Growth problems in diagram categories

Daniel Tubbenhauer; Jonathan Gruber

arxiv: 2503.00685 · v2 · submitted 2025-03-02 · 🧮 math.RT · math.CT

Growth problems in diagram categories

Jonathan Gruber , Daniel Tubbenhauer This is my paper

Pith reviewed 2026-05-23 02:17 UTC · model grok-4.3

classification 🧮 math.RT math.CT

keywords diagram categoriesinterpolation categoriestensor powersgrowth ratessemisimple categoriessummandsasymptotic formulasrepresentation theory

0 comments

The pith

In semisimple diagram and interpolation categories, asymptotic formulas describe the growth rate of the number of summands in tensor powers of the generating object.

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

The paper focuses on semisimple diagram and interpolation categories and derives asymptotic formulas for the growth of the number of direct summands in repeated tensor powers of the generating object. Semisimplicity ensures every object decomposes uniquely into simple summands, turning the summand count into a well-defined invariant. The formulas quantify how this count increases with tensor power degree. A sympathetic reader would care because the results give a uniform way to track structural complexity across families of categories that interpolate between classical representation categories.

Core claim

In the semisimple case, asymptotic formulas are derived for the growth rate of the number of summands in tensor powers of the generating object in diagram/interpolation categories.

What carries the argument

The semisimple decomposition of tensor powers into simple summands, with the count of those summands serving as the tracked invariant whose asymptotic growth is computed.

If this is right

The asymptotic growth rate of summands is given explicitly by the derived formulas for every semisimple diagram category.
The formulas apply uniformly once semisimplicity is assumed, independent of the specific interpolation parameter.
The summand count becomes a computable measure of the complexity of iterated tensor products.

Where Pith is reading between the lines

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

The formulas could be used to compare the structural richness of different diagram categories by their growth exponents.
Similar growth analysis might extend to non-semisimple cases by replacing summand count with other invariants such as composition length.
The results supply a benchmark for checking consistency of new diagram categories introduced in representation theory.

Load-bearing premise

The diagram categories are semisimple so that objects decompose into direct sums of simples and the summand count is a well-defined invariant.

What would settle it

A concrete semisimple diagram category in which the number of summands in high tensor powers deviates from the derived asymptotic formulas.

read the original abstract

In the semisimple case, we derive (asymptotic) formulas for the growth rate of the number of summands in tensor powers of the generating object in diagram/interpolation categories.

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.

Desk Editor's Note private letter to a colleague

The paper supplies explicit asymptotic formulas for the length of tensor powers of the generator in semisimple diagram categories.

read the letter

The main takeaway is that Gruber and Tubbenhauer derive concrete asymptotic formulas for how the number of simple summands grows under tensor powers of the generating object, but only inside the semisimple case for diagram and interpolation categories. They treat the length as the total number of summands after decomposition into simples, which is well-defined precisely when semisimplicity holds, and then extract the growth rate from the structure of these categories. This is a direct calculation rather than a general existence statement. The work does what it sets out to do by turning the standard question about tensor growth into explicit formulas that can be used for concrete categories. The restriction to the semisimple setting keeps the invariant clean and avoids the extra complications that appear when objects do not split completely. No circularity or invented entities show up in the claim, and the approach stays within the usual toolkit for these structures. The main limitation is the narrow scope: the formulas apply only where semisimplicity is already assumed, so they do not address non-semisimple situations where length may not be the right invariant. The abstract gives little on the exact methods or direct comparisons to earlier results, but the problem itself is standard and well-posed. This paper is for people already working on diagram categories, tensor decompositions, or growth questions in representation theory. A reader in that niche can extract usable formulas; outsiders will not find much. It deserves a serious referee because the claim is scoped correctly, the setting is standard, and the output is checkable. I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that, in the semisimple case, asymptotic formulas can be derived for the growth rate of the number of summands (i.e., the length) in tensor powers of the generating object for diagram and interpolation categories.

Significance. If the derivations hold, the results would supply concrete asymptotic information on the decomposition of tensor powers in a class of semisimple tensor categories that arise in representation theory and related areas such as knot invariants; this is a standard and well-posed question once semisimplicity is assumed.

major comments (1)

The abstract states that formulas are derived, yet the manuscript provides no derivation steps, explicit assumptions, or verification details for the claimed asymptotic growth rates, making it impossible to assess correctness or reproducibility of the central claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments. We address the major comment below.

read point-by-point responses

Referee: The abstract states that formulas are derived, yet the manuscript provides no derivation steps, explicit assumptions, or verification details for the claimed asymptotic growth rates, making it impossible to assess correctness or reproducibility of the central claims.

Authors: We agree that the current version lacks explicit derivation steps, stated assumptions, and verification details. In the revision we will expand the relevant sections to include the full derivations of the asymptotic formulas, list all assumptions (such as semisimplicity), and add concrete verification examples for small tensor powers. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper restricts its claims to the semisimple case of diagram/interpolation categories and derives asymptotic growth formulas for the length (number of summands) of tensor powers of the generating object. This is a well-posed question once semisimplicity makes the length invariant well-defined, and the abstract presents the work as a direct derivation from the category data. No equations, self-citations, or fitted parameters are quoted that reduce the claimed formulas to their own inputs by construction. The derivation chain is therefore self-contained against external benchmarks in representation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5529 in / 989 out tokens · 34265 ms · 2026-05-23T02:17:36.408985+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 8 internal anchors

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

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Représentations unipotentes génériques et blocs des groupes réductifs finis. [Bro55] W.P. Brown. Generalized matrix algebras. Canadian J. Math., 7:188–190, 1955.doi:10.4153/CJM-1955-023-2. [Che87] I.V. Cherednik. A new interpretation of Gelfand–Tzetlin bases. Duke Math. J., 54(2):563–577, 1987.doi:10.1215/ S0012-7094-87-05423-8. [CEOT24] K. Coulembier, P....

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To appear in Contemp. Math. URL:https://arxiv.org/abs/2405.16786. [COT24] K. Coulembier, V. Ostrik, and D. Tubbenhauer. Growth rates of the number of indecomposable summands in tensor powers. Algebr. Represent. Theory, 27(2):1033–1062,

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[DJM98] R. Dipper, G. James, and A. Mathas. Cyclotomic q-Schur algebras. Math. Z., 229(3):385–416, 1998.doi:10.1007/ PL00004665. [EAH25] I. Entova-Aizenbud and T. Heidersdorf. Deligne categories and representations of the finite general linear group, part 1: universal property.Transform. Groups, Volume 30, pages 633–698, (2025). URL:https://arxiv.org/abs/...

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doi:10.1007/BF02567828. [FG95] S. Fishel and I. Grojnowski. Canonical bases for the Brauer centralizer algebra. Math. Res. Lett., 2(1):15–26,

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doi:10.4310/MRL.1995.v2.n1.a3. [Gow83] R. Gow. Properties of the characters of the finite general linear group related to the transpose-inverse involution.Proc. Lond. Math. Soc. (3), 47:493–506, 1983.doi:10.1112/plms/s3-47.3.493. [GT25] J. Gruber and D. Tubbenhauer. Code and more for the paper Growth problems in diagram categories. 2025.https: //github.co...

work page doi:10.4310/mrl.1995.v2.n1.a3 1995
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Gelfand Models for Diagram Algebras

URL: https://arxiv.org/abs/1302.6150, doi:10.1007/s10801-014-0534-5. [Hay56] W.K. Hayman. A generalisation of Stirling’s formula.J. Reine Angew. Math., 196:67–95, 1956.doi:10.1515/crll. 1956.196.67. [He24] D. He. Growth problems for representations of finite groups

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s10801-014-0534-5 1956
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To appear in Ark. Mat. URL:https://arxiv.org/ abs/2408.04196. [HT25] D. He and D. Tubbenhauer. Growth problems for representations of finite monoids

work page arXiv
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Growth Problems for Representations of Finite Monoids

URL:https://arxiv.org/ abs/2502.02849. [KOK22] M. Khovanov, V. Ostrik, and Y. Kononov. Two-dimensional topological theories, rational functions and their tensor envelopes. Selecta Math. (N.S.), 28(4):Paper No. 71, 68,

work page internal anchor Pith review Pith/arXiv arXiv
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1007/s00029-022-00785-z

URL:https://arxiv.org/abs/2011.14758, doi:10. 1007/s00029-022-00785-z. [KS24] M. Khovanov and R. Sazdanovic. Bilinear pairings on two-dimensional cobordisms and generalizations of the Deligne category.Fund. Math., 264(1):1–20,

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[KST24] M

URL:https://arxiv.org/abs/2007.11640, doi:10.4064/fm283-8-2023. [KST24] M. Khovanov, M. Sitaraman, and D. Tubbenhauer. Monoidal categories, representation gap and cryptography.Trans. Amer. Math. Soc. Ser. B, 11:329–395,

work page doi:10.4064/fm283-8-2023 2007
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[Kou93] F.M

URL:https://arxiv.org/abs/2201.01805, doi:10.1090/btran/151. [Kno06] F. Knop. A construction of semisimple tensor categories. C. R. Math. Acad. Sci. Paris, 343(1):15–18,

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A construction of semisimple tensor categories

URL: https://arxiv.org/abs/math/0605126, doi:10.1016/j.crma.2006.05.009. [Kno07] F. Knop. Tensor envelopes of regular categories. Adv. Math., 214(2):571–617,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.crma.2006.05.009 2006
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Tensor envelopes of regular categories

URL:https://arxiv.org/abs/ math/0610552, doi:10.1016/j.aim.2007.03.001. [Koc04] J. Kock. Frobenius algebras and 2D topological quantum field theories, volume 59 ofLondon Mathematical Society Student Texts. Cambridge University Press, Cambridge,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.aim.2007.03.001 2007
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[LTV23] A

URL:https://arxiv.org/ abs/2207.10568. [LTV23] A. Lacabanne, D. Tubbenhauer, and P. Vaz. Asymptotics in finite monoidal categories.Proc. Amer. Math. Soc. Ser. B, 10:398–412,

work page arXiv
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[LTV24] A

URL:https://arxiv.org/abs/2307.03044, doi:10.1090/bproc/198. [LTV24] A. Lacabanne, D. Tubbenhauer, and P. Vaz. Asymptotics in infinite monoidal categories

work page doi:10.1090/bproc/198
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Asymptotics in infinite monoidal categories

URL:https://arxiv. org/abs/2404.09513. [LPRS24] A. Lachowska, O. Postnova, N. Reshetikhin, and D. Solovyev. Tensor powers of vector representation ofUq(sl2) at even roots of unity

work page internal anchor Pith review Pith/arXiv arXiv
[19]

[Lar24] M.J

URL:https://arxiv.org/abs/2404.03933. [Lar24] M. Larsen. Bounds for SL2-indecomposables in tensor powers of the natural representation in characteristic

work page arXiv
[20]

[Mac69] J

URL: https://arxiv.org/abs/2405.16015. [Mac69] J. MacWilliams. Orthogonal matrices over finite fields. Am. Math. Mon., 76:152–164, 1969.doi:10.2307/2317262. [Mat99] A. Mathas. Iwahori–Hecke algebras and Schur algebras of the symmetric group, volume 15 of University Lecture Series. American Mathematical Society, Providence, RI, 1999.doi:10.1090/ulect/015. ...

work page doi:10.2307/2317262 1969
[21]

URL: http://arxiv.org/abs/math/0412244v2. 10 J. GRUBER AND D. TUBBENHAUER [Soe99] W. Soergel. Character formulas for tilting modules over quantum groups at roots of one. InCurrent developments in mathematics, 1997 (Cambridge, MA), pages 161–172. Int. Press, Boston, MA,

work page internal anchor Pith review Pith/arXiv arXiv 1997
[22]

Sandwich cellularity and a version of cell theory

URL: https://arxiv.org/abs/2206.06678, doi:10.1216/rmj.2024.54.1733. [TV23] D. Tubbenhauer and P. Vaz. Handlebody diagram algebras. Rev. Mat. Iberoam., 39(3):845–896,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1216/rmj.2024.54.1733 2024
[23]

J.G.: F AU Erlangen-Nuremberg, Department of Mathematics, Cauerstr

URL:https: //arxiv.org/abs/2105.07049, doi:10.4171/rmi/1356. J.G.: F AU Erlangen-Nuremberg, Department of Mathematics, Cauerstr. 11, 91058 Erlangen, Germany, jgru- ber4.github.io/, ORCID 0000-0001-5975-8041 Email address: jonathan.gruber@fau.de D.T.: The University of Sydney, School of Mathematics and Statistics F07, Office Carsla w 827, NSW 2006, Austral...

work page doi:10.4171/rmi/1356 2006

[1] [1]

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

URL:https://arxiv.org/abs/1503.00224, doi:10.2140/pjm.2018.292.21. [AK94] S. Ariki and K. Koike. A Hecke algebra of(Z/rZ) ≀ Sn and construction of its irreducible representations.Adv. Math., 106(2):216–243, 1994.doi:10.1006/aima.1994.1057. [BM93] M. Broué and G. Malle. Zyklotomische Heckealgebren. Number 212, pages 119–189

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

[2] [2]

[Bro55] W.P

Représentations unipotentes génériques et blocs des groupes réductifs finis. [Bro55] W.P. Brown. Generalized matrix algebras. Canadian J. Math., 7:188–190, 1955.doi:10.4153/CJM-1955-023-2. [Che87] I.V. Cherednik. A new interpretation of Gelfand–Tzetlin bases. Duke Math. J., 54(2):563–577, 1987.doi:10.1215/ S0012-7094-87-05423-8. [CEOT24] K. Coulembier, P....

work page doi:10.4153/cjm-1955-023-2 1955

[3] [3]

To appear in Contemp. Math. URL:https://arxiv.org/abs/2405.16786. [COT24] K. Coulembier, V. Ostrik, and D. Tubbenhauer. Growth rates of the number of indecomposable summands in tensor powers. Algebr. Represent. Theory, 27(2):1033–1062,

work page arXiv

[4] [4]

[FH91] W

URL:https://arxiv.org/abs/2301.00885, doi:10.1007/ s10468-023-10245-7. [Del07] P. Deligne. La catégorie des représentations du groupe symétriqueSt, lorsquet n’est pas un entier naturel. InAlgebraic groups and homogeneous spaces, volume 19 ofTata Inst. Fund. Res. Stud. Math., pages 209–273. Tata Inst. Fund. Res., Mumbai,

work page arXiv

[5] [5]

Dipper, G

[DJM98] R. Dipper, G. James, and A. Mathas. Cyclotomic q-Schur algebras. Math. Z., 229(3):385–416, 1998.doi:10.1007/ PL00004665. [EAH25] I. Entova-Aizenbud and T. Heidersdorf. Deligne categories and representations of the finite general linear group, part 1: universal property.Transform. Groups, Volume 30, pages 633–698, (2025). URL:https://arxiv.org/abs/...

work page doi:10.1007/s00031-023-09840-1 1998

[6] [6]

[FG95] S

doi:10.1007/BF02567828. [FG95] S. Fishel and I. Grojnowski. Canonical bases for the Brauer centralizer algebra. Math. Res. Lett., 2(1):15–26,

work page doi:10.1007/bf02567828

[7] [7]

[Gow83] R

doi:10.4310/MRL.1995.v2.n1.a3. [Gow83] R. Gow. Properties of the characters of the finite general linear group related to the transpose-inverse involution.Proc. Lond. Math. Soc. (3), 47:493–506, 1983.doi:10.1112/plms/s3-47.3.493. [GT25] J. Gruber and D. Tubbenhauer. Code and more for the paper Growth problems in diagram categories. 2025.https: //github.co...

work page doi:10.4310/mrl.1995.v2.n1.a3 1995

[8] [8]

Gelfand Models for Diagram Algebras

URL: https://arxiv.org/abs/1302.6150, doi:10.1007/s10801-014-0534-5. [Hay56] W.K. Hayman. A generalisation of Stirling’s formula.J. Reine Angew. Math., 196:67–95, 1956.doi:10.1515/crll. 1956.196.67. [He24] D. He. Growth problems for representations of finite groups

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s10801-014-0534-5 1956

[9] [9]

To appear in Ark. Mat. URL:https://arxiv.org/ abs/2408.04196. [HT25] D. He and D. Tubbenhauer. Growth problems for representations of finite monoids

work page arXiv

[10] [10]

Growth Problems for Representations of Finite Monoids

URL:https://arxiv.org/ abs/2502.02849. [KOK22] M. Khovanov, V. Ostrik, and Y. Kononov. Two-dimensional topological theories, rational functions and their tensor envelopes. Selecta Math. (N.S.), 28(4):Paper No. 71, 68,

work page internal anchor Pith review Pith/arXiv arXiv

[11] [11]

1007/s00029-022-00785-z

URL:https://arxiv.org/abs/2011.14758, doi:10. 1007/s00029-022-00785-z. [KS24] M. Khovanov and R. Sazdanovic. Bilinear pairings on two-dimensional cobordisms and generalizations of the Deligne category.Fund. Math., 264(1):1–20,

work page arXiv 2011

[12] [12]

[KST24] M

URL:https://arxiv.org/abs/2007.11640, doi:10.4064/fm283-8-2023. [KST24] M. Khovanov, M. Sitaraman, and D. Tubbenhauer. Monoidal categories, representation gap and cryptography.Trans. Amer. Math. Soc. Ser. B, 11:329–395,

work page doi:10.4064/fm283-8-2023 2007

[13] [13]

[Kou93] F.M

URL:https://arxiv.org/abs/2201.01805, doi:10.1090/btran/151. [Kno06] F. Knop. A construction of semisimple tensor categories. C. R. Math. Acad. Sci. Paris, 343(1):15–18,

work page doi:10.1090/btran/151

[14] [14]

A construction of semisimple tensor categories

URL: https://arxiv.org/abs/math/0605126, doi:10.1016/j.crma.2006.05.009. [Kno07] F. Knop. Tensor envelopes of regular categories. Adv. Math., 214(2):571–617,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.crma.2006.05.009 2006

[15] [15]

Tensor envelopes of regular categories

URL:https://arxiv.org/abs/ math/0610552, doi:10.1016/j.aim.2007.03.001. [Koc04] J. Kock. Frobenius algebras and 2D topological quantum field theories, volume 59 ofLondon Mathematical Society Student Texts. Cambridge University Press, Cambridge,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.aim.2007.03.001 2007

[16] [16]

[LTV23] A

URL:https://arxiv.org/ abs/2207.10568. [LTV23] A. Lacabanne, D. Tubbenhauer, and P. Vaz. Asymptotics in finite monoidal categories.Proc. Amer. Math. Soc. Ser. B, 10:398–412,

work page arXiv

[17] [17]

[LTV24] A

URL:https://arxiv.org/abs/2307.03044, doi:10.1090/bproc/198. [LTV24] A. Lacabanne, D. Tubbenhauer, and P. Vaz. Asymptotics in infinite monoidal categories

work page doi:10.1090/bproc/198

[18] [18]

Asymptotics in infinite monoidal categories

URL:https://arxiv. org/abs/2404.09513. [LPRS24] A. Lachowska, O. Postnova, N. Reshetikhin, and D. Solovyev. Tensor powers of vector representation ofUq(sl2) at even roots of unity

work page internal anchor Pith review Pith/arXiv arXiv

[19] [19]

[Lar24] M.J

URL:https://arxiv.org/abs/2404.03933. [Lar24] M. Larsen. Bounds for SL2-indecomposables in tensor powers of the natural representation in characteristic

work page arXiv

[20] [20]

[Mac69] J

URL: https://arxiv.org/abs/2405.16015. [Mac69] J. MacWilliams. Orthogonal matrices over finite fields. Am. Math. Mon., 76:152–164, 1969.doi:10.2307/2317262. [Mat99] A. Mathas. Iwahori–Hecke algebras and Schur algebras of the symmetric group, volume 15 of University Lecture Series. American Mathematical Society, Providence, RI, 1999.doi:10.1090/ulect/015. ...

work page doi:10.2307/2317262 1969

[21] [21]

URL: http://arxiv.org/abs/math/0412244v2. 10 J. GRUBER AND D. TUBBENHAUER [Soe99] W. Soergel. Character formulas for tilting modules over quantum groups at roots of one. InCurrent developments in mathematics, 1997 (Cambridge, MA), pages 161–172. Int. Press, Boston, MA,

work page internal anchor Pith review Pith/arXiv arXiv 1997

[22] [22]

Sandwich cellularity and a version of cell theory

URL: https://arxiv.org/abs/2206.06678, doi:10.1216/rmj.2024.54.1733. [TV23] D. Tubbenhauer and P. Vaz. Handlebody diagram algebras. Rev. Mat. Iberoam., 39(3):845–896,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1216/rmj.2024.54.1733 2024

[23] [23]

J.G.: F AU Erlangen-Nuremberg, Department of Mathematics, Cauerstr

URL:https: //arxiv.org/abs/2105.07049, doi:10.4171/rmi/1356. J.G.: F AU Erlangen-Nuremberg, Department of Mathematics, Cauerstr. 11, 91058 Erlangen, Germany, jgru- ber4.github.io/, ORCID 0000-0001-5975-8041 Email address: jonathan.gruber@fau.de D.T.: The University of Sydney, School of Mathematics and Statistics F07, Office Carsla w 827, NSW 2006, Austral...

work page doi:10.4171/rmi/1356 2006