Stability conditions and Artin--Tits groups

Anthony M. Licata; Edmund Heng

REVIEW 3 major objections 5 minor 1 cited by

Stability conditions and Artin--Tits groups

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Pith's one-line read For any Coxeter graph, the paper constructs a triangulated category whose fusion-equivariant stability component is a covering of a hyperplane complement, with the Artin–Tits group and the Serre shift acting as deck transformations.

desk verdict A genuinely new generalization of the Bridgeland–Ikeda stability picture to arbitrary Coxeter systems, with a strong construction and an honest write-up, but the main covering theorem currently rests on one delegated proof (Lemma 7.21) that is not routine. read the letter →

arxiv 2412.15919 v1 pith:7Q7LAJG7 submitted 2024-12-20 math.RT math.GRmath.QA

Edmund Heng , Anthony M. Licata This is my paper · ORCID

classification math.RTmath.GRmath.QA MSC 16G2018G8020F3620F55

keywords BridgelandstabilityconditionsArtin-TitsgroupsCoxetersystemszigzagalgebrasfusioncategoriessphericaltwistshyperplanecomplementsK(pi1)conjecture

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 constructs, for every Coxeter graph, a triangulated category that carries a natural action of the associated Artin–Tits group, and then studies the space of Bridgeland stability conditions on it that are equivariant for a fusion-category action. The central result is that a distinguished connected component of this equivariant stability space is a covering space of the hyperplane complement Upsilon_reg, and that the deck transformations of the quotient covering to Upsilon_reg modulo the Coxeter group are exactly the Artin–Tits braid group together with the Serre shift [2]. If correct, this realizes the fundamental group of the quotient hyperplane complement as autoequivalences of a triangulated category, giving a geometric K(pi,1)-style model for Artin–Tits groups. The construction works for arbitrary Coxeter systems, going beyond the symmetric Kac-Moody cases studied earlier.

What carries the argument

The central object is the zigzag algebra zig(Gamma), built as a graded Frobenius algebra object in a fusion category C(Gamma) associated to Gamma; each edge labeled m contributes an object of Frobenius-Perron dimension 2cos(pi/m). Its category of graded projective modules, after passing to bounded homotopy categories, gives the triangulated 2-Calabi-Yau category D(Gamma). The action of the Artin-Tits group is generated by C-spherical twists, two-term complexes of bimodules whose braiding relations mirror the Coxeter relations. The covering theorem is proved by showing that the central-charge map restricted to the C-equivariant component is a local homeomorphism onto Upsilon_reg, that the group actions are free and properly discontinuous, and that the quotient is Upsilon_reg/W; the identification of the image rests on a semistable-object algorithm for every positive real root.

What would settle it

Compute the image of the central-charge map for a Coxeter system where the transferred algorithm is not verified, and look for a positive real root alpha whose hyperplane is hit by a central charge in the image; such a point would contradict Proposition 7.23 and Theorem 7.7. More directly, search for a C-equivariant stability condition whose central charge vanishes on a positive real root or on a nonzero element of the closed imaginary cone, which Lemmas 7.21 and 7.22 assert never happens.

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Extended reading notes

Core claim

Theorem 7.7 states that for any Coxeter graph with Artin-Tits group B and Coxeter group W, the C-equivariant stability manifold of the category D contains a connected component Stab^dagger_C(D) for which the central-charge map is a covering onto the hyperplane complement Upsilon_reg. The composite covering to Upsilon_reg/W has deck-transformation group generated by the image of B, acting through C-spherical twists, and by the degree-two Serre shift [2]; when W is infinite, the fundamental group of Upsilon_reg/W is Z times B, and the Z factor is realized by [2]. For symmetric Kac-Moody Coxeter graphs, the construction recovers the earlier stability-space covering theorems as a special case. The paper also proves that the larger non-equivariant component Stab^dagger(D) covers the hyperplane complement of an unfolded Coxeter graph, with the equivariant component embedded inside it, yielding a geometric realization of LCM-homomorphism embeddings between Artin-Tits groups.

Load-bearing premise

The proof that the image of the central-charge map is exactly Upsilon_reg assumes that for every positive real root alpha the fusion-enhanced category has a semistable object of class alpha; the paper transfers an algorithm from the symmetric Kac-Moody setting without giving a full proof of that transfer.

Editorial extensions

If this is right

If the Artin-Tits group acts faithfully on D, then Stab^dagger_C(D) is simply connected and is the universal cover of Upsilon_reg; faithfulness is known for finite Coxeter groups, so in finite type Stab^dagger_C(D) is contractible and Upsilon_reg/W is a K(pi,1) space.
The deck-transformation description realizes pi_1(Upsilon_reg/W), which is B or Z times B, as autoequivalences of D, with spherical twists matching Coxeter reflections at the level of central charges.
The closed embedding of covering spaces from the equivariant component to the unfolded component realizes LCM-homomorphism embeddings geometrically, and faithfulness of the B-action on D would imply injectivity of those embeddings.
The construction extends the known stability-space covering descriptions from symmetric Kac-Moody types to arbitrary Coxeter systems, with the fusion-category action playing an essential role outside that class.

Reading between the lines

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

A testable check that goes beyond the paper is to verify, in a small infinite Coxeter example, that the transferred algorithm really produces a semistable object for every positive real root; failure would localize the main gap in the image identification.
If the covering theorem holds, the equivariant stability component gives a candidate intrinsic Teichmuller space for arbitrary Artin-Tits groups, and proving its contractibility would settle the K(pi,1) conjecture for all Coxeter types, not just finite ones.
Because the equivalence between D(Gamma) and the unfolded simply-laced category identifies the non-equivariant stability spaces, any faithfulness statement for D(Gamma) would embed B(Gamma) into a simply-laced Artin-Tits group, an open question the paper leaves.

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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

First thing to know: this is a plausible and genuinely new generalization of the Bridgeland–Ikeda picture to arbitrary Coxeter systems. The construction of zigzag algebras as algebra objects in fusion categories built from Temperley–Lieb–Jones categories, with Frobenius–Perron dimensions matching 2cos(pi/m), is a real achievement and likely of independent interest. The paper is honest, well-structured, and it explicitly corrects a gap in Ikeda's work (Remark 7.28). If the main theorem holds, it gives candidate K(pi,1) spaces for all Artin–Tits groups and ties the K(pi,1) conjecture to faithfulness of the B-action on D(Γ). That is a worthwhile target.

The soft spots are concentrated exactly where the reader's report says. Lemma 7.21 is load-bearing: it is the only step ruling out root hyperplanes in the image of Z^†_C, and the proof is one sentence saying the algorithm from [BDL23, Section 4] goes over verbatim. That transfer is not obviously routine. In [BDL23] root classes live in an integral lattice; here they live in a K0(C)-module with coefficients in Z[2cos(pi/m)], and you need to re-verify that the constructed complexes are semistable and C-equivariant. The stress-test note is also right that the lemma as stated is automatic for τ in Stab_C(H), while Proposition 7.23 needs it for the whole component. The reduction to H in Lemma 7.20 only works for points already in (Z^†_C)^{-1}(Υ_reg), which is what the proof is trying to establish, so as written there is a real circularity risk. This is the main gap. Proposition 5.3 is also delegated to the first author's thesis, but it is a rank-two reduction with an explicit sketch, so I am less worried. Proposition 8.9's 'best checked independently by the reader' is a hand-wave, but it is a minor point in the unfolding section.

I want to give credit where it is due: the paper does not hide these gaps, the categorical setup is coherent, and the claimed theorem is not fitted to a target answer. No red flags on data or circular fitting. The unfolding story and the connection to LCM-homomorphisms are genuinely interesting and should be preserved.

Who is this for? Representation theorists working on stability conditions, braid groups, and categorical actions. A serious referee should engage with it, but the referee should demand a complete proof of Lemma 7.21 and a clarification of the reduction in Proposition 7.23. My recommendation: send it to peer review, and treat the current version as conditionally acceptable pending those additions.

Referee Report

3 major / 5 minor

Summary. The paper associates to an arbitrary Coxeter graph Γ a fusion category C(Γ), a graded Frobenius zigzag algebra zig(Γ) inside C(Γ), and a triangulated 2-Calabi–Yau category D(Γ) = Komb(zig(Γ)-prmod). It constructs an action of the Artin–Tits group B(Γ) on D(Γ) by C-spherical twists, decategorifies this action to a faithful action of the Coxeter group W(Γ) on a K0(C(Γ))-lattice Λ, and studies C(Γ)-equivariant Bridgeland stability conditions. The main result (Theorem 7.7) claims that the connected component Stab†_C(D) containing the linear heart is a covering space of the hyperplane complement Υ_reg(Γ), and that the deck transformation action of π1(Υ_reg/W) is realized by autoequivalences generated by B(Γ) and the shift [2]. Corollary 8.10 and Theorem 8.5 describe the full component Stab†(D) in terms of a Coxeter unfolding ˇΓ and connect the resulting embeddings to LCM-homomorphisms. The paper also formulates faithfulness and K(π,1) conjectures and states a finite-type version (Theorem 7.13).

Significance. If correct, the main theorem substantially extends the Thomas–Bridgeland–Ikeda picture from symmetric Kac–Moody types to arbitrary Coxeter systems, providing a new geometric presentation of the hyperplane complement Υ_reg/W as the deck group action on a stability manifold. The construction of zigzag algebras as distinguished algebra objects in fusion categories, the explicit computation of Hom spaces, and the correction of gaps in Ikeda's argument are valuable contributions in their own right. However, the central image identification and the braid-relation verification are delegated to external sources, so the significance is conditional on those points being supplied.

major comments (3)

[§7.6, Lemma 7.21 and Proposition 7.23] The only argument ruling out root hyperplanes in the image of Z†_C is Lemma 7.21, whose proof is a one-sentence citation to [BDL23, Section 4] with the assertion that the algorithm 'goes over verbatim'. This transfer is not routine: in [BDL23] the root classes live in a free Z-lattice, whereas here they have coefficients in K0(C(Γ)), and the constructed objects' classes, semistability, and C-equivariance must be re-verified with fusion-enhanced spherical twists. More importantly, the lemma as stated is for τ ∈ Stab_C(H), where it is automatic by positivity: for Z ∈ C, every positive real root α has Z(α) ≠ 0 because α is a nonnegative combination of simple roots. The case needed in Proposition 7.23 is an arbitrary τ ∈ Stab†_C(D) whose central charge could a priori lie on a root hyperplane; Lemma 7.20 cannot reduce to Stab_C(H) for such τ because it presupposes that the central charge lies in Υ_N^reg. Thus the proof of im(Z†_C) = Υ_reg is incomplete as written and appears to rely circularly on the very identification being proved.
[§5.2, Proposition 5.3] The braid relations among the C-spherical twists are load-bearing: they give the well-defined homomorphism B(Γ) → Br_ST in Theorem 5.6 and hence the deck-group identification in Theorem 7.7. The proof says 'exactly as in the rank two case' and delegates the key computation (21) to [Hen, §2.2], without verifying in this paper that the fusion-enhanced context satisfies all hypotheses of that reference. Please provide a complete proof of (21), or a precise reduction to the cited rank-two computation with every hypothesis checked.
[§7.6, Lemma 7.22] The proof of Lemma 7.22 claims that the support property together with Lemma 7.21 implies that Z′(v) ≠ 0 for every nonzero v on an accumulation ray of positive root rays, and that the statement for all v in the closed imaginary cone follows by convexity. This is not justified as written: the support property constrains only classes of semistable objects, not arbitrary accumulation points of root rays, and convexity of Z(I) does not by itself prevent a positive linear combination of nonzero vectors in a cone from being zero. Since Lemma 7.22 is needed to exclude the imaginary-cone hyperplanes in Proposition 7.23, this step needs a detailed argument.

minor comments (5)

[§7.3, Theorem 7.13] Theorem 7.13 and Proposition 7.14 are stated as results but their proofs are omitted with a sketch and a reference to similar arguments. If they are not needed for the main theorem, they should be explicitly labeled as conditional or deferred to future work.
[§2.2 and §7.1] The notation H is used both for the upper half-plane in Definition 2.13 and for the linear heart in Section 7.1. This is a recurring source of potential confusion; consider renaming one of them.
[§1 and §3] The notation Stab_C(D) uses C both for a fusion category and for a complexified chamber. The authors do note the convention, but a different symbol for the chamber (for example, C_ch) would improve readability.
[§7.3, Proposition 7.11] The proof of Proposition 7.11 invokes Proposition 7.25, but Proposition 7.25 only concerns Ψ(H) ⊂ H, not Ψ(P_s) ≅ P_s for all s. The intended implication can be obtained from Proposition 7.26, whose proof is stated to be 'essentially identical' and omitted. Please clarify the logical dependence.
[Throughout] The notation Komb appears alongside Kom^b; please standardize.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found; the main covering theorem is an independent geometric statement, though it contains delegated proof steps that are correctness gaps rather than circularity.

full rationale

The main theorem (Theorem 7.7) is not circular: Stab^†_C(D) is defined as the connected component containing the linear-heart chamber, while Υ_reg is defined independently from the Coxeter root system and imaginary cone (Definitions 2.13 and 7.6). The image identification im(Z†_C)=Υ_reg is a substantive claim proved from the group actions and chamber structure, not assumed. Lemma 7.5 identifies Stab_C(H) with the complexified chamber C by a direct computation of K0(C)-linear stability functions; Lemma 7.16/Proposition 6.8 gives the W-equivariance of the central-charge map; Lemma 7.20 and Proposition 7.24 reduce points to the chamber; Proposition 7.25 proves freeness of the spherical-twist action. None of these steps presupposes the covering statement. The main caveat is Lemma 7.21, whose proof cites [BDL23, Section 4] and asserts that the semistable-object algorithm 'goes over verbatim in our setting.' This is a self-citation (Licata is a common author) and an unwritten transfer. However, as stated the lemma is automatic for τ ∈ Stab_C(H): each Z'(α_s) lies in H ∪ R_{<0}, so any positive root, being a nonnegative linear combination of simple roots, has nonzero image (positive imaginary part or negative real value). Thus the cited algorithm is not load-bearing for the lemma itself. The genuine issue is that Proposition 7.23 applies Lemma 7.21 and Lemma 7.22, which are stated only for Stab_C(H), to arbitrary limit points in Stab†_C(D) without explicitly invoking Lemma 7.20 to reduce to the chamber. That is a potential correctness gap, not a circular reduction: it does not make the equality im(Z†_C)=Υ_reg true by definition. Proposition 5.3 also delegates the braid-relation verification to the first author's thesis [Hen, §2.2], but with an explicit rank-two sketch; this is a self-citation, not a circular import of the main theorem.

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

The central claim rests on standard Coxeter theory and stability theory, plus two delegated results from prior work by the same group: the semistable-object algorithm and the rank-two braiding relations. No fitted numerical parameters appear; the choice of fusion category C(Gamma) is a construction choice, not a free parameter. The new objects such as C(Gamma) and zig(Gamma) are explicitly constructed rather than postulated.

assumptions (6)

standard math The Tits representation of a Coxeter group is faithful.
Used in Section 2 and Corollary 6.7 to conclude faithfulness of the W-action on the lattice Lambda.
standard math Dyer's imaginary cone properties for Coxeter systems (Proposition 2.5 and Lemma 2.6).
Used to define Upsilon_reg and to prove openness of Upsilon and Upsilon_reg in Section 2.2.
standard math Van der Lek's theorem: pi_1(Omega_reg/W) is isomorphic to the Artin-Tits group B(Gamma).
Used in Corollary 2.24 to compute pi_1(Upsilon_reg/W) as B(Gamma) for finite W and Z x B(Gamma) for infinite W.
standard math Bridgeland's deformation theorem and the local homeomorphism property of the stability manifold.
Used in Section 7 to give Stab_C(D) the structure of a complex manifold, following [Bri07], [Bay19], and [DHL23, Theorem A].
ad hoc to paper The algorithm from [BDL23, Section 4] constructing semistable objects for every positive real root class transfers verbatim to the fusion-enhanced setting (Lemma 7.21).
Stated without proof in Section 7.6; it is load-bearing for proving that the central charge avoids the root hyperplanes and hence the image is Upsilon_reg.
ad hoc to paper The rank-two braiding relations for fusion spherical twists from [Hen, Section 2.2] extend to arbitrary pairs of vertices in the Coxeter diagram (Proposition 5.3).
Proof is delegated to [Hen]; needed to define the Artin-Tits group action on D(Gamma) via the spherical twist generators.

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Pith. "Pith review of Stability conditions and Artin--Tits groups." pith.science (2026). https://pith.science/paper/7Q7LAJG7

@misc{pith2026241215919,
  author       = {Pith},
  title        = {Pith review of: Stability conditions and Artin--Tits groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Q7LAJG7}},
  note         = {Machine review of arXiv:2412.15919}
}

abstract

We describe spaces of Bridgeland stability conditions on certain triangulated categories associated to Coxeter systems. These categories are defined algebraically using the category of modules for zigzag algebras associated to Coxeter systems, which we construct as distinguished (quadratic, graded) algebra objects in fusion categories. The resulting stability spaces are closely related to conjectural K($\pi$,1) spaces for Artin--Tits groups.

Figures

Figures reproduced from arXiv: 2412.15919 by the authors.

**Figure 1.** The images Z(I), Z′ (I) ⊂ C of the imaginary cone when I 6= {0}, shaded in blue. Z ′ (right) is normalised, whereas Z (left) is not. e i(−φ I (Z)+π/2) · Z. It follows that (I = {0} case is by convention): Υreg ∼= ( S 1 × ΥN reg, if I 6= {0}; ΥN reg, if I = {0}. (8) Lemma 2.18. The actions of S 1 and W on Υreg commute with each other. Moreover, ΥN reg is a W-invariant subset of Υreg, and the homeomorphism in (8) can … view at source ↗

**Figure 2.** Coxeter graphs Γ and their corresponding unfolded Coxeter graphs Γ. ˇ is given by rep(S3) (chosen as the Deligne tensor of C(Γ) associated to the label ∞), where Π(e) is taken to be the standard two dimensional irreducible representation. We claim that the two categories zig(Γ)-prmod and zig(Γ)- ˇ prmod are equivalent as additive categories, and hence their bounded homotopy categories are equivalent as triangulated… view at source ↗

**Figure 3.** The real slice of Υreg(Γ) := Υreg(I2(5)) viewed as a subspace of Υreg(Γ) := Υ ˇ reg(A4). In this subspace, the (removed) 10 reflection hyperplanes of type A4 coincides with the 5 reflection lines of type I2(5), as mentioned in example 8.6. To describe the embedding Υreg(Γ) ֒→ Υreg(Γ) in more detail, consider the following ˇ C-linear subspace of HomZ(Λ(Γ) ˇ , C): HomC(Γ) Z (Λ(Γ) ˇ , C) := {Z ∈ HomZ(Λ(Γ) ˇ , C) | ∀(s,… view at source ↗

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The Thurston compactification of the stability manifold of a generic analytic K3 surface
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For an analytic K3 surface with Picard group zero, the mass map from the projective stability manifold to the space of masses of semi-rigid objects is a homeomorphism onto an open disk whose closure is a closed disk.

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6 extracted references · 3 canonical work pages · cited by 1 Pith paper

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