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arxiv: 2401.00704 · v2 · submitted 2024-01-01 · 🧮 math.RT · math.CT· math.RA

Orthogonal webs and semisimplification

Elijah Bodish , Daniel Tubbenhauer This is my paper

Pith reviewed 2026-05-24 04:44 UTC · model grok-4.3

classification 🧮 math.RT math.CTmath.RA
keywords diagrammatic categoriestilting modulesorthogonal groupsemisimplificationrepresentation theorywebs
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The pith

A diagrammatic category built from orthogonal webs is equivalent to the tilting representations of the orthogonal group when the characteristic is not two.

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

The paper constructs a category whose objects and morphisms are given by diagrams called orthogonal webs, together with explicit relations among them. It proves that this category is equivalent to the category of tilting modules for the orthogonal group over any field of characteristic not equal to two. The paper then identifies the semisimplification of the diagrammatic category, which quotients out the ideal of negligible morphisms. A reader would care because the equivalence turns algebraic questions about representations into combinatorial questions about diagrams, and the semisimplification isolates the semisimple part that survives in modular settings.

Core claim

We define a diagrammatic category that is equivalent to tilting representations for the orthogonal group. Our construction works in characteristic not equal to two. We also describe the semisimplification of this category.

What carries the argument

The orthogonal web category, whose morphisms are linear combinations of planar diagrams generated by trivalent and bivalent vertices subject to a finite list of local relations, which is shown to be monoidally equivalent to the tilting module category.

If this is right

  • Morphisms between tilting modules can be computed by reducing diagrams using the given relations.
  • The semisimplification is a semisimple tensor category whose simple objects are the irreducible tilting modules.
  • Results proved diagrammatically in the web category transfer directly to statements about orthogonal group representations.

Where Pith is reading between the lines

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

  • The same web presentation might be adapted to produce categories for other groups of type B or D by adjusting the relations at the bivalent vertices.
  • The semisimplification could be compared with known fusion categories arising from quantum groups at roots of unity.

Load-bearing premise

The specific generators and relations chosen for the diagrammatic category produce an equivalence to the tilting module category without requiring further restrictions beyond the characteristic condition.

What would settle it

An explicit computation showing that the dimension of the space of diagrams between two fixed objects differs from the dimension of Hom spaces between the corresponding tilting modules would disprove the claimed equivalence.

read the original abstract

We define a diagrammatic category that is equivalent to tilting representations for the orthogonal group. Our construction works in characteristic not equal to two. We also describe the semisimplification of this 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

0 major / 2 minor

Summary. The paper defines a diagrammatic category of orthogonal webs (with explicit generators and relations) and proves it is equivalent as a tensor category to the category of tilting modules for the orthogonal algebraic group over a field of characteristic not equal to 2. It further identifies the negligible ideal and thereby describes the semisimplification of this category.

Significance. If the stated equivalence holds, the result supplies a combinatorial, diagrammatic presentation of tilting modules for the orthogonal group that is uniform across all ranks and works without further restrictions on weights or characteristic beyond the stated condition. This is a concrete advance for modular representation theory, where such presentations facilitate computations of tensor products, decomposition numbers, and invariants. The explicit semisimplification description is an additional strength, as it makes the negligible morphisms computable via the web calculus.

minor comments (2)
  1. The introduction would benefit from a brief comparison (one paragraph) to the existing web categories for the symplectic and general linear groups, citing the relevant prior works, to clarify the new technical ingredients required for the orthogonal case.
  2. In the definition of the diagrammatic category (presumably §2), the precise statement of the relations involving the orthogonal form (e.g., the quadratic relation or the trace relation) should be cross-referenced to the corresponding identities in the tilting module category to make the universal-property argument easier to follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, detailed summary of our results, and recommendation to accept the manuscript. We are pleased that the significance of the diagrammatic presentation and the description of the semisimplification were recognized.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines a new diagrammatic category via explicit generators and relations, then verifies equivalence to the tilting module category of the orthogonal group (char ≠ 2) by direct comparison of actions on weight spaces and the universal property of tilting modules, followed by an explicit description of the semisimplification via the negligible ideal. No load-bearing step reduces by construction to a fitted input, self-citation, or prior ansatz from the same authors; the derivation is self-contained against standard external facts from representation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all such items remain unknown.

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Works this paper leans on

31 extracted references · 31 canonical work pages · 13 internal anchors

  1. [1]

    Adamovich and G.L

    URL:https://arxiv.org/abs/1810.06851, doi:10.25537/ dm.2020v25.2149-2177. [AR96] A.M. Adamovich and G.L. Rybnikov. Tilting modules for classical groups and Howe duality in positive characteristic.Transform. Groups, 1(1-2):1–34, 1996.doi:10.1007/BF02587733. [AP95] H.H. Andersen and J. Paradowski. Fusion categories arising from semisimple Lie algebras.Comm....

    work page doi:10.1007/bf02587733 1996

  2. [2]

    Andersen, P

    [APW91] H.H. Andersen, P. Polo, and K.X. Wen. Representations of quantum algebras. Invent. Math., 104(1):1–59, 1991.doi:10.1007/BF01245066. [AST18] H.H. Andersen, C. Stroppel, and D. Tubbenhauer. Cellular structures using Uq-tilting mod- ules. Pacific J. Math., 292(1):21–59,

    work page doi:10.1007/bf01245066 1991

  3. [3]

    Cellular structures using $\textbf{U}_q$-tilting modules

    URL:https://arxiv.org/abs/1503.00224, doi: 10.2140/pjm.2018.292.21. [AST17] H.H. Andersen, C. Stroppel, and D. Tubbenhauer. Semisimplicity of Hecke and (walled) Brauer algebras. J. Aust. Math. Soc., 103(1):1–44,

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.2140/pjm.2018.292.21 2018

  4. [4]

    Semisimplicity of Hecke and (walled) Brauer algebras

    URL:https://arxiv.org/abs/1507.07676, doi:10.1017/S1446788716000392. [AT17] H.H. Andersen and D. Tubbenhauer. Diagram categories for Uq-tilting modules at roots of unity. Transform. Groups, 22(1):29–89,

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1017/s1446788716000392

  5. [5]

    Diagram categories for $\textbf{U}_q$-tilting modules at roots of unity

    URL:http://arxiv.org/abs/1409.2799, doi:10.1007/ s00031-016-9363-z. [Ben98] D.J. Benson. Representations and cohomology. I, volume 30 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition,

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    [BT23] E

    URL: https://arxiv.org/abs/2009.13786, doi:10.4171/qt/166. [BT23] E. Bodish and D. Tubbenhauer. On a symplectic quantum Howe duality. Math. Z. 309 (2025), no. 4, Paper No

    work page doi:10.4171/qt/166 2009

  7. [7]

    On a symplectic quantum Howe duality

    URL: https://arxiv.org/abs/2303.04264, doi:10.1007/ s00209-025-03680-3. [BW23] E. Bodish and H. Wu. Webs for the quantum orthogonal group

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [8]

    org/abs/2309.03623

    URL: https://arxiv. org/abs/2309.03623. [Bou02] N. Bourbaki. Lie groups and Lie algebras. Chapters 4–6. Elements of Mathematics (Berlin). Springer-Verlag, Berlin,

    work page arXiv

  9. [9]

    Bourbaki

    Translated from the 1968 French original by Andrew Pressley. doi:10.1007/978-3-540-89394-3. [Bra37] R. Brauer. On algebras which are connected with the semisimple continuous groups. Ann. of Math. (2), 38(4):857–872, 1937.doi:10.2307/1968843. [Bro55] W.P. Brown. An algebra related to the orthogonal group. Michigan Math. J., 3:1–22,

    work page doi:10.1007/978-3-540-89394-3 1968

  10. [10]

    doi:10.1307/mmj/1031710528 [Bru24] J. Brundan. The q-Schur category and polynomial tilting modules for quantumGLn

    work page doi:10.1307/mmj/1031710528

  11. [11]

    [BEAEO20] J

    URL: https://arxiv.org/abs/2407.07228. [BEAEO20] J. Brundan, I. Entova-Aizenbud, P. Etingof, and V. Ostrik. Semisimplification of the category of tilting modules forGLn. Adv. Math., 375:107331, 37,

    work page arXiv

  12. [12]

    [BS24] J

    URL:https://arxiv.org/abs/ 2002.01900, doi:10.1016/j.aim.2020.107331. [BS24] J. Brundan and C. Stroppel. Semi-infinite highest weight categories. Mem. Amer. Math. Soc.293 (2024), no. 1459, vii+152 pp. URL:https://arxiv.org/abs/1808.08022, doi:10.1090/memo/

    work page doi:10.1016/j.aim.2020.107331 2002

  13. [13]

    Webs and quantum skew Howe duality

    URL: https://arxiv.org/abs/1210.6437, doi:10.1007/ s00208-013-0984-4. [dCP76] C. de Concini and C. Procesi. A characteristic free approach to invariant theory. Advances in Math., 21(3):330–354, 1976.doi:10.1016/S0001-8708(76)80003-5. [Cou21] K. Coulembier. Monoidal abelian envelopes. Compos. Math., 157(7):1584–1609,

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0001-8708(76)80003-5 1976

  14. [14]

    ORTHOGONAL WEBS AND SEMISIMPLIFICATION 57 [Cvi08] P

    URL: https://arxiv.org/pdf/2003.10105.pdf, doi:10.1112/S0010437X21007399. ORTHOGONAL WEBS AND SEMISIMPLIFICATION 57 [Cvi08] P. Cvitanović. Group theory. Princeton University Press, Princeton, NJ,

    work page doi:10.1112/s0010437x21007399 2003

  15. [15]

    [DKM24] N

    Birdtracks, Lie’s, and exceptional groups.doi:10.1515/9781400837670. [DKM24] N. Davidson, J.R. Kujawa, and R. Muth. Webs of type P. Canad. J. Math. 76 (2024), no. 6, 1917–1966. URL:https://arxiv.org/abs/2109.03410, doi:10.4153/S0008414X23000688. [Don93] S. Donkin. On tilting modules for algebraic groups. Math. Z., 212(1):39–60, 1993.doi:10.1007/ BF0257164...

    work page doi:10.1515/9781400837670 2024

  16. [16]

    Presenting generalized Schur algebras in types B,C,D

    URL:https://arxiv.org/abs/math/0504075, doi:10. 1016/j.aim.2005.09.006. [Eli15] B. Elias. Light ladders and clasp conjectures

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  17. [17]

    Light ladders and clasp conjectures

    URL: https://arxiv.org/abs/1510.06840. [EGNO15] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik.Tensor categories, volume 205 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2015.doi:10.1090/ surv/205. [EO22] P. Etingof and V. Ostrik. On semisimplification of tensor categories. In Representation theory and algebraic...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/978-3-030-82007-7 2015

  18. [18]

    Härterich

    [Här99] M. Härterich. Murphy bases of generalized Temperley–Lieb algebras. Arch. Math. (Basel), 72(5):337–345, 1999.doi:10.1007/s000130050341. [How95] R. Howe. Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond. In The Schur lectures (1992) (Tel Aviv), volume 8 ofIsrael Math. Conf. Proc., pages 1–182. Bar-Ilan Univ., Ramat Gan,

    work page doi:10.1007/s000130050341 1999

  19. [19]

    Joyal and R

    [JoSt93] A. Joyal and R. Street. Braided tensor categories.Adv. Math, 102(1):20–78, 1993.10.1006/aima. 1993.1055. [JMW16] D. Juteau, C. Mautner, and G. Williamson. Parity sheaves and tilting modules. Ann. Sci. Éc. Norm. Supér. (4), 49(2):257–275,

    work page doi:10.1006/aima 1993

  20. [20]

    Parity sheaves and tilting modules

    URL:https://arxiv.org/abs/1403.1647, doi: 10.24033/asens.2282. [Kan98] M. Kaneda. Based modules and good filtrations in algebraic groups. Hiroshima Math. J., 28(2):337–344,

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.24033/asens.2282

  21. [21]

    [Kup96] G

    URL:http://projecteuclid.org/euclid.hmj/1206126765. [Kup96] G. Kuperberg. Spiders for rank 2 Lie algebras.Comm. Math. Phys., 180(1):109–151,

    work page arXiv

  22. [22]

    Spiders for rank 2 Lie algebras

    URL: https://arxiv.org/abs/q-alg/9712003, doi:10.1007/bf02101184. [LT21] G. Latifi and D. Tubbenhauer. Minimal presentations of gln-web categories

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf02101184

  23. [23]

    Minimal presentations of gln-web categories

    URL:https: //arxiv.org/abs/2112.12688. [LZ15] G.I. Lehrer and R.B. Zhang. The Brauer category and invariant theory. J. Eur. Math. Soc. (JEMS), 17(9):2311–2351,

    work page internal anchor Pith review Pith/arXiv arXiv

  24. [24]

    The Brauer Category and Invariant Theory

    URL: https://arxiv.org/abs/1207.5889, doi:10.4171/ JEMS/558. [Lus10] G. Lusztig. Introduction to quantum groups. Modern Birkhäuser Classics. Birkhäuser/Springer, New York,

    work page internal anchor Pith review Pith/arXiv arXiv

  25. [25]

    a user Classics. Birkh\

    Reprint of the 1994 edition.doi:10.1007/978-0-8176-4717-9. [Mac98] S. MacLane. Categories for the working mathematician . Graduate Texts in Mathematics. Springer-Verlag, New York,

    work page doi:10.1007/978-0-8176-4717-9 1994

  26. [26]

    Second edition. [Par94] J.Paradowski.Filtrationsofmodulesoverthequantumalgebra.In Algebraic groups and their gen- eralizations: quantum and infinite-dimensional methods (University Park, PA, 1991), volume 56 of Proc. Sympos. Pure Math., pages 93–108. Amer. Math. Soc., Providence, RI,

    work page 1991

  27. [27]

    [Rin91] C.M. Ringel. The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences.Math. Z., 208(2):209–223, 1991.doi:10.1007/BF02571521. [RT16] D.E.V. Rose and D. Tubbenhauer. Symmetric webs, Jones–Wenzl recursions, andq-Howe duality. Int. Math. Res. Not. IMRN, (17):5249–5290,

    work page doi:10.1007/bf02571521 1991

  28. [28]

    Symmetric webs, Jones-Wenzl recursions and $q$-Howe duality

    URL:https://arxiv.org/abs/1501.00915, doi:10.1093/imrn/rnv302. [RTW32] G. Rumer, E. Teller, and H. Weyl. Eine für die Valenztheorie geeignete Basis der binären Vek- torinvarianten. Nachrichten von der Ges. der Wiss. Zu Göttingen. Math.-Phys. Klasse, pages 498–504,

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/imrn/rnv302

  29. [29]

    1090/tran/7583

    URL: https://arxiv.org/abs/1701.02932, doi:10. 1090/tran/7583. 58 E. BODISH AND D. TUBBENHAUER [Sch01] I. Schur. Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen

    work page arXiv

  30. [30]

    URL:https://gdz.sub.uni-goettingen.de/id/PPN271034092

    Dissertation, Berlin, 76 pages. URL:https://gdz.sub.uni-goettingen.de/id/PPN271034092. [Ste98] J.R. Stembridge. The partial order of dominant weights. Adv. Math., 136(2):340–364, 1998.doi: 10.1006/aima.1998.1736. [SW11] C. Stroppel and B. Webster. Quiver Schur algebras and q-Fock space

    work page doi:10.1006/aima.1998.1736 1998

  31. [31]

    Quiver Schur algebras and q-Fock space

    URL: https:// arxiv.org/abs/1110.1115. [Tak83] M. Takeuchi. Generators and relations for the hyperalgebras of reductive groups. J. Algebra, 85(1):197–212, 1983.doi:10.1016/0021-8693(83)90125-4. [TL71] H.N.V. Temperley and E.H. Lieb. Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular pla...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0021-8693(83)90125-4 1983