pith. sign in

arxiv: 2502.02849 · v2 · submitted 2025-02-05 · 🧮 math.RT · math.CT

Growth Problems for Representations of Finite Monoids

David He , Daniel Tubbenhauer This is my paper

Pith reviewed 2026-05-23 04:34 UTC · model grok-4.3

classification 🧮 math.RT math.CT
keywords finite monoidsrepresentationstensor powersindecomposable summandsasymptotic growthBrauer character tablegroup of unitsgood characteristic
0
0 comments X

The pith

The number of indecomposable summands in tensor powers of finite monoid representations grows asymptotically according to the Brauer character table of the monoid's group of units.

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

The paper conjectures an explicit formula for the asymptotic growth rate of indecomposable direct summands appearing in the successive tensor powers of representations of a finite monoid. The formula is expressed directly in terms of the Brauer character table of the monoid's group of units. The conjecture is established when an extra hypothesis on the monoid is satisfied. Separate exact and asymptotic formulas are supplied for the growth of the total length of these tensor powers when the base field has good characteristic. Concrete computations are carried out for the full transformation monoid, the symmetric inverse monoid, and the monoid of 2-by-2 matrices.

Core claim

A conjecture states that the asymptotic growth rate of the number of indecomposable summands in the tensor powers of representations of finite monoids is given by an expression involving the Brauer character table of the monoid's group of units. The statement is proved under an additional hypothesis. Exact and asymptotic formulas are also derived for the growth rate of the length of the tensor powers over a field of good characteristic, with explicit calculations supplied for the full transformation monoid, the symmetric inverse monoid, and the monoid of all 2-by-2 matrices.

What carries the argument

The Brauer character table of the monoid's group of units, which supplies the numerical data for the conjectured growth rate of indecomposable summands.

If this is right

  • The growth rate of indecomposable summands is computable from the character table once the extra hypothesis holds.
  • The length of tensor powers admits both exact and asymptotic formulas in good characteristic.
  • Explicit numerical growth rates are obtained for the full transformation monoid, the symmetric inverse monoid, and the 2-by-2 matrix monoid.

Where Pith is reading between the lines

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

  • The same character-table method might be tested on other families of finite monoids whose representations are already classified.
  • If the hypothesis can be removed or weakened, the conjecture would apply to a wider class of monoids without further restrictions.
  • The length formulas could be used to bound the dimension of the representation ring generated by a single module under tensor product.

Load-bearing premise

The conjecture requires an additional hypothesis on the monoid whose precise statement is not given in the abstract.

What would settle it

Compute the actual number of indecomposable summands in high tensor powers for a finite monoid that satisfies all stated conditions yet produces a growth rate different from the value predicted by its group of units' Brauer character table.

Figures

Figures reproduced from arXiv: 2502.02849 by Daniel Tubbenhauer, David He.

Figure 1
Figure 1. Figure 1: The fusion graph and action matrix for V = Y ⊕ X. The vertex colored in cyan is the null representation. In this case a(n) = 3n. ✸ 3. Main results 3.1. Counting summands. This section focuses on the study of b(n). As before, if V is a kM-module let ResG(V ) denote the kG-module that comes from restriction of V . If W is a kG-module, let IndG(W) denote the induced kM-module on which elements in M \ G act as… view at source ↗
Figure 3
Figure 3. Figure 3: The ratio b(n)/a(n) for the 12-dimensional projective kT4-module in characteristic 0. Here a(n) = 5/12 · 12n and λ sec = 2 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: The fusion graph and action matrix for the 12-dimensional projective kT4-module in characteristic 0. The modules are labelled with their dimensions, and the projective cell Γ G is coloured in cyan. 4.2. Matrix monoids M(2, q). Let M(2, q) = M(2, Fq) denote the monoid of 2 × 2 matrices with entries in Fq, and let G = GL(2, q) be its group of units. We know from [Kov92] that when p ∤ |G| = (q 2 − 1)(q 2 − q)… view at source ↗
Figure 4
Figure 4. Figure 4: The fusion graph and action matrix for the four-dimensional projective k M(2, 3)- module in characteristic 3. The modules are labelled with their dimensions, and the projective cell Γ G is coloured in cyan [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The ratio b(n)/a(n) for the four-dimensional projective k M(2, 3)-module in char￾acteristic 3. Here a(n) = (1/4 + 1/12(−1)n) · 4 n and λ sec = 1. 4.3. An example where Theorem 1 does not apply. Condition ii) in the statement of Theorem 1 is somewhat restrictive: it is already not satisfied when the monoid is N = {1, x, 0} where x 2 = 0, and of the 35 nonisomorphic monoids of order 4, 9 do not satisfy the c… view at source ↗
Figure 6
Figure 6. Figure 6: The fusion graph and action matrix for the four-dimensional projective k(T3×N)- module V , in characteristic 31. The modules are labelled with their dimensions, and the projective cell Γ G is coloured in cyan. References [BK72] R.M. Bryant and L.G. Kovács. Tensor products of representations of finite groups. Bull. London Math. Soc., 4:133–135, 1972. doi:10.1112/blms/4.2.133. [CEO24] K. Coulembier, P. Eting… view at source ↗
read the original abstract

We give a conjecture for the asymptotic growth rate of the number of indecomposable summands in the tensor powers of representations of finite monoids, expressing it in terms of the (Brauer) character table of the monoid's group of units. We prove it under an additional hypothesis. We also give (exact and asymptotic) formulas for the growth rate of the length of the tensor powers when working over a good characteristic. As examples, we compute the growth rates for the full transformation monoid, the symmetric inverse monoid, and the monoid of 2 by 2 matrices. We also provide code used for our calculation.

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

2 major / 2 minor

Summary. The manuscript conjectures an explicit formula, in terms of the Brauer character table of the group of units, for the asymptotic growth rate of the number of indecomposable summands appearing in tensor powers of representations of finite monoids. The conjecture is proved under an additional hypothesis whose precise statement is supplied in the body of the paper. Exact and asymptotic formulas are derived for the growth of the length of these tensor powers when the base field has good characteristic. Explicit computations of the growth rates are carried out for the full transformation monoid, the symmetric inverse monoid, and the monoid of 2-by-2 matrices, and the code used for the calculations is included.

Significance. If the conjecture holds in the generality claimed, the result would supply a practical computational bridge between the representation theory of monoids and the ordinary character theory of their groups of units. The length formulas and the three worked examples already furnish concrete data that can be checked independently, and the provision of code supports reproducibility.

major comments (2)
  1. [statement of the conjecture and the examples section] The additional hypothesis required for the proof of the main conjecture is stated in the body of the paper, but its verification for the three explicit examples (full transformation monoid, symmetric inverse monoid, 2-by-2 matrix monoid) is not carried out or even asserted; without this check the computational evidence does not directly corroborate the conjecture.
  2. [conjecture statement] The conjecture expresses the growth rate directly in terms of the Brauer character table, yet the manuscript does not supply a self-contained algorithm or worked numerical example showing how the table entries are converted into the predicted growth rate when the hypothesis is assumed; this step is load-bearing for any reader wishing to test the formula on further monoids.
minor comments (2)
  1. The term 'good characteristic' is used without an early definition or reference; a one-sentence clarification in the introduction would help readers outside the immediate subfield.
  2. The code is a positive feature; ensure that any repository link or supplementary file is permanently archived rather than relying on a transient URL.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address the two major comments point by point below and will revise the manuscript accordingly to improve clarity and strengthen the presentation of the evidence.

read point-by-point responses

  1. Referee: The additional hypothesis required for the proof of the main conjecture is stated in the body of the paper, but its verification for the three explicit examples (full transformation monoid, symmetric inverse monoid, 2-by-2 matrix monoid) is not carried out or even asserted; without this check the computational evidence does not directly corroborate the conjecture.

    Authors: We agree that the manuscript would be strengthened by explicitly verifying the additional hypothesis for each of the three examples. In the revised version we will add a dedicated subsection confirming that the hypothesis holds for the full transformation monoid, the symmetric inverse monoid, and the monoid of 2-by-2 matrices, thereby making the computational evidence directly supportive of the conjecture. revision: yes

  2. Referee: The conjecture expresses the growth rate directly in terms of the Brauer character table, yet the manuscript does not supply a self-contained algorithm or worked numerical example showing how the table entries are converted into the predicted growth rate when the hypothesis is assumed; this step is load-bearing for any reader wishing to test the formula on further monoids.

    Authors: We acknowledge that an explicit, self-contained procedure for converting Brauer character table entries into the predicted growth rate would improve accessibility. In the revision we will insert a new subsection containing a step-by-step algorithm for this conversion (under the standing hypothesis) together with a short worked numerical example drawn from one of the computed monoids. revision: yes

Circularity Check

0 steps flagged

Conjecture stated via independent Brauer character table; no reduction to inputs by construction

full rationale

The paper states a conjecture expressing asymptotic growth rates in terms of the Brauer character table of the monoid's group of units, an externally defined object. It provides a conditional proof under an additional hypothesis and computational examples for specific monoids, along with exact/asymptotic formulas over good characteristic. No step equates a claimed prediction or result to a fitted parameter, self-defined quantity, or load-bearing self-citation chain. The derivation chain remains independent of its target outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the conjecture is expressed directly in terms of the already-known Brauer character table.

pith-pipeline@v0.9.0 · 5623 in / 1216 out tokens · 19899 ms · 2026-05-23T04:34:22.740339+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Growth problems in diagram categories

    math.RT 2025-03 unverdicted novelty 4.0

    Derives asymptotic formulas for the growth rate of the number of summands in tensor powers of the generating object in semisimple diagram/interpolation categories.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper · 2 internal anchors

  1. [1]

    London Math

    [BK72] R.M.BryantandL.G.Kovács.Tensorproductsofrepresentationsoffinitegroups.Bull. London Math. Soc., 4:133–135, 1972.doi:10.1112/blms/4.2.133. [CEO24] K. Coulembier, P. Etingof, and V. Ostrik. Asymptotic properties of tensor powers in symmetric tensor categories. Pure Appl. Math. Q., 20(3):1141–1179,

    work page doi:10.1112/blms/4.2.133 1972

  2. [2]

    v20.n3.a4

    URL:https://arxiv.org/abs/2301.09804,doi:10.4310/pamq.2024. v20.n3.a4. [CEOT25] K. Coulembier, P. Etingof, V. Ostrik, and D. Tubbenhauer. Fractal behavior of tensor powers of the two dimen- sional space in prime characteristic.Contemp. Math., Volume: 829; 2025; 291 pp, American Mathematical Society, [Providence], RI, [2025],©2025.https://arxiv.org/abs/240...

    work page doi:10.4310/pamq.2024 2024

  3. [3]

    [FH91] W

    URL:https://arxiv.org/abs/2301.00885,doi:10.1007/ s10468-023-10245-7. [FH91] W. Fulton and J. Harris.Representation theory, volume 129 ofGraduate Texts in Mathematics. Springer-Verlag, New York,

    work page arXiv

  4. [4]

    [He25] D

    A first course, Readings in Mathematics.doi:10.1007/978-1-4612-0979-9. [He25] D. He. Growth problems for representations of finite groups

    work page doi:10.1007/978-1-4612-0979-9

  5. [5]

    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

  6. [6]

    [Kou93] F.M

    URL:https://arxiv.org/abs/2201.01805,doi:10.1090/btran/151. [Kou93] F.M. Kouwenhoven. Indecomposable representations ofM(2,F q)overF q.J. Algebra, 155(2):369–396, 1993.doi: 10.1006/jabr.1993.1050. GROWTH PROBLEMS FOR REPRESENTATIONS OF FINITE MONOIDS 9 [Kov92] L.G. Kovács. Semigroup algebras of the full matrix semigroup over a finite field.Proc. Amer. Mat...

    work page doi:10.1090/btran/151 1993

  7. [7]

    [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

  8. [8]

    Asymptotics in infinite monoidal categories

    To appear in High. Struct.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

  9. [9]

    [Lar24] M.J

    URL:https://arxiv.org/abs/2404.03933. [Lar24] M.J. Larsen. Bounds forSL 2-indecomposables in tensor powers of the natural representation in characteristic2

    work page arXiv

  10. [10]

    [Ste16b] B

    URL:https://arxiv.org/abs/2405.16015. [Ste16b] B. Steinberg.Representation theory of finite monoids. Universitext. Springer, Cham, 2016.doi:10.1007/ 978-3-319-43932-7. [Ste16a] B. Steinberg. The global dimension of the full transformation monoid (with an appendix by V. Mazorchuk and B. Steinberg).Algebr. Represent. Theory, 19(3):731–747,

    work page arXiv 2016

  11. [11]

    The global dimension of the full transformation monoid with an appendix by V. Mazorchuk and B. Steinberg

    URL:https://arxiv.org/abs/1502.00959,doi:10. 1007/s10468-016-9597-4. [Ste23] B. Steinberg. The modular representation theory of monoids

    work page internal anchor Pith review Pith/arXiv arXiv

  12. [12]

    URL:https://arxiv.org/abs/2305.08251. D. H.: The University of Sydney, School of Mathematics and Statistics, Australia Email address:d.he@sydney.edu.au D.T.: The University of Sydney, School of Mathematics and Statistics F07, Office Carsla w 827, NSW 2006, Australia, www.dtubbenhauer.com, ORCID 0000-0001-7265-5047 Email address:daniel.tubbenhauer@sydney.edu.au

    work page arXiv 2006