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arxiv: 2605.22440 · v1 · pith:BY4UQLIInew · submitted 2026-05-21 · 🧮 math.RT · math.CO· math.SG

Categorical Lusztig cycles and weave schobers

Roger Casals , Merlin Christ This is my paper

Pith reviewed 2026-05-22 01:53 UTC · model grok-4.3

classification 🧮 math.RT math.COmath.SG
keywords categorical Lusztig cyclesDemazure weavesperverse sheaves of categoriessimple-minded collectionssilting collectionsbraid varietiesCalabi-Yau triangulated categoriescluster tilting
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The pith

Categorical Lusztig cycles form simple-minded and silting collections in the global sections of sheaves attached to Demazure weaves.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper associates a perverse sheaf of triangulated categories to each Demazure weave. It constructs categorical Lusztig cycles and their duals using the diagrammatics of these weaves. These cycles are shown to form simple-minded and silting collections in the category of global sections. The collections tilt under weave mutations. A sympathetic reader would care because this supplies a diagrammatic handle on homological properties in Calabi-Yau categories and cluster tilting theory.

Core claim

By associating a perverse sheaf of triangulated categories to each Demazure weave, the categorical Lusztig cycles and their duals, built using the diagrammatics of weaves, form simple-minded and silting collections in the category of global sections of such a sheaf of categories. These collections undergo tilts under weave mutations. The paper also develops categorical weighted braid words as canonical rigid filtered dg modules over derived preprojective algebras and the categorical incarnation of the tropical Lusztig rules as a gluing mechanism for such filtered objects.

What carries the argument

The perverse sheaf of triangulated categories associated to a Demazure weave, which provides the ambient setting in which the categorical Lusztig cycles form simple-minded and silting collections.

If this is right

  • The collections undergo tilts under weave mutations.
  • Categorical weighted braid words arise as rigid filtered dg modules over derived preprojective algebras.
  • The tropical Lusztig rules receive a categorical incarnation through gluing of filtered objects.
  • A novel construction of simple-minded and silting collections from full exceptional collections is supplied via highest weight structures.
  • The diagrammatic changes of weaves correspond to controlled homological operations on the collections.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction could supply new categorical invariants for braid varieties by tracking the cycles across different weaves.
  • Explicit low-rank computations might test whether the tilting matches known equivalences in cluster categories.
  • The approach may link to other combinatorial models in representation theory by replacing weaves with different diagrams.
  • Global sections of these sheaves might yield new examples of Calabi-Yau categories with controlled silting theory.

Load-bearing premise

A perverse sheaf of triangulated categories can be associated to each Demazure weave in a way that makes the global sections well-defined and allows the Lusztig cycles to satisfy the simple-minded and silting properties under weave changes.

What would settle it

A concrete Demazure weave for which the constructed categorical Lusztig cycles fail to form a silting collection in the global sections or fail to undergo the predicted tilt after a weave mutation.

Figures

Figures reproduced from arXiv: 2605.22440 by Merlin Christ, Roger Casals.

Figure 1
Figure 1. Figure 1: Two instances of Demazure weaves w, w′ : β −→ β ′ from β = s2s1(s1s2) 3 ∈ Br+ 3 to β ′ = s2s1s2, where blue weave edges are labeled by s1 ∈ S3 and red weave edges by s2 ∈ S3. (Left) The mutable Lusztig cycles γ1, γ2, γ3 for w are highlighted in yellow, green and pink respectively, and their intersection quiver is displayed in the square box at the lower right of w. (Right) The weave w′ obtained from w by p… view at source ↗
Figure 2
Figure 2. Figure 2: A Demazure weave w : β −→ δ(β) from the 3-stranded braid word β = s2s1(s1s2) 3 to β ′ = s2s1s2, where blue weave edges are labeled by s1 ∈ S3 and red weave edges by s2 ∈ S3. Generic horizontal slices are depicted in yellow dashed lines, and the positive braid word associated to each such slice is written to its right. We have highlighted in purple the letters of each braid word to which the next move will … view at source ↗
Figure 3
Figure 3. Figure 3: The three types of vertices allowed in the diagrammatic version of a weave [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A few weave equivalences, in all of them blue indicates [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (Left) An elementary weave mutation, between the only two Demazure weaves [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A redrawing of Figure [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Two instances of Lusztig cycles on a Demazure weave [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A sequence of spans of contractions realizing the clockwise [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Similar to Figure [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: A weave w, depicted left, and its associated graph Gw, drawn to its right. The trivalent vertices p1, . . . , pm of w and the associated vertices v1, . . . , vm are also depicted, with vi at the same height as pi , i ∈ [1, m]. The edges e1, . . . , em of Gw are also drawn, with ei denoting the edge between vi−1 and vi . The weave schober Fw is a perverse schober on Gw. 22 [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 11
Figure 11. Figure 11: The local model for a 6-valent vertex at a weave [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The 3-stranded weave w for Example 3.4. The positive braid word at the top is β = s2s1(s1s2) 3 , the braid word at the bottom is its Demazure product δ(β) = σ2σ1σ2. The graph Gw over which the weave schober Fw is depicted to the right of w. The vertices of Gw are labeled v1, v2, v3, v4, v5 bottom to top. The south point v−∞ is depicted in a round green circle. Example 3.4. Consider the 3-stranded weave w … view at source ↗
Figure 13
Figure 13. Figure 13: The weave w from [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The local weave equivalence that exchanges the height of trivalent vertices, [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: (i) A height exchange for a 6-valent vertex, in [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: (a) The pushthrough move, where blue color for a weave line indicates [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The graph GD in the case of n = 3, with its four vertices v1, v2, v3, v4 (gray) and its three edges e1, e2, e3 (black). There are three πD-matching paths γ1, γ2, γ3, depicted in yellow, purple and gray respectively. Let GD be the linear ribbon graph with n+ 1 vertices, i.e. an An+1-Dynkin diagram. See [PITH_FULL_IMAGE:figures/full_fig_p039_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The six different types of local pieces for a [PITH_FULL_IMAGE:figures/full_fig_p040_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: A summary of the key cases in Proposition [PITH_FULL_IMAGE:figures/full_fig_p044_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The intersections of πD-matching paths and their contribution to morphisms between the associated matching objects in the case of the canonical grading of Re￾mark 5.11. For instance, by Example 5.21 the first row here must coincide with the first row of [PITH_FULL_IMAGE:figures/full_fig_p046_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The resolution in Remark 5.24, describing fibers as matching objects. Lemma 5.25 (semi-twists preserve M). Let γ, γ′ be graded πD-matching paths and Xγ, Xγ ′ ∈ M their associated matching objects. Suppose that γ ′ is embedded. Then both semi-twists T −,≤0 Xγ′ (Xγ) and T +,≥0 Xγ′ (Xγ) are matching, i.e. they are objects in M. Proof. By definition, T +,≥0 Xγ′ (Xγ) = cof(τ≥0(MorD(Xγ ′, Xγ)) ⊗ Xγ ′ → Xγ). By … view at source ↗
Figure 22
Figure 22. Figure 22: (Left) Pieces of πD-matching paths for Example 5.37, with γ1 indicating only a piece that continues to other parts of the graph GD, and e1, e2 as πD-matching paths inside a neighborhood containing v1, v2 and v3. The morphism α : Xe1 [−1] −→ Xe2 of degree 1 is indicated in green. (Right) The endpoint resolution of the morphism α, which yields a crossing intersection with the matching object Xγ1 . Note that… view at source ↗
Figure 23
Figure 23. Figure 23: Four types of w-matching segments, cf. Definition 7.3 and Remark 7.4. Definition 7.3. Given i ∈ [1, m], let Dvi ⊂ D be a closed horizontal band whose upper boundary, resp. lower boundary, coincides with a horizontal slices immediately above, resp. below, the vertex vi . (1) For i ∈ [1, m−1], a w-matching segment in Dvi is defined to be an embedded curve γ : [0, 1] → Dvi such that • γ is smoothly isotopic … view at source ↗
Figure 24
Figure 24. Figure 24: A general Demazure weave with its weave vanishing paths, as introduced in [PITH_FULL_IMAGE:figures/full_fig_p066_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: (1) The start of the path for a left vanishing path at the trivalent vertex of a [PITH_FULL_IMAGE:figures/full_fig_p067_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: (1) An illustration of the fact that Figure [PITH_FULL_IMAGE:figures/full_fig_p067_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Two 2-stranded weaves discussed in Example [PITH_FULL_IMAGE:figures/full_fig_p069_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: The weave w from Example 3.4 and Example 7.19, cf. also [PITH_FULL_IMAGE:figures/full_fig_p071_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: The weave cycles {ρi} realizing the silting collection dual to the simple-minded collection of Lusztig cycles for this right-inductive weave, cf. Example 9.16.(1). These weave cycles {ρi} are highlighted in pink. Example 9.16. (1) Consider the right-inductive weave w for β = σ 4 1 , as in [PITH_FULL_IMAGE:figures/full_fig_p094_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: The dual Lusztig weave cycles {ρi} from Example 9.16.(2), highlighted in pink. 95 [PITH_FULL_IMAGE:figures/full_fig_p095_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: A few of the weave cycles {ρi} from Example 9.16.(3), highlighted in pink, with the corresponding (absolute) Lusztig cycles {γi} highlighted in yellow. (3) Consider the weave w depicted in [PITH_FULL_IMAGE:figures/full_fig_p096_31.png] view at source ↗
read the original abstract

We establish the foundations of categorical weave calculus, developing the diagrammatic calculus of weaves and braid varieties within the study of Calabi-Yau triangulated categories and cluster tilting theory. This is achieved by associating a perverse sheaf of triangulated categories to each Demazure weave. A central contribution is the construction and study of the categorical Lusztig cycles and their duals, which we show form simple-minded and silting collections in the category of global sections of such a sheaf of categories. These categorical collections are built using the diagrammatics of weaves and we study their behavior under changes of weaves. For instance, we show that they undergo tilts under weave mutations. En route, we develop the study of categorical weighted braid words, as canonical rigid filtered dg modules over derived preprojective algebras, and the categorical incarnation of the tropical Lusztig rules, as a gluing mechanism for such filtered objects. Appendix A contains homological results, providing a novel construction of simple-minded and silting collections from full exceptional collections, and characterizing when these arise from a highest weight structure on an abelian category.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper establishes foundations for categorical weave calculus by associating a perverse sheaf of triangulated categories to each Demazure weave. It constructs categorical Lusztig cycles and their duals, proving they form simple-minded and silting collections in the global sections of this sheaf. The work examines their behavior under weave changes, including tilts under mutations, and develops categorical weighted braid words as rigid filtered dg-modules over derived preprojective algebras together with a categorical version of tropical Lusztig rules as a gluing mechanism. Appendix A supplies general homological results on constructing simple-minded and silting collections from full exceptional collections and characterizing those arising from highest-weight structures on abelian categories.

Significance. If the constructions and verifications hold, the paper would supply a new diagrammatic and sheaf-theoretic framework connecting weave combinatorics and braid varieties to Calabi-Yau triangulated categories and cluster-tilting theory. The categorical Lusztig cycles would furnish concrete silting collections whose mutation behavior is controlled by weave operations, while the appendix offers a general homological tool that may apply beyond the present setting. The use of perverse sheaves and filtered dg-modules provides a systematic way to encode tropical rules categorically.

major comments (3)
  1. [§3] §3: The association of a perverse sheaf of triangulated categories to a Demazure weave is presented via categorical weighted braid words and gluing along tropical Lusztig rules, but the manuscript does not supply an explicit check that the resulting sheaf is constructible with respect to the weave stratification and that its global sections form a triangulated category in which the Lusztig cycles are well-defined objects.
  2. [§5.2] §5.2: The claim that the categorical Lusztig cycles and duals undergo tilts under weave mutations and remain silting relies on the interaction between the perverse sheaf structure and the mutation functors; the verification that Hom-vanishing and generation properties are preserved under these changes is only sketched and requires a detailed computation for at least one non-trivial mutation.
  3. [Appendix A, Theorem A.3] Appendix A, Theorem A.3: The general construction of silting collections from full exceptional collections is applied to the global sections, yet the manuscript does not verify that the highest-weight structure on the abelian heart is compatible with the weave stratification or that the resulting silting collection coincides with the categorical Lusztig cycles.
minor comments (2)
  1. [§2] The notation for filtered dg-modules over derived preprojective algebras in §2 could be accompanied by a small table of examples for low-rank weaves to clarify the rigid filtered structure.
  2. [§4] Several diagrams in §4 illustrating weave mutations would benefit from explicit labels indicating which strands correspond to the categorical Lusztig cycles before and after the mutation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below with clarifications and indicate revisions that will be incorporated to strengthen the explicitness of the arguments.

read point-by-point responses

  1. Referee: [§3] §3: The association of a perverse sheaf of triangulated categories to a Demazure weave is presented via categorical weighted braid words and gluing along tropical Lusztig rules, but the manuscript does not supply an explicit check that the resulting sheaf is constructible with respect to the weave stratification and that its global sections form a triangulated category in which the Lusztig cycles are well-defined objects.

    Authors: We agree that an explicit verification would improve clarity. The construction via categorical weighted braid words and gluing along tropical Lusztig rules is intended to ensure constructibility with respect to the weave stratification by design of the gluing data. In the revised version we will add a dedicated paragraph in §3 that verifies constructibility directly and confirms that the global sections form a triangulated category (as the global sections of a sheaf of triangulated categories) in which the categorical Lusztig cycles are well-defined objects. revision: yes

  2. Referee: [§5.2] §5.2: The claim that the categorical Lusztig cycles and duals undergo tilts under weave mutations and remain silting relies on the interaction between the perverse sheaf structure and the mutation functors; the verification that Hom-vanishing and generation properties are preserved under these changes is only sketched and requires a detailed computation for at least one non-trivial mutation.

    Authors: We acknowledge that the verification in §5.2 is schematic. We will expand this section to include a detailed computation for one concrete non-trivial mutation (e.g., the mutation corresponding to a single crossing change in a Demazure weave). The computation will explicitly track the action of the mutation functors on the perverse sheaf, verify preservation of Hom-vanishing between cycles and duals, and confirm that the silting property is retained. revision: yes

  3. Referee: [Appendix A, Theorem A.3] Appendix A, Theorem A.3: The general construction of silting collections from full exceptional collections is applied to the global sections, yet the manuscript does not verify that the highest-weight structure on the abelian heart is compatible with the weave stratification or that the resulting silting collection coincides with the categorical Lusztig cycles.

    Authors: We agree that the compatibility and coincidence statements require explicit confirmation in the specific setting. In the revised appendix we will add a short verification subsection showing that the highest-weight structure on the abelian heart is compatible with the weave stratification and that the silting collection produced by Theorem A.3 coincides with the categorical Lusztig cycles constructed in the main body. revision: yes

Circularity Check

0 steps flagged

No circularity: constructions of perverse sheaves and Lusztig cycles are independent of target properties

full rationale

The paper defines an association of a perverse sheaf of triangulated categories to each Demazure weave using categorical weighted braid words and tropical Lusztig rules for gluing. Categorical Lusztig cycles are then built directly from the weave diagrammatics and their behavior under mutations is studied. The claim that these form simple-minded and silting collections is supported by an explicit general construction in Appendix A that produces such collections from full exceptional collections on an abelian category with highest weight structure. This appendix result is stated and proved independently of the specific weave or cycle data. No equation or definition reduces the target silting property to a fitted input, self-referential definition, or load-bearing self-citation; the derivation chain consists of explicit constructions and homological verifications that remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on background properties of triangulated and Calabi-Yau categories together with the new association of perverse sheaves to weaves; no numerical parameters are mentioned.

axioms (2)
  • standard math Standard properties of triangulated categories, Calabi-Yau structures, and cluster tilting theory hold.
    Invoked throughout the construction of sheaves and collections.
  • domain assumption A perverse sheaf of triangulated categories can be canonically associated to each Demazure weave.
    This association is the starting point for all subsequent categorical objects.
invented entities (2)
  • Categorical Lusztig cycles no independent evidence
    purpose: To serve as simple-minded and silting collections inside the global sections of the weave sheaf.
    Newly defined objects whose properties are proved in the paper.
  • Perverse sheaf of triangulated categories no independent evidence
    purpose: To encode the categorical data attached to each Demazure weave.
    Central new structure introduced to support the weave calculus.

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Reference graph

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