REVIEW 2 major objections 4 minor 17 references
Verification of the conjugacy classes and ordinary character table of the Monster
T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read Under its defining hypotheses, the Monster is shown to have exactly 194 conjugacy classes and the ATLAS character table.
desk verdict A serious, largely independent re-derivation of the Monster's class list and character table, with one unresolved ambiguity about the role of the ATLAS oracle in the final verification step. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the ordinary character $\chi$ of degree $196883 = 47\cdot59\cdot71$, together with its restrictions to the involution centralizer $C_M(z) \cong 2.\mathbb{B}$ (the double cover of the Baby Monster) and to other p-local subgroups. The paper computes the permutation character of $M$ on the cosets of $2.\mathbb{B}$ from suborbit data, which yields centralizer orders for many elements via Frobenius reciprocity. For each prime dividing the group order, the class list is built by fusing classes of computed centralizer and normalizer character tables, using Sylow's theorem and rationality arguments to pin down normalizer structures. The same restricted characters then determine the values of $\chi$ on all 194 classes, and the full table of irreducibles is verified from induced characters.
What would settle it
The claim would be refuted by exhibiting an element order, centralizer order, or character value on any of the 194 classes that differs from the ATLAS entry, assuming the Monster is the group under test. A more direct test is to inspect the computer code for Section 5.2 and check whether the oracle call returns the full row of character values from the ATLAS table (circular) or only the list of candidate labels to be checked for irreducibility (non-circular).
Extended reading notes
Core claim
Under the uniqueness hypotheses for the Monster and the existence of an ordinary faithful representation of degree 196883, the paper proves that the Monster has exactly 194 conjugacy classes, with element orders, centralizer orders, and centralizer structures exactly as listed in the ATLAS (Proposition 1). It further shows that the character of degree 196883 has the ATLAS values on all classes and that the complete ordinary character table, including power maps and class fusions, coincides with the ATLAS table. The centralizer structures of prime-order elements are determined case by case, including the $7^5$ extraspecial centralizer of 7B-elements, the $13^{1+2}$ Sylow normalizer structure, and the cyclic normalizers for 29, 41, 59, and 71.
Load-bearing premise
The argument assumes as given that the Monster is unique and possesses a faithful 196883-dimensional ordinary representation, and in the final verification step it consults the ATLAS table as an oracle; if the oracle is used to supply character values rather than just candidate labels, the proof of the character table's correctness assumes part of the statement being proved.
Editorial extensions
If this is right
- The ATLAS character table of the Monster is reproducible from the stated hypotheses, closing the last gap in the independent verification of all ATLAS character tables.
- All conjugacy classes of the Monster, including the power maps and the fusion of every local subgroup class, are now certified; these data are available for future computations.
- The centralizer structures of all prime-order elements are determined, including the exotic $7^5$ and $13^{1+2}$ local structures, without invoking an explicit matrix construction of the Monster.
- The degree-196883 character is shown to have its values on the 194 classes forced by the local restrictions and congruence conditions, so the displayed table of values is not merely an artifact of the ATLAS oracle.
Reading between the lines
- A natural next step would be to eliminate the 'oracle' use of the ATLAS table in Section 5.2, for instance by proving the irreducibility of the induced characters without consulting the target table; until then, the verification's final step is not fully self-contained.
- The certified local subgroup character tables could be used to compute modular character tables or cohomological invariants of the Monster, since the necessary fusion data are now explicit.
- The same restriction-and-fusion strategy is portable to other large sporadic groups whose uniqueness is known but whose tables were only computed once.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to verify, under explicit uniqueness hypotheses and assuming the existence of a degree-196883 ordinary irreducible character of the Monster group M, that M has exactly 194 conjugacy classes with the orders and centralizer structures listed in the ATLAS, and that the ordinary character table of M agrees with the ATLAS table. Section 4 obtains the class list by analyzing p-local subgroups, permutation characters on suborbits of the involution centralizer, and Sylow normalizers for each relevant prime. Section 5 computes the values of the degree-196883 character from subgroup restrictions and congruence constraints, and then sketches how the full character table is completed by inducing from known centralizer subgroups and verifying irreducibility. The detailed machine computations are deferred to the companion paper [5].
Significance. If the result holds, it completes the programme of independent verification of all ATLAS character tables, which is a substantial reliability contribution to computational group theory. The local analysis in Section 4 is structured around explicit p-local and Sylow computations, and the incremental class-counting strategy is a credible argumentative framework. A notable strength is that the paper states its hypotheses clearly and relies on previously verified character tables and explicit representations rather than on an unstated construction of M. However, the final verification step in Section 5.2 is not fully specified, and the manuscript explicitly defers the details to a companion paper; because the central claim is the correctness of the ATLAS table, the role of the ATLAS 'oracle' must be made precise for the argument to be non-circular.
major comments (2)
- [5.2] The manuscript states that 'the ATLAS table of M is used as an "oracle"' when verifying the irreducibles, but it does not specify what data the oracle supplies. If the oracle provides the irreducible character values or the full set of irreducible rows, the verification of the full character table is circular relative to the paper's central claim that the table matches the ATLAS. If, instead, the oracle supplies only candidate labels or degrees, or a class skeleton, the verification can be sound, provided irreducibility and completeness are checked by independent computations such as norm-one inner products and the sum-of-squares identity using the class head. The authors should explicitly state the input and output of the oracle and confirm that no ATLAS character values are used in the verification; if the details reside in [5], the present paper should at least include a precise statement of this independence.
- [5.2] The sentence 'The class fusions of these subgroups are determined by the given data, we can induce from them, and from cyclic subgroups. This suffices to verify the irreducibles' is a nontrivial computational claim that is not demonstrated in the paper. The reader cannot tell from the text how the induced characters are decomposed, how the 194 irreducible characters are obtained, or how completeness is established from the given class fusions and power maps. Since the goal is reproducibility, the authors should describe the algorithm at least schematically, including the exact inputs (which character tables, which fusions, which power maps) and the exact checks (e.g., inner products, orthogonality, sum of squared degrees equal to |M|), or clearly indicate where in [5] each of these steps appears.
minor comments (4)
- [5.1] In the sentence 'Now we compute the values of the 199883 on all classes', the number '199883' should be '196883'.
- [5.1] The phrase 'their congruence modulop to the value at p-th powers' contains a typo; 'modulop' should be 'modulo p'.
- [5.2] The term 'character table head' is used without definition; it should be stated explicitly whether this means the class names, element orders, centralizer orders, and power maps only, as opposed to any character values.
- [2] The notational dependence on the companion paper [5] is heavy; for instance, the proof of Lemma 1 and Lemma 2 is summarized as 'Full details of the calculations are in Section 2 of [5]'. This is acceptable for an outline, but the authors should ensure the statements of the lemmas themselves are self-contained enough for the reader to follow the subsequent class-counting arguments.
Circularity Check
No demonstrated circularity; the ATLAS 'oracle' passage is a candidate-source caveat, not a value-forcing input, and the central derivation chain is independent of the table being verified.
full rationale
We reviewed the derivation chain. The paper's hypotheses are explicit: existence and uniqueness of M, two involution centralizer structures, and an ordinary irreducible character of degree 196883. From these, Section 4 derives the conjugacy classes and centralizer orders using subgroup character tables, permutation characters, Sylow arguments, and prior structural results; Proposition 1 is stated as a comparison with the ATLAS, not as an input. Section 5.1 computes the values of chi from restrictions, power maps, congruence conditions, block theory, and scalar products, again with the ATLAS used only as a stated reference for the final comparison. The only potentially circular passage is Section 5.2, where the text says 'the ATLAS table of M is used as an "oracle"' in verifying the irreducibles. If the oracle supplied the 194 irreducible character rows as already-trusted values, the verification would reduce to checking the table against itself. However, the surrounding text says the class fusions are determined from the given subgroup data and that induced characters from 2.B, 3.Fi24, 2^{1+24}.Co1, 3^{1+12}.2.Suz.2, and cyclic subgroups are used to verify the irreducibles. Read in the context of the cited Baby Monster verification, where the ATLAS table was used in the same way to propose candidate rows that are then independently verified as genuine characters via induction and orthogonality, the oracle is a candidate generator rather than a truth input. The paper defers full computational detail to the companion paper [5], which is a completeness limitation rather than a circular dependency. The self-citations [2,3,4,13,16] point to earlier verifications and constructions that do not assume the Monster character table under test, so they are not load-bearing circular references. No specific equation, fitted parameter, or constructed character row is shown to reduce by definition to its own input. We therefore find no significant circularity, while noting that the oracle sentence is terse and would be clearer if it explicitly stated that the ATLAS rows are only candidate lists to be verified.
Assumptions & free parameters
assumptions (5)
- domain assumption Hypothesis 1: M is a finite group with exactly two involution classes, centralizers 2.B and 2^{1+24}.Co1, and an irreducible character of degree 196883.
- domain assumption Fact 1 suborbit information: the suborbit lengths and point stabilizers of 2.B on z^M are as in Table 1.
- domain assumption The character tables of 2.B, 3.Fi24, HN, and other Atlas groups used as inputs are correct.
- domain assumption Existence of subgroup 3^8.O_8^-(3) from [16].
- ad hoc to paper The ATLAS table of M is a valid oracle for verifying the irreducibles.
Cite this review
Pith. "Pith review of Verification of the conjugacy classes and ordinary character table of the Monster." pith.science (2026). https://pith.science/paper/6EO6ZNFC
@misc{pith2026241212182,
author = {Pith},
title = {Pith review of: Verification of the conjugacy classes and ordinary character table of the Monster},
year = {2026},
howpublished = {\url{https://pith.science/paper/6EO6ZNFC}},
note = {Machine review of arXiv:2412.12182}
}
abstract
As part of the programme to re-compute the character tables of all the groups in the Atlas we re-compute the character table of $\mathbb M$, the Monster simple group. We operate under the uniqueness hypotheses of $\mathbb M$ and the existence of an ordinary faithful representation of degree $196883 = 47.59.71$ and determine the conjugacy classes and centralizer orders of the elements of $\mathbb M$. Along the way we re-compute the character tables of centralizers of $p$-elements for $p < 11$ as well as fusions of conjugacy classes of these centralizers in $\mathbb M$.
Reference graph
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Reviewed August 11, 2026 · model on record in the stance chip above.
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