Stability for socle-projective categories of type $\mathbb{A}$

Gabriel Bravo Rios; Kostiantyn Iusenko; Robinson-Julian Serna

REVIEW 4 major objections 5 minor 28 references

Stability for socle-projective categories of type $\mathbb{A}$

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that every indecomposable peak P-space over a type A poset is stable—for a weight from the incidence matrix and for an angle from a polygon model.

desk verdict Plausible and interesting results, but the proof of the main bilinear-form stability theorem has a real gap (missing isolated maximal-point cases) and the geometric stability proof is too sketchy; deserves review but needs major revision. read the letter →

arxiv 2501.13578 v1 pith:44XMTYUC submitted 2025-01-23 math.RT

Kostiantyn Iusenko , Gabriel Bravo Rios , Robinson-Julian Serna This is my paper · ORCID

classification math.RT MSC 16G2016G70

keywords stabilitysocle-projectiverepresentationspeakposetsoftypeAbilinearformgeometricmodellinesegmentsAuslander-Reitenquiver

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

The paper extends a stability notion for representations of posets to the wider category of socle-projective representations, which in this setting are called peak P-spaces: systems of finite-dimensional vector spaces indexed by the points of a poset, one subspace for each point inside an ambient space built from the maximal points. The main result is that when the poset has type A, every indecomposable peak P-space is stable for some weight. This is established twice, once with an explicit weight derived from the poset's incidence matrix and once with a stability function that measures the angle of an oriented line segment in a polygon model. The paper also gives a new geometric realization of the whole category as a category of such line segments and shows the two stability constructions agree in explicit form. If the result is right, the stability machinery previously available for one-peak poset representations now covers all type A socle-projective categories, with weights that can be written down.

What carries the argument

The argument rests on four pieces. First, proper peak P-subspaces: in type A every indecomposable peak space has one-dimensional subspaces, and Lemma 2.1 identifies each proper peak P-subspace with a proper subset I of the maximal points in its support, with support I^△ \ (I^c)^△; this turns the stability check into finitely many support computations. Second, the bilinear form b_P(α,β) = α·$C_P^{{-1}}$·$β^{{tr}}$ associated with the poset P through its incidence matrix C_P; the stability weight is the antisymmetrization of this form against the dimension vector of U. Third, the category of sp-segments: inside the polygon of a type A Dynkin quiver, line segments that are not frozen by the added alien arrows and not swept into a certain forbidden set form a subcategory of line segments; the paper constructs a functor from this category to the socle-projective modules and proves it is an equivalence. Fourth, the angle stability function: each sp-segment is an oriented vector in the plane, so its direction defines a central charge and a phase φ(M) = (1/π) arg(Z[M]); the geometric proof shows that every proper subrepresentation decomposes into consecutive blocks whose angle vectors add to a smaller phase, giving stability.

What would settle it

Compute the weight θ(W_X) = b_P(dim U, dim W_X) - b_P(dim W_X, dim U) directly for the sincere peak P-spaces of $S_2^{{(r)}}$ and $S_3^{{(r)}}$ using the incidence matrices displayed in Proposition 2.5, for r = 2 and r = 3. If any proper peak P-subspace W_X of the sincere space has θ(W_X) ≥ 0, Theorem 2.6 is false; a negative value on every W_X would confirm the missing step.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.6: for any poset P of type A, every indecomposable peak P-space U is θ-stable with a weight θ built from the bilinear form of P. Specifically, θ(W) = b_P(dim U, dim W) − b_P(dim W, dim U), where b_P is defined from the incidence matrix of P and its inverse. The proof reduces to sincere objects: every indecomposable is the image under a subposet-induced lift functor of a sincere peak space on a sincere peak-subposet, and stability is preserved by that lift. For the three sincere families of type A posets, the paper checks that every proper peak P-subspace W_X has θ(W_X) < 0, using the fact that such subspaces are indexed by subsets of the maximal points in the support. The geometric half proves Theorem 4.3: in the line-segment model, the oriented segment representing an indecomposable M gives a central charge Z([M]) and a stability function φ(M) = (1/π)arg Z([M]); every indecomposable socle-projective representation is φ-stable because every proper subrepresentation is a direct sum of consecutive blocks whose angle vectors sum to a strictly smaller angle than the original. Section 4 closes the circle by writing the bilinear-form weight as θ = κ(M)w − w(M)κ and giving explicit φ_m stability functions on the sincere families.

Load-bearing premise

The proof that the bilinear-form weight is negative on every proper subspace is written out only for the sincere family $S_1^{{(r)}}$; for the other two sincere families it says the same computation works without displaying it, so the stability theorem for all type A posets rests on those unstated computations being correct.

Editorial extensions

If this is right

In a type A socle-projective category every indecomposable object lies in the stable locus for an explicit weight, so θ-stability is a structural feature rather than a rare condition.
The categorical equivalence identifies the Auslander–Reiten quiver of mod_sp kP with the translation quiver of sp-segments, giving a complete combinatorial description of irreducible morphisms and mesh relations.
Lemma 1.4 turns each θ-stable object into a µ-stable object for every positive slope function κ, so the Harder–Narasimhan and Jordan–Hölder filtrations exist for all peak P-spaces of type A.
The angle-based central charge lands in the strict right half-plane and attaches a real phase to every indecomposable, connecting the additive-category stability with the usual complex-stability language.
The explicit φ_m functions of Theorem 4.4 give a one-parameter family of stability functions on sincere type A posets, all of which certify stability of every indecomposable.

Reading between the lines

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

The reduction to sincere objects plus a classification of sincere posets is a template: if a larger class of posets of finite prinjective type has a sincere-poset classification, the same two proofs would go through whenever the bilinear-form computation and the line-segment model are available.
The angle stability suggests a comparison question the paper does not ask: as the polygon vertices move, the phases of indecomposables vary, so one could look for walls in polygon-coordinate space where two indecomposables exchange order and stability changes, giving a concrete stability-space picture for the category.
A direct symbolic check of the S_2^{(r)} and S_3^{(r)} weight computations, which the paper leaves to 'the same arguments', would settle the only unexhibited step in Theorem 2.6; if a counterexample appeared there, Theorem 4.3 would still provide stability through the geometric route.

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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 has a clear, valuable goal: it extends Futorny–Iusenko stability from one-peak posets to all socle-projective categories of type A, and it adds a new geometric model via sp-segments. If the main theorem is right, it completes a natural story, and the geometric equivalence in Section 3 is a genuine new construction, not just a repackaging of BGMS. The authors position it honestly relative to [7] and [21].

The soft spots are in the proofs, and they are real. The stress-test note is correct. In Proposition 2.5, the reduction to consecutive blocks X ⊆ A does not cover proper peak P-subspaces whose support contains isolated maximal points. Example: P = S1^(3), I = {z1,z3}. Then A = ∅, so the W_X sum is empty and the proof never evaluates θ on this subspace. Direct computation gives θ = −2, so the theorem survives this example, but the proof as written does not establish it. This gap occurs in the fully displayed S1 family, not only in the delegated S2/S3 computations.

I also have trouble with the displayed formulas (2.2)–(2.4). They appear to conflate the dimension vector of the auxiliary W_j with that of the sincere U. Computing directly from the incidence matrix for S1^(3) gives θ(e_z1) = −1, θ(e_z2) = −2, θ(e_x1) = 2, which do not match what the formulas suggest. The final estimates in cases (i)–(iii) seem to use the correct numbers, so the intended argument is probably salvageable, but the text is internally inconsistent.

Section 3 is better. The category of sp-segments and the functor Ω are carefully defined, and the proof of the equivalence, though long, is mostly checkable. The weak spot is Theorem 4.3: the proof is only about a page, and the central angle inequality is asserted rather than shown. The claim that M^I decomposes as a direct sum of indecomposable M^{I_j} needs justification, and the statement that each Φ_n(M^{I_j}) is an sp-segment at some translate needs detail. This is a moderate gap, less serious than the Section 2 issue.

Bottom line: the paper is for representation theorists working on poset stability and geometric models. It deserves a serious referee; the results are plausible and the geometric model is a real contribution. But the proof of the main bilinear-form theorem is not complete, and Theorem 4.3 is too sketchy. I would send it out, with the expectation of a major revision.

Referee Report

4 major / 5 minor

Summary. The paper extends the Futorny–Iusenko stability notion from ordinary poset representations to the category P-spr of peak P-spaces, i.e., socle-projective modules over incidence algebras, and proves that for every poset of type A every indecomposable object is stable. The first proof uses a weight derived from the incidence-algebra bilinear form (Proposition 2.5 and Theorem 2.6). The second proof constructs a geometric model: inspired by Schiffler–Serna and the BGMS polygon model, the authors define a category of sp-segments, prove it is equivalent to the category of socle-projective modules (Theorem 3.22), and then define a stability function from the angles of oriented line segments (Theorem 4.3). A final result (Theorem 4.4) connects the geometric stability function to the bilinear-form weights for sincere type A posets.

Significance. If the proofs are completed, the paper gives a genuine extension of stability theory to socle-projective categories for all type A posets, with two complementary mechanisms: explicit bilinear-form weights and a geometric angle stability function. The categorical equivalence in Theorem 3.22 is an independently interesting structural result, and the paper rightly credits the prior geometric work of Schiffler–Serna and BGMS. The bilinear-form construction is explicit and algorithmic, and the geometric model gives a uniform reason why all indecomposables are stable. The significance is currently conditional, because several load-bearing computations and inequalities are delegated or asserted rather than demonstrated.

major comments (4)

[§2, Proposition 2.5, proof after eq. (2.5)] The case analysis reduces a proper peak P-subspace W = U_{K_I} to a sum over sets X of consecutive points in A = {x_i in min P | z_i, z_{i+1} in I}, and verifies θ(W_X) < 0 for each X. This does not cover proper subspaces supported on maximal points that are not adjacent through A. For P = S_1^(3) and I = {z_1, z_3}, one has A = ∅ and supp W = {z_1, z_3}, so the sum over the W_X is empty and no inequality is checked. The missing case is harmless in this example because θ(e_{z_1}+e_{z_3}) = -2, but the written proof of Theorem 2.6 therefore does not establish strict negativity for every proper peak P-subspace, and the gap already occurs in the fully displayed family S_1^(r). The proof needs an explicit treatment of isolated maximal points of I or a different decomposition that accounts for them.
[§2, Proposition 2.5, eqs. (2.3)–(2.4)] The formulas for the families S_2^(r) and S_3^(r) are delegated: the text states that 'the same arguments can be used' without displaying the analogous computations with the inverse incidence matrices, the endpoint corrections, or the resulting signs. These two families have different boundary conventions (the extra points x_r, x_0, and x_{r+1}), and the subsequent case analysis in items (ii) and (iii) depends on exactly those endpoint terms. Since Theorem 2.6 relies on Proposition 2.5 for all sincere type A posets, the omitted S_2^(r) and S_3^(r) computations are load-bearing and should be written out.
[§4, Theorem 4.3, proof] The proof asserts that after partitioning I into consecutive subsets I_1, ..., I_s, each summand Φ_n(M^{I_j}) has argument below θ(M), and 'hence' the argument of the sum is below θ(M). This implication is not valid without an additional sector argument: vectors whose arguments are individually smaller than θ(M) can sum to a vector whose argument is larger than θ(M). The proof must show that all vectors Φ_n(M^{I_j}) lie in a common half-plane bounded by the line of angle θ(M), or otherwise bound the argument of the sum directly. Because Theorem 4.3 is one of the two central stability theorems, this assertion cannot remain as it is.
[§4, Theorem 4.4 and eqs. (4.2)–(4.3)] The proof claims that for every m ≥ 1 the vectors w+η and κ+η have positive entries. This is false for m = 1: in case (i) of (4.2), the entry of w+η at an interior maximal point z_i with 1 < i < r equals zero, and the entry of κ+η at every minimal point x_j equals zero. Concretely, for P = S_1^(3) and M the simple representation supported at the middle maximal point z_2, one obtains Z_1(M) = 2i, whose argument is π/2, i.e., on the boundary of the half-plane H in Definition 4.1. Thus the central-charge condition of Definition 4.1 fails for m = 1. The statement should be changed to m sufficiently large, or a separate treatment of the boundary cases should be supplied.

minor comments (5)

[General] The manuscript contains numerous typographical errors, including 'objet', 'dence', 'subpace', 'asocciated', 'Auslander-Reiter', and inconsistent hyphenation of 'peak P-space'; the paper should be carefully copyedited.
[§2, Lemma 2.1] The lemma speaks of 'proper subsets' of supp U ∩ max P, which includes the empty subset; the corresponding admissible subspace K = 0 gives the zero subrepresentation, for which θ(U_0) = 0, not < 0. The stability arguments should explicitly restrict to nonempty proper subsets or otherwise exclude the zero subrepresentation.
[§3, Theorem 3.22, proof] In the density part of the proof the text refers to 'Lemma 3.10 (b)' and 'Theorem 4.4 [21]'; with the numbering in this preprint, Lemma 3.10 has no part (b) and the intended statement in the current paper is not clear. The cross-reference should be corrected.
[§4, Example 4.5] The supports of the proper subrepresentations N and L are stated as {1} and {3,4}; this appears inconsistent with Lemma 2.2 for the displayed poset, where the two proper subsets of the relevant maximal set {3,5} give supports {3,2} and {5,4}. The example should be checked and the supports corrected.
[General notation] The notation overloads P(Q) and P(QF): the same symbol is used for a polygon, a set of non-frozen line segments, and a category of sp-segments. A consistent notation or an index of symbols would greatly improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: stability weights are constructed and directly verified, and the geometric stability argument rests on an external BGMS model. A non-circular proof gap exists in Proposition 2.5's case analysis.

full rationale

The central derivations are not circular. In Section 2, the stability weight is defined explicitly by theta(W)=bP(dim U,dim W)-bP(dim W,dim U), a skew-symmetric expression that automatically gives theta(U)=0; the remaining work is a direct verification that theta(W)<0 for proper peak subrepresentations. This is a construction plus check, not a fitted parameter renamed as a prediction, and the conclusion is not encoded in the definition of theta. The geometric stability argument in Section 4 is likewise independent: Theorem 3.22 establishes a categorical equivalence by an explicit functor Omega, using the external BGMS model [1], and Theorem 4.3 verifies angle-stability from the geometry of subsegments rather than from the desired conclusion. Self-citations to [7] and [21] provide background material (the stability formalism and the type-A classification/geometric realization), but those are published results and the new stability checks do not reduce to them. The one issue that should be flagged is a non-circular proof gap in Proposition 2.5: after equation (2.5), the reduction 'dim W is the sum of all dimensions dim W_X, where X runs over all of subsets of consecutive points in A' does not cover proper subrepresentations supported on non-adjacent maximal points, such as I={z1,z3} in S1^(3), where A is empty but W is nonzero; the analogous cases for S2^(r) and S3^(r) are also delegated to 'the same arguments can be used'. This is an omitted-case correctness concern, not a self-referential or definitional circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The central claim rests on standard background, the external classification of sincere type A posets, and the paper's own definitions of the geometric model. There are no fitted numerical parameters. The newly introduced geometric constructs are definitions whose correctness is established only through the paper's own proofs.

assumptions (4)

domain assumption Classification of sincere type A posets into S1^(r), S2^(r), S3^(r) (Kleiner, Kosakowska)
Used in Proposition 2.5 and Theorem 2.6 to reduce stability to three families.
domain assumption Every indecomposable peak P-space of type A has U_x isomorphic to k for x in its support, and is a lift of a sincere representation of a sincere peak-subposet via the subposet induced functor T_S (from [21], [26])
Used to reduce Theorem 2.6 to sincere posets.
domain assumption The BGMS categorical equivalence between the category of line segments and representations of a type A quiver (Theorem 3.5, cited from [1])
Basis for the sp-segment model and the geometric stability proof.
standard math Standard linear algebra and category theory (kernels, quotients, additive categories, Auslander-Reiten theory)
Used throughout, no special status.

invented entities (2)

Category CPsp(QF) of sp-segments
purpose: To model the category modsp(kP) geometrically and to define an angle stability function
New mathematical definition; its validity rests on Theorem 3.22 proved in the paper, no external evidence.
Frozen line segments and principal subchains
purpose: To carve out the subcategory of sp-segments from the full category of line segments
Defined in Section 3.2 to identify the correct subcategory.

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Pith. "Pith review of Stability for socle-projective categories of type $\mathbb{A}$." pith.science (2026). https://pith.science/paper/44XMTYUC

@misc{pith2026250113578,
  author       = {Pith},
  title        = {Pith review of: Stability for socle-projective categories of type $\mathbbA$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/44XMTYUC}},
  note         = {Machine review of arXiv:2501.13578}
}

abstract

We extend the notion of stability in the non-abelian category of poset representations (introduced by Futorny and Iusenko) to the category of socle-projective representations of a given $r$-peak poset $\P$. When $\P$ is a poset of type $\mathbb{A}$, we demonstrate in two distinct ways that every indecomposable peak $\P$-space is stable. First, this is shown using a bilinear form associated with the poset. Second, we prove it by observing that a stability function derived from a geometric model ensures that all indecomposable objects are stable. Along the way, we provide a new geometric realization of the category of socle-projective representations, inspired by the work of Schiffler and Serna [\textit{J. Pure Appl. Algebra} \textbf{224} (2020), no.~12, 106436, 23 pp.; MR4101480]. Finally, we establish a connection between the geometric perspective and the bilinear form approach.

Figures

Figures reproduced from arXiv: 2501.13578 by the authors.

**Figure 1.** Polygon P(Q) given by Example 3.1 The Category of line segments. Given a polygon P(Q), E is the set of all line segments γ(i, j) of P(Q) defined as follows E = {γ(i, j) | 0 ≤ i < j ≤ n + 2} where γ(i, j) denotes the line segment between the vertices i and j. For any vertex ℓ of P(Q), denote by R(ℓ) and by R−1 (ℓ), respectively, the clockwise and counterclockwise neighbor of ℓ on the boundary of P(Q) [PITH_FULL_IMA… view at source ↗

**Figure 2.** Set of all suitable line segments in P(Q). Note that the suitable line segments correspond to simple projective and simple injective objects in Ind Q. Proposition 3.8. Let Q be a quiver of type An+2, let P(Q) be its corresponding polygon and let F be the functor in Definition 3.1. The segment γ(i−1, i) is a suitable line segment in P(Q) if and only if F(γ(i−1, i)) is a simple projective or a simple injective represe… view at source ↗

**Figure 3.** Configuration of suitable line segments. The socle of the representation M(s+ 1, m) is given by Lh−1 r=1 M(s 1 ir , s1 ir ). Proposition 3.8 implies that M(s 1 ir , s1 ir ) is a simple projective, therefore F(γ(s, m)) is a socle-projective representation. In the other direction, suppose that F(γ(s, m)) is a socle-projective representation, that is, soc M(s + 1, m) is a direct sum of the simple projective representat… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Diagram of the mesh relations in CPsp(QF ) 3.3. The categorical equivalence. Actually, we are going to show that the category of sp-segments is equivalent to the category of peak P-spaces, where P is the poset asocciated to the quiver QF . However, for an easier explan…

**Figure 5.** Figure 5: Auslander-Reiter quiver of modsp(kP) [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]

**Figure 6.** Figure 6: Auslander-Reiten quiver of the category of segments CP (Q). The indecomposable objets of the category of sp-segments CP (QF ) corresponds to the blue segments 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6…

**Figure 7.** Figure 7: Auslander-Reiten quiver of the category of sp-segments CP (QF ) 4. STABILITY ARISING FROM THE GEOMETRIC MODEL OF BGMS In this section, we prove Theorem 4.3, which establishes a stability function for the category modsp(kP), derived from the geometric model introduced i…

**Figure 8.** Figure 8: Polygons P ′ (Q) associated with a poset P (left m = 1, center m = 2, right m = 3) The sp-segments are represented in the right part of [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]

**Figure 9.** Figure 9: Some line segments on the polygon P ′ (Q), along with their associated vectors and their reflections across the line y = x REFERENCES [1] Emily Barnard, Emily Gunawan, Emily Meehan, and Ralf Schiffler, Cambrian combinatorics on quiver representations (type A), Adv. in …

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Reviewed August 10, 2026 · model on record in the stance chip above.

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Pure Appl

Vyacheslav Futorny and Kostiantyn Iusenko, Stable representations of posets, J. Pure Appl. Algebra223 (2019), no. 12, 5251–

work page 2019

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Ralf Schiffler and Robinson-Julian Serna, A geometric realization of socle-projective categories for posets of typeA, J. Pure Appl. Algebra 224 (2020), no. 12, 106436, 23. MR4101480

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