Metadata
bibkey: he2025tensor
authors: David He and Daniel Tubbenhauer
year: 2025
title: Tensor powers of representations of (diagram) monoids
doi: 10.48550/arXiv.2508.04054
url: https://arxiv.org/abs/2508.04054v1
claim: Conjecture 4.2 asserts that every singular finite-semigroup module has a one-dimensional null direct summand in some positive tensor power, and its indecomposable-summand count is asymptotic to the tensor dimension.
strata_touched:
- D5/S3/HomologicalAlgebra/TensorNullSummand
license: citation-only
triage: anchor
Verified locator
DOI: https://doi.org/10.48550/arXiv.2508.04054
Source: https://arxiv.org/abs/2508.04054v1
Source statements
Conjecture 4.2, page 17:
Let V be a kS-module on which no element acts invertibly, then there is some m ≥ 1 such that Z is a summand of V^{⊗m}, and consequently a(n)=(dim V)ⁿ.
Section 1, page 1:
b(n)=b^{M,V}(n)=# M-indecomposable summands of V^{⊗n} (counted with multiplicity).
Section 4, page 17:
Unlike in the monoid case, the null representation Z with all elements in S acting as 0 is admitted as a kS-module, and it forms a tensor ideal (in the sense that V⊗Z=Z^{⊕dim V} for all V.)
The field is algebraically closed with arbitrary characteristic. Semigroups are finite and representations are finite dimensional. Section 4 permits nonunital modules for semigroups without identity. The null representation Z is the one-dimensional module with zero action. Section 1 uses a(n) for an asymptotic equivalent of the number b(n) of indecomposable summands, including multiplicity; the displayed formula for a(n) is a ratio-limit assertion.
Relation to the formal statements
D5/S3/HomologicalAlgebra/TensorNullSummand expresses a null direct summand by a one-dimensional zero-action subspace with an invariant complementary subspace. Indecomposability is stated in the ambient submodule lattice, and the number of summands is expressed through an internal direct sum indexed by Fin b. The diagonal tensor action uses PiTensorProduct.map. The claim includes both existence of a count sequence and the limit for every count sequence.
The proof uses a complement of the intersection of the common kernel and common image, bounds their dimensions by sums of operator ranks, and uses geometric decay of tensor ranks. Its private dimension and convergence bounds require neither algebraic closure nor absence of a semigroup identity. The universal Jordan-family sharpness formula, a general infinite-semigroup extension and the separate invertible-action monoid conjecture remain open here.
Literature boundary
The bounded literature screen recorded in issue #22531 found no settlement of Conjecture 4.2. Global publication and citation-index completeness are ASSUMED-UNVERIFIED. The restricted right-idempotent-generation results in arXiv:2512.05510v3 do not establish the general assertion.
Independent finite check
The exact checker is in the-omega-institute/trureturing-experiments, commit 93806b3096b542f57f6884b2c30f723a225ccb43, at docs/reports/semigroup-tensor-null-summand/check.py (SHA-256 50e965a51db43be71be3771da00239828e289983585bc4cba17f24d22db5b0b6). The command python3 docs/reports/semigroup-tensor-null-summand/check.py exits 0 and ends with ALL_CHECKS_OK. It checks 1023 nonempty singular GF(2) matrix families in 3069 tensor cases and 25 Jordan-chain cases. These finite results support the construction; the universal conclusion is supplied by Lean.
Tensor-dimension supplier
The private tensor-dimension proof specializes CFSG.LinearAlgebra.finrank_piTensorProduct from AxiomMath/CFSG, immutable revision 69cb972f587a3ecc035e092253167ff2e303c962. Its field and vector spaces are instantiated with k and the constant family on Fin n, and the finite product is normalized to a power. The source leaf has no author or copyright header. Its root license is Apache-2.0, reproduced below; there is no root NOTICE file at this revision. This attribution is for the dimension supplier, not for the null-summand theorem. Replace the adapted supplier by an import when this repository’s pinned Mathlib contains an equivalent finite tensor-dimension theorem.
Supplier license
Apache License
Version 2.0, January 2004
http://www.apache.org/licenses/
TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION
1. Definitions.
"License" shall mean the terms and conditions for use, reproduction,
and distribution as defined by Sections 1 through 9 of this document.
"Licensor" shall mean the copyright owner or entity authorized by
the copyright owner that is granting the License.
"Legal Entity" shall mean the union of the acting entity and all
other entities that control, are controlled by, or are under common
control with that entity. For the purposes of this definition,
"control" means (i) the power, direct or indirect, to cause the
direction or management of such entity, whether by contract or
otherwise, or (ii) ownership of fifty percent (50%) or more of the
outstanding shares, or (iii) beneficial ownership of such entity.
"You" (or "Your") shall mean an individual or Legal Entity
exercising permissions granted by this License.
"Source" form shall mean the preferred form for making modifications,
including but not limited to software source code, documentation
source, and configuration files.
"Object" form shall mean any form resulting from mechanical
transformation or translation of a Source form, including but
not limited to compiled object code, generated documentation,
and conversions to other media types.
"Work" shall mean the work of authorship, whether in Source or
Object form, made available under the License, as indicated by a
copyright notice that is included in or attached to the work
(an example is provided in the Appendix below).
"Derivative Works" shall mean any work, whether in Source or Object
form, that is based on (or derived from) the Work and for which the
editorial revisions, annotations, elaborations, or other modifications
represent, as a whole, an original work of authorship. For the purposes
of this License, Derivative Works shall not include works that remain
separable from, or merely link (or bind by name) to the interfaces of,
the Work and Derivative Works thereof.
"Contribution" shall mean any work of authorship, including
the original version of the Work and any modifications or additions
to that Work or Derivative Works thereof, that is intentionally
submitted to Licensor for inclusion in the Work by the copyright owner
or by an individual or Legal Entity authorized to submit on behalf of
the copyright owner. For the purposes of this definition, "submitted"
means any form of electronic, verbal, or written communication sent
to the Licensor or its representatives, including but not limited to
communication on electronic mailing lists, source code control systems,
and issue tracking systems that are managed by, or on behalf of, the
Licensor for the purpose of discussing and improving the Work, but
excluding communication that is conspicuously marked or otherwise
designated in writing by the copyright owner as "Not a Contribution."
"Contributor" shall mean Licensor and any individual or Legal Entity
on behalf of whom a Contribution has been received by Licensor and
subsequently incorporated within the Work.
2. Grant of Copyright License. Subject to the terms and conditions of
this License, each Contributor hereby grants to You a perpetual,
worldwide, non-exclusive, no-charge, royalty-free, irrevocable
copyright license to reproduce, prepare Derivative Works of,
publicly display, publicly perform, sublicense, and distribute the
Work and such Derivative Works in Source or Object form.
3. Grant of Patent License. Subject to the terms and conditions of
this License, each Contributor hereby grants to You a perpetual,
worldwide, non-exclusive, no-charge, royalty-free, irrevocable
(except as stated in this section) patent license to make, have made,
use, offer to sell, sell, import, and otherwise transfer the Work,
where such license applies only to those patent claims licensable
by such Contributor that are necessarily infringed by their
Contribution(s) alone or by combination of their Contribution(s)
with the Work to which such Contribution(s) was submitted. If You
institute patent litigation against any entity (including a
cross-claim or counterclaim in a lawsuit) alleging that the Work
or a Contribution incorporated within the Work constitutes direct
or contributory patent infringement, then any patent licenses
granted to You under this License for that Work shall terminate
as of the date such litigation is filed.
4. Redistribution. You may reproduce and distribute copies of the
Work or Derivative Works thereof in any medium, with or without
modifications, and in Source or Object form, provided that You
meet the following conditions:
(a) You must give any other recipients of the Work or
Derivative Works a copy of this License; and
(b) You must cause any modified files to carry prominent notices
stating that You changed the files; and
(c) You must retain, in the Source form of any Derivative Works
that You distribute, all copyright, patent, trademark, and
attribution notices from the Source form of the Work,
excluding those notices that do not pertain to any part of
the Derivative Works; and
(d) If the Work includes a "NOTICE" text file as part of its
distribution, then any Derivative Works that You distribute must
include a readable copy of the attribution notices contained
within such NOTICE file, excluding those notices that do not
pertain to any part of the Derivative Works, in at least one
of the following places: within a NOTICE text file distributed
as part of the Derivative Works; within the Source form or
documentation, if provided along with the Derivative Works; or,
within a display generated by the Derivative Works, if and
wherever such third-party notices normally appear. The contents
of the NOTICE file are for informational purposes only and
do not modify the License. You may add Your own attribution
notices within Derivative Works that You distribute, alongside
or as an addendum to the NOTICE text from the Work, provided
that such additional attribution notices cannot be construed
as modifying the License.
You may add Your own copyright statement to Your modifications and
may provide additional or different license terms and conditions
for use, reproduction, or distribution of Your modifications, or
for any such Derivative Works as a whole, provided Your use,
reproduction, and distribution of the Work otherwise complies with
the conditions stated in this License.
5. Submission of Contributions. Unless You explicitly state otherwise,
any Contribution intentionally submitted for inclusion in the Work
by You to the Licensor shall be under the terms and conditions of
this License, without any additional terms or conditions.
Notwithstanding the above, nothing herein shall supersede or modify
the terms of any separate license agreement you may have executed
with Licensor regarding such Contributions.
6. Trademarks. This License does not grant permission to use the trade
names, trademarks, service marks, or product names of the Licensor,
except as required for reasonable and customary use in describing the
origin of the Work and reproducing the content of the NOTICE file.
7. Disclaimer of Warranty. Unless required by applicable law or
agreed to in writing, Licensor provides the Work (and each
Contributor provides its Contributions) on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or
implied, including, without limitation, any warranties or conditions
of TITLE, NON-INFRINGEMENT, MERCHANTABILITY, or FITNESS FOR A
PARTICULAR PURPOSE. You are solely responsible for determining the
appropriateness of using or redistributing the Work and assume any
risks associated with Your exercise of permissions under this License.
8. Limitation of Liability. In no event and under no legal theory,
whether in tort (including negligence), contract, or otherwise,
unless required by applicable law (such as deliberate and grossly
negligent acts) or agreed to in writing, shall any Contributor be
liable to You for damages, including any direct, indirect, special,
incidental, or consequential damages of any character arising as a
result of this License or out of the use or inability to use the
Work (including but not limited to damages for loss of goodwill,
work stoppage, computer failure or malfunction, or any and all
other commercial damages or losses), even if such Contributor
has been advised of the possibility of such damages.
9. Accepting Warranty or Additional Liability. While redistributing
the Work or Derivative Works thereof, You may choose to offer,
and charge a fee for, acceptance of support, warranty, indemnity,
or other liability obligations and/or rights consistent with this
License. However, in accepting such obligations, You may act only
on Your own behalf and on Your sole responsibility, not on behalf
of any other Contributor, and only if You agree to indemnify,
defend, and hold each Contributor harmless for any liability
incurred by, or claims asserted against, such Contributor by reason
of your accepting any such warranty or additional liability.
END OF TERMS AND CONDITIONS
APPENDIX: How to apply the Apache License to your work.
To apply the Apache License to your work, attach the following
boilerplate notice, with the fields enclosed by brackets "[]"
replaced with your own identifying information. (Don't include
the brackets!) The text should be enclosed in the appropriate
comment syntax for the file format. We also recommend that a
file or class name and description of purpose be included on the
same "printed page" as the copyright notice for easier
identification within third-party archives.
Copyright [yyyy] [name of copyright owner]
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.