Rational SU(3)-equivariant cohomology theories

J.P.C.Greenlees

REVIEW 2 major objections 5 minor 16 references

Rational SU(3)-equivariant cohomology theories

T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Rational SU(3)-equivariant cohomology theories are Quillen equivalent to differential graded objects of the abelian category A(SU(3)), built from an 18-block partition of subgroup conjugacy classes.

desk verdict A real milestone for rational SU(3)-spectra, but the printed central theorem has an inconsistent exceptional subgroup list (A5×C3 vs A6) that must be fixed before the claimed equivalence is well-defined. read the letter →

arxiv 2502.06017 v1 pith:OZRHYBKQ submitted 2025-02-09 math.AT

J.P.C.Greenlees This is my paper · ORCID

classification math.AT MSC 55P4255P9155N91

keywords rationalequivariantcohomologySU(3)-spectraQuillenequivalencealgebraicmodelsubgroupconjugacyclassessheavesofringscotoralorder18-blockdecomposition

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that rational SU(3)-equivariant cohomology theories are completely described by an explicit abelian category $\mathcal{A}(\mathrm{SU}(3))$, so that the homotopy theory of rational $\mathrm{SU}(3)$-spectra becomes a piece of ordinary homological algebra. The category is assembled from the space of conjugacy classes of closed subgroups of $\mathrm{SU}(3)$, which is partitioned into 18 Zariski-clopen blocks, each dominated by a subgroup of the form $(H_e, F)$. For every block the paper specifies a sheaf of rings and a component structure, and it assembles results from companion papers to conclude that rational $\mathrm{SU}(3)$-spectra are Quillen equivalent to differential graded objects of $\mathcal{A}(\mathrm{SU}(3))$. If correct, this settles the proposed algebraic-model classification for $\mathrm{SU}(3)$ and all of its subgroups.

What carries the argument

The load-bearing object is the partition of $X_G = \mathrm{Sub}(G)/G$ into blocks dominated by pairs $(H_e,F)$ with finite Weyl group, stated as Theorem 4.3. A block $V^G_{(H_e,F)}$ is the closure under cotoral specialization of a neighbourhood of the dominating subgroup; a subgroup $K$ is cotoral in $H$ when $K$ is normal in $H$ with torus quotient. The algebraic model over each block is a sheaf of modules over a sheaf of rings with stalk $H^*(B\,W^e_G(K))[W^d_G(K)]$, together with a component structure recording the action of the discrete part of the Weyl group. These pieces multiply together, $\mathcal{A}(\mathrm{SU}(3)) = \prod_i \mathcal{A}(\mathrm{SU}(3)|V_i)$, and the sheaf data over cotoral inclusions encodes the Localization Theorem for equivariant cohomology.

What would settle it

Check the claimed 18-block partition directly: if any closed subgroup of $\mathrm{SU}(3)$ is not conjugate to a subgroup of $\mathrm{U}(2)$ and not conjugate to one of the seven listed exceptional finite groups, then Theorem 4.3 is incomplete. A second check is to compute a known rational $\mathrm{SU}(3)$-equivariant cohomology group, for example Borel cohomology of $\mathrm{SU}(3)/T$ with constant coefficients, from the sheaf-ring data and compare it with the standard answer.

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Extended reading notes

Core claim

The central claim is a complete structural description of rational $\mathrm{SU}(3)$-equivariant cohomology. The space $X_{\mathrm{SU}(3)} = \mathrm{Sub}(\mathrm{SU}(3))/\mathrm{SU}(3)$ of conjugacy classes of closed subgroups is shown to split as a disjoint union of 18 Zariski-clopen blocks $V^G_{(H_e,F)}$, each dominated by a connected subgroup $H_e$ together with a finite subgroup $F$ of its Weyl group. Over each block the model is a sheaf of modules over a sheaf of rings whose stalk at a subgroup $K$ is the twisted group ring $H^*(B\,W^e_G(K))[W^d_G(K)]$, equipped with a component structure coming from the finite group $\pi_0(W_G(K))$. Joining these blocks yields an abelian category $\mathcal{A}(\mathrm{SU}(3))$, and the paper assembles the supporting models to conclude that the category of rational $\mathrm{SU}(3)$-spectra is Quillen equivalent to the category of differential graded objects of $\mathcal{A}(\mathrm{SU}(3))$.

Load-bearing premise

The load-bearing premise is that the algebraic models for the one-dimensional and toral blocks, taken from companion papers, are correct, together with the classical classification of finite subgroups of $\mathrm{SU}(3)$ that fixes the seven exceptional 0-dimensional blocks; the present paper assembles these ingredients rather than reproving them.

Editorial extensions

If this is right

Rational $\mathrm{SU}(3)$-equivariant cohomology theories are classified by algebraic objects, so computations can be carried out as sheaf cohomology and Ext over the 18-block space.
The model factorizes as a product of 18 blocks, so every rational $\mathrm{SU}(3)$-spectrum is the product of its pieces of geometric isotropy supported on single blocks.
Blocks inherited from $\mathrm{U}(2)$ survive in $\mathrm{SU}(3)$ with only a small amount of fusion; restriction from $\mathrm{SU}(3)$ to $\mathrm{U}(2)$ induces category isomorphisms on those blocks.
Nine of the blocks are 0-dimensional singletons, where the model reduces to modules over the group ring of a finite Weyl group, i.e. ordinary finite-group cohomology.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An implicit consequence is that maps between rational $\mathrm{SU}(3)$-equivariant cohomology theories, and the spectral sequences that compute them, should become Ext computations inside $\mathcal{A}(\mathrm{SU}(3))$; the paper does not spell this out.
The same fusion-and-block method suggests an explicit route to algebraic models for nearby rank-2 groups such as $\mathrm{PSU}(3)$ or $\mathrm{U}(3)$, whose subgroup spaces differ mainly through normalizers and fusion of conjugacy classes.
A testable extension is to compute a concrete rational $\mathrm{SU}(3)$-equivariant cohomology theory, such as Borel cohomology of a homogeneous space, directly from the sheaf stalks and compare with classical calculations; a mismatch would pinpoint which block's sheaf data needs adjustment.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this is a genuine milestone: the first explicit 18-block decomposition of X_{SU(3)} and a proposed abelian category A(SU(3)) that should model rational SU(3)-spectra. The subgroup analysis is careful, the block count is arithmetically consistent, and the normalizer lemmas are convincing. Second, the paper as printed has a load-bearing inconsistency: Section 4.C lists the exceptional finite subgroup A5×C3, while Theorem 4.3 replaces it with (1,A6). A6 has order 360 and no faithful 3-dimensional complex representation, so it cannot be a subgroup of SU(3). Since Theorem 4.3 defines the blocks that Section 5 assembles into A(SU(3)), the target category is ambiguous until this is fixed. This is not a cosmetic typo.

The genuinely new content is the fusion analysis specific to SU(3) and the identification of the 11 blocks not inherited from U(2). The paper is also honest about what it does not prove: the Quillen equivalence is assembled from four same-author preprints, all on arXiv. That is standard for a capstone paper, but it means the central theorem is not self-contained and a referee will need to inspect those preprints.

Soft spots in proportion: the A5×C3/A6 issue is the main one; it should be corrected and the Weyl group for the affected block re-derived. There are also minor typos, e.g., a malformed sentence in the proof of Lemma 3.6. The reliance on companion papers is a structural feature rather than a flaw, though it does cap the paper's independence.

Who this is for: equivariant homotopy theorists working on algebraic models for rational G-spectra. They will want to read this after (or with) the companion papers. It deserves a serious referee: the framework is right and the result is important, but the printed version needs correction before the claimed equivalence is determinate. I would send it to review, with the expectation of a substantive revision fixing the subgroup list and clarifying the status of the imported results.

Referee Report

2 major / 5 minor

Summary. The paper describes the space of conjugacy classes of subgroups of SU(3) as a disjoint union of 18 Zariski clopen blocks, each dominated by a pair (H_e, F). For every block it identifies the algebraic input (sheaf of rings, component structure, and Weyl-group data) needed for a rational algebraic model, and then assembles results from companion papers to assert a Quillen equivalence between rational SU(3)-spectra and the category of differential graded objects of the abelian category A(SU(3)). Sections 2 and 3 recall the block decompositions for proper subgroups and analyze fusion from U(2) to SU(3); Section 4 performs the new subgroup classification and states the 18-block partition in Theorem 4.3; Section 5 indicates the model structures for the blocks.

Significance. If correct, the paper would give a complete algebraic model for rational SU(3)-equivariant cohomology, a substantial non-toral example of the general Greenlees conjecture. The block decomposition and the normalizer computations in Sections 3 and 5 are concrete and useful, and the explicit 18-block structure is a meaningful contribution. However, the central Quillen equivalence is not proved here; it is imported from the same-author preprints [8,9,10] and [5,2], so the result is conditional on those works. Moreover, the internal inconsistency in the list of exceptional finite subgroups between §4.C and Theorem 4.3 makes the definition of A(SU(3)) ambiguous as printed, and must be corrected before the main claim is well posed.

major comments (2)

[§4.C and Theorem 4.3] The list of Type E finite subgroups in §4.C is inconsistent with the table in Theorem 4.3. §4.C states the seven exceptional groups are PSL2(7), A5×C3, PSL2(7)×C3, A6·3, G36·3, G72·3, G216·3, whereas Theorem 4.3 replaces A5×C3 by a row (1,A6) with W_G(H)=C3 while retaining (1,A6·3). These are not notational variants: A5×C3 has order 180, while A6 has order 360 and admits no faithful 3-dimensional complex representation, so A6 cannot be a subgroup of SU(3). Since Theorem 4.3 is the input that Section 5 assembles into the model A(SU(3)), the definition of the target category is ambiguous and the central Quillen equivalence is not well posed until the list is corrected.
[§1.E, §5] The central claim, that rational SU(3)-spectra are Quillen equivalent to DG-A(SU(3)), is not proved in this paper. Section 5 assembles models block by block, but the actual equivalences for the 1-dimensional blocks are delegated to the same-author preprints [8], [9], [10], and the toral block to [5, 2]. These are not peer-reviewed and are not included, so a reader cannot verify the main theorem from this manuscript alone. The authors should either include the necessary proofs or explicitly state that the main result is conditional on these companion papers and make them publicly available.

minor comments (5)

[§1.G] Typo: 'matices' should be 'matrices'.
[Theorem 4.3] In the table, the row (T^2, C1) uses 'C1' for the trivial group, whereas elsewhere the trivial group is denoted '1'; please unify the notation.
[References] The reference [15] is incomplete; it gives volume and page range but no journal or book title.
[§5.C] The phrase 'The one dominated by the maximal torus A(G|toral) is treated in [5, 2]' is grammatically awkward and should be rephrased for clarity.
[§4.C] The sentence 'Those with a two dimensional faithful representations' contains an agreement error; it should be 'a two-dimensional faithful representation'.

Circularity Check

1 steps flagged · score 2.0 of 10

No substantive circularity: the main SU(3) equivalence is imported from the author's own independent preprints, but the block analysis itself is new and no fitted quantity is renamed a prediction.

self citation load bearing [Abstract; Section 1.E; Section 5]
"Taken together, this gives an abelian category A(SU(3)) designed to reﬂect the structure of rational SU(3)-equivariant cohomology theories, and we assemble the results from elsewhere to show that the category of rational SU(3)-spectra is Quillen equivalent to the category of diﬀerential graded objects of A(SU(3))."

The paper's central Quillen equivalence is not proved inside the paper; it is assembled from the author's own preceding preprints [5,2,8,9,10]. In Section 5 each block model is delegated by citation: 'We proved in [8] that each 1-dimensional block ... has A(G|V) as a model,' 'A(G|toral) is treated in [5,2],' and 'The block dominated by T2⋊C2 in the toral case is treated in [10].' Thus the main theorem is justified only by a chain of same-author citations, which is load-bearing. However, the cited results are independent theorems about U(2), toral groups, or general 1-dimensional blocks; they do not assume the SU(3) equivalence. The circularity is therefore evidentiary/dependence-based rather than a reduction of the target to itself.

full rationale

The paper's own contribution is the 18-block decomposition and the fusion analysis of Sections 3–4, obtained from the external Blichfeldt classification and elementary normalizer arguments, not from the model being asserted. No parameter is fitted to a subset of cohomology data and then renamed a prediction; no ansatz is smuggled in via citation; no uniqueness theorem from the author is invoked to forbid alternatives. The only load-bearing self-citation is the assembly of the Quillen equivalence from the same-author preprints, which is a proof-dependency rather than a constructed identity. I therefore find no self-definitional or statistical circularity. The apparent A5×C3 versus A6 discrepancy between Section 4.C and Theorem 4.3 is a mathematical consistency issue, not a circularity issue, and does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters: the paper introduces no fitted constants or ad hoc scalings; the only numerical data are group orders and Weyl groups determined by the subgroup classification. The axioms are cited external results: the Blichfeldt classification of finite subgroups of SU(3), the standard centralizer formula, and the model theorems for toral, one-dimensional, and U(2) cases from the author's own series. These are not assumptions of the target result but independent theorems about smaller groups; however, several are not peer-reviewed. The construct A(SU(3)) is the paper's main object, not a postulated entity with independent physical evidence, so no invented entities are listed.

assumptions (5)

standard math The Blichfeldt classification of finite subgroups of SU(3) (Miller-Blichfeldt-Dickson [16], Ludl [15])
Used in Section 4.C to enumerate the 7 additional finite singleton blocks in Type E; the count of 18 blocks depends on this classification.
domain assumption Models for rational G-spectra over toral blocks are Quillen equivalent as shown in [5, 2]
Invoked in Section 5.C for the (T^2, C1) block; the present paper does not prove this.
domain assumption Models for one-dimensional blocks from [8] and for U(2) blocks from [10] are correct
The central Quillen equivalence for 8 of the 18 blocks is inherited from these companion papers by the same author.
standard math Formula for centralizers in compact Lie groups ([3, V.2.3])
Used in Lemma 5.3 to identify C_SU(3)(a) = U(2).
standard math Uniqueness of maximal tori and standard facts about Weyl groups (e.g., N_G(T^2)/T^2 = Sigma_3)
Used throughout Sections 3 and 4 for the block decomposition.

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Cite this review

Pith. "Pith review of Rational SU(3)-equivariant cohomology theories." pith.science (2026). https://pith.science/paper/OZRHYBKQ

@misc{pith2026250206017,
  author       = {Pith},
  title        = {Pith review of: Rational SU(3)-equivariant cohomology theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OZRHYBKQ}},
  note         = {Machine review of arXiv:2502.06017}
}

read the original abstract

We describe the spectral space of conjugacy classes of subgroups of SU(3), together with the additional structure of a sheaf of rings and a component structure. It is a disjoint union of 18 blocks each dominated by a subgroup. For each of these blocks we identify a sheaf of rings and component structure. Taken together, this gives an abelian category A(SU(3)) designed to reflect the structure of rational SU(3)-equivariant cohomology theories, and we assemble the results from elsewhere to show that the category of rational SU(3)-spectra is Quillen equivalent to the category of differential graded objects of A(SU(3)).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 11 canonical work pages

[8]

J. P. C. Greenlees. Algebraic models for one-dimensional catego ries of rational G-spectra. Preprint, 28pp, arXiv:2501.11200, 2025

arXiv 2025
[9]

J. P. C. Greenlees. Rational G-spectra for rank 2 toral groups of mixed type. Preprint, 24pp, arXiv:2501.15584, 2025

arXiv 2025
[10]

J. P. C. Greenlees. An algebraic model for rational U (2)-spectra. Preprint, 15pp, arXiv:2502.00959 , 2025

arXiv 2025
[1]

Balchin, T

S. Balchin, T. Barthel, and J. P. C. Greenlees. Prismatic decompo sitions and rational G-spectra. Preprint, 59pp, arXiv 2311.18808 , 2023

arXiv 2023
[2]

Greenlees, and Magdalena K¸ edziorek

David Barnes, J.P.C. Greenlees, and Magdalena K¸ edziorek. An alg ebraic model for rational toral G- spectra. Algebr. Geom. Topol., 19(7):3541–3599, 2019

work page 2019
[3]

Br¨ ocker and T

T. Br¨ ocker and T. tom Dieck. Representations of compact Lie groups , volume 98 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1995. Translated from the German m anuscript, Corrected reprint of the 1985 translation

work page 1995
[4]

J. P. C. Greenlees. Triangulated categories of rational equivar iant cohomology theories. Oberwolfach Reports, pages 480–488, 2006. (cit. on p. 2) , 2006

work page 2006
[5]

J. P. C. Greenlees. Rational equivariant cohomology theories wit h toral support. Algebr. Geom. Topol., 16(4):1953–2019, 2016

work page 1953

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J. P. C. Greenlees. The Balmer spectrum of rational equivariant cohomology theories. J. Pure Appl. Algebra, 223(7):2845–2871, 2019

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J. P. C. Greenlees. Spaces of subgroups of toral groups. Preprint, 38pp, arXiv:2501.06914 , 2025

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J. P. C. Greenlees. Rational G-spectra for components with ﬁnite Weyl groups. In preparation, 4pp , 2025

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J. P. C. Greenlees. An abelian model for rational G-spectra for a compact Lie group G. In preparation, 27pp
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J. P. C. Greenlees. An algebraic model for rational G-spectra for ﬁnite central extensions of a torus. In preparation, 18pp
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P.O. Ludl. The ﬁnite subgroups of SU (3). (7):177–183
[16]

G. A. Miller, H. F. Blichfeldt, and L. E. Dickson. Theory and applications of ﬁnite groups . Dover Publications, Inc., New York, 1961. Mathematics Institute, Zeeman Building, Coventry CV4, 7AL , UK Email address : john.greenlees@warwick.ac.uk 14

1961

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [8]

J. P. C. Greenlees. Algebraic models for one-dimensional catego ries of rational G-spectra. Preprint, 28pp, arXiv:2501.11200, 2025

arXiv 2025

[2] [9]

J. P. C. Greenlees. Rational G-spectra for rank 2 toral groups of mixed type. Preprint, 24pp, arXiv:2501.15584, 2025

arXiv 2025

[3] [10]

J. P. C. Greenlees. An algebraic model for rational U (2)-spectra. Preprint, 15pp, arXiv:2502.00959 , 2025

arXiv 2025

[4] [1]

Balchin, T

S. Balchin, T. Barthel, and J. P. C. Greenlees. Prismatic decompo sitions and rational G-spectra. Preprint, 59pp, arXiv 2311.18808 , 2023

arXiv 2023

[5] [2]

Greenlees, and Magdalena K¸ edziorek

David Barnes, J.P.C. Greenlees, and Magdalena K¸ edziorek. An alg ebraic model for rational toral G- spectra. Algebr. Geom. Topol., 19(7):3541–3599, 2019

work page 2019

[6] [3]

Br¨ ocker and T

T. Br¨ ocker and T. tom Dieck. Representations of compact Lie groups , volume 98 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1995. Translated from the German m anuscript, Corrected reprint of the 1985 translation

work page 1995

[7] [4]

J. P. C. Greenlees. Triangulated categories of rational equivar iant cohomology theories. Oberwolfach Reports, pages 480–488, 2006. (cit. on p. 2) , 2006

work page 2006

[8] [5]

J. P. C. Greenlees. Rational equivariant cohomology theories wit h toral support. Algebr. Geom. Topol., 16(4):1953–2019, 2016

work page 1953