background; also reproved in Corollary 6.3 via Theorem B.
standard math Crawley-Boevey's basis theorem for Hom-sets of tree modules.
Cited as Theorem 1.4; it motivates the definition of graph maps and frames the generalisation to generalised graph maps.
standard math Assem-Simson-Skowronski criterion: a finite-dimensional module is indecomposable iff its endomorphism algebra has only two idempotents.
Used as Proposition 1.8 in the proofs of Theorem B and Theorem 6.4.
standard math Ringel's theorem: all exceptional modules over the path algebra of a finite quiver are generalised tree modules.
Used in Section 7 to conclude that all indecomposable modules over Dynkin type D quivers are generalised tree modules.
ad hoc to paper The pair (M1,M2) is ghost-free.
Introduced in Definition 5.1; needed for Lemma 5.7 and hence for Theorem A. The paper asks in Question 5.8 whether this condition can be dropped.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Generalised tree modules: Hom-sets and indecomposability." pith.science (2026). https://pith.science/paper/3ILMIZCY

@misc{pith2026250418996,
  author       = {Pith},
  title        = {Pith review of: Generalised tree modules: Hom-sets and indecomposability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3ILMIZCY}},
  note         = {Machine review of arXiv:2504.18996}
}

abstract

For a zero-relation algebra over a field $\mathcal K$, Crawley-Boevey introduced the concept of a tree module and provided a combinatorial description of a basis for the space of homomorphisms between two tree modules--the basis elements are called graph maps. The indecomposability of tree modules is essentially due to Gabriel. We relax a condition in the definition of a tree module to define generalised tree modules and when $\mathrm{char}(\mathcal K)\neq2$, under a certain condition, provide a combinatorial description of a finite generating set for the space of homomorphisms between two such modules--we call the generators generalised graph maps. As an application, we provide a sufficient condition for the (in)decomposability of certain generalised tree modules. We also show that all indecomposable modules over a Dynkin quiver of type $\mathbf D$ are isomorphic to generalised tree modules--this result also follows from a theorem of Ringel which states that all exceptional modules over the path algebra $\mathcal KQ$ of a finite quiver $Q$ are generalised tree modules.

Figures

Figures reproduced from arXiv: 2504.18996 by the authors.

**Figure 1.** Two trees T 1 (in the left) and T 2 (in the right) The assignment v1 7→ 0, v2 7→ w4, v3 7→ −w4 describes a “sign-flip” homomorphism as the only element in a basis of HomKA2 (M1, M2). ♢ Our goal is to find a finite generating set for HomΛ(M1, M2) by choosing generators as certain elements of P0 × {−1, 0, 1}. Examples like the above motivate us to record sign-flips as (undirected) edges between some vertices of P–we c… view at source ↗

**Figure 2.** A network N used in Example 2.4 Example 2.4. Consider the network N in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Pullback diagram used in the definition of pullback network Explicitly, the quiver P := (P0,P1, s, t) is defined by P0 := {(n, m) ∈ T 1 0 × T 2 0 | F1(n) = F2(m)}; and P1 := {(n, m) (a,b) −−−→ (n ′ , m′ ) | (n a−→ n ′ ) ∈ T 1 1 ,(m b −→ m′ ) ∈ T 2 1 , F1(a) = F2(b)}. There is a quiver morphism F : P → Q defined by F((x, y)) := F1(x) = F2(y) for (x, y) ∈ P0 ∪ P1. Recall from Example 1.9 that there is a basis homomorp… view at source ↗

Figures from the paper (17 more)

**Figure 4.** Figure 4: Two situations showing the existence of an edge in the pullback network, where portions of T 1 , T 2 and N [1] are being shown respectively from left to right. We mention below some observations about N [1]. Remark 2.5. Since T 1 and T 2 are trees, there is at most one…

**Figure 5.** Figure 5: Four types of triangles We will see now that these triangles have an interesting structure, a useful result which will be used in § 4. Proposition 2.9. Let V1 := {(n1, m1),(n2, m2),(n3, m3)}, V2 := {(n1, m1),(n2, m2),(n4, m4)} be subsets of N [1]0 such that ⟨V1⟩,⟨V2⟩ ∈…

**Figure 6.** Figure 6: The structure of triangles used in the proof of Proposition 2.9 m m′ m′′ m′′′ m a b c d [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: Subtree of T 2 used in Case II of Proposition 2.9 [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 8.** Figure 8: A triangle in N [1] and the corresponding hexagon in N [2] 3. Generalised graph maps If M1 = F1λ(VT1 ) and M2 = F2λ(VT2 ) are tree modules, recall from Theorem 1.4 the description of a graph map (T fac, Tim, Φ) from M1 to M2. The quiver isomorphism Φ can be treated as …

**Figure 9.** Figure 9: Two trees T 1 and T 2 used in Example 3.1 (1, 4, 1) (2, 5, 1) (3, 5, 1) (3, 5, −1) (2, 5, −1) (1, 4, −1) (a,c,1) (b,c,1) (b,c,−1) (a,c,−1) [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 11.** Figure 11: A perfect matching (in red) for an R[2]-system used in the proof of Proposition 4.5 5. Generalised graph maps span the Hom-set Given a homomorphism H : M1 → M2, we aim at expressing it as a finite K-linear combination of homomorphisms HGi corresponding to generalised…

**Figure 12.** Figure 12: Trees T 1 and T 2 used in Example 5.4 [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]

**Figure 13.** Figure 13: The subnetwork N [1] in Example 5.4 such that π −1 (N [1]) = N [2] contains a ghost (1, 6, 1) (1, 8, 1) (4, 5, 1) (4, 7, 1) (4, 9, 1) (2, 6, 1) (2, 8, 1) (2, 8, −1) (2, 6, −1) (4, 9, −1) (4, 7, −1) (4, 5, −1) (1, 8, −1) (1, 6, −1) (a,e,1) (a,d,1) (a,f,1) (a,g,1) (b,d,…

**Figure 14.** Figure 14: A ghost (shown in black) in N [2] used in Example 5.4 We begin our journey to Theorem A by proving an important property of the support of a homomorphism. Proposition 5.5. Let H : M1 → M2 be a homomorphism. Then M := ⟨π −1 (supp(H))⟩ is complete. Proof. Let M =: (M0,M…

**Figure 15.** Figure 15: Tree T used in Theorem 6.4(A) Proof. (A) In view of Proposition 1.8, it is sufficient to produce a non-trivial idempotent homomorphism I : M → M. By the hypotheses, we have a generalised graph map G and distinct n1, n2 ∈ {1, · · · , k} such that (n1, n2) ∈ π(G0). In v…

**Figure 16.** Figure 16: Dynkin diagram Dn Consider a Dynkin quiver Q = (Q0, Q1, s, t) of type Dn (see [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]

**Figure 17.** Figure 17: Underlying undirected trees of the generalised tree modules over Dynkin quivers of type Dn, with F(a) = F(aj ) = a, F(b) = F(bj ) = b and F(c) = F(cj ) = c for j ∈ {1, 2} Since the arguments for the (in)decomposability of the generalised tree modules sketched in [PIT…

**Figure 18.** Figure 18: A particular Dynkin quiver Q of type Dn If T is as in [PITH_FULL_IMAGE:figures/full_fig_p020_18.png]

**Figure 19.** Figure 19: Sketch of a tree T of type described in [PITH_FULL_IMAGE:figures/full_fig_p021_19.png]

**Figure 20.** Figure 20: Pullback network associated with the pair (M, M) There is only one subtree of T (upto relabelling) of the form n1 a ′′ ←− n b ′′ −→ n2 or n1 a ′′ −→ n b ′′ ←− n2 with n1 ̸= n2 and satisfying F(a ′′) = F(b ′′), namely (n − 2)′ ←− b1 (n − 1) b2 −→ (n − 2)′′. On the one …

**Figure 21.** Figure 21: Sketch of a tree T ′ of type described in [PITH_FULL_IMAGE:figures/full_fig_p022_21.png]

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

Rooted tree modules
math.RT 2025-08 conditional novelty 6.0 of 10

A rooted tree module over a zero-relation algebra is indecomposable (char K not 2) exactly when the defining tree has no nontrivial idempotent self-map, giving checkable splitting and construction algorithms.

Reference graph

Works this paper leans on

13 extracted references · 7 canonical work pages · cited by 1 Pith paper

[2]

Covering spaces in representation-theory

Klaus Bongartz and Peter Gabriel. Covering spaces in representation-theory. Invent. Math. , 65:331--378, 1982

work page 1982
[1]

Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory

Ibrahim Assem, Daniel Simson, and Andrzej Skowronski. Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory . Cambridge University Press, 2006

2006
[3]

Crawley-Boevey

William W. Crawley-Boevey. Maps between representations of zero-relation algebras. Journal of Algebra , 126(2):259--263, 1989

1989
[4]

Unzerlegbare D arstellungen

Peter Gabriel. Unzerlegbare D arstellungen. I . Manuscripta Math. , 6:71--103; correction, ibid. 6 (1972), 309, 1972

1972
[5]

The universal cover of a representation-finite algebra

Peter Gabriel. The universal cover of a representation-finite algebra. In Representations of algebras ( P uebla, 1980) , volume 903 of Lecture Notes in Math. , pages 68--105. Springer, Berlin-New York, 1981

1980
[6]

Enumeration of mixed graphs

Frank Harary and Ed Palmer. Enumeration of mixed graphs. Proceedings of the American Mathematical Society , 17(3):682--687, 1966

work page 1966
[7]

Rank functions on rooted tree quivers

Ryan Kinser. Rank functions on rooted tree quivers. Duke mathematical journal , 152(1):27 -- 92, 2010

work page 2010
[8]

Reduced representations of rooted trees

Valentin Katter and Nils Mahrt. Reduced representations of rooted trees. Journal of Algebra , 413:41--49, 2014

2014

Show all 13 references

[9]

Combinatorics on words , volume 17

Monsieur Lothaire. Combinatorics on words , volume 17. Cambridge University Press, 1997

1997
[10]

Exceptional modules are tree modules

Claus Michael Ringel. Exceptional modules are tree modules. Linear Algebra and its Applications , 275-276:471--493, 1998

1998
[11]

Tame algebras and integral quadratic forms , volume 1099

Claus M Ringel. Tame algebras and integral quadratic forms , volume 1099. Springer, 2006

2006
[12]

Quiver representations

Ralf Schiffler. Quiver representations . CMS Books in Mathematics. Springer, Cham, 2014

2014
[13]

Characterisation of band bricks over certain string algebras and a variant of perfectly clustering words

Annoy Sengupta and Amit Kuber. Characterisation of band bricks over certain string algebras and a variant of perfectly clustering words. arXiv preprint arXiv:2402.05656 , 2024

2024 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [2]

Covering spaces in representation-theory

Klaus Bongartz and Peter Gabriel. Covering spaces in representation-theory. Invent. Math. , 65:331--378, 1982

work page 1982

[2] [1]

Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory

Ibrahim Assem, Daniel Simson, and Andrzej Skowronski. Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory . Cambridge University Press, 2006

2006

[3] [3]

Crawley-Boevey

William W. Crawley-Boevey. Maps between representations of zero-relation algebras. Journal of Algebra , 126(2):259--263, 1989

1989

[4] [4]

Unzerlegbare D arstellungen

Peter Gabriel. Unzerlegbare D arstellungen. I . Manuscripta Math. , 6:71--103; correction, ibid. 6 (1972), 309, 1972

1972

[5] [5]

The universal cover of a representation-finite algebra

Peter Gabriel. The universal cover of a representation-finite algebra. In Representations of algebras ( P uebla, 1980) , volume 903 of Lecture Notes in Math. , pages 68--105. Springer, Berlin-New York, 1981

1980

[6] [6]

Enumeration of mixed graphs

Frank Harary and Ed Palmer. Enumeration of mixed graphs. Proceedings of the American Mathematical Society , 17(3):682--687, 1966

work page 1966

[7] [7]

Rank functions on rooted tree quivers

Ryan Kinser. Rank functions on rooted tree quivers. Duke mathematical journal , 152(1):27 -- 92, 2010

work page 2010

[8] [8]

Reduced representations of rooted trees

Valentin Katter and Nils Mahrt. Reduced representations of rooted trees. Journal of Algebra , 413:41--49, 2014

2014