Some steps in the verification of the ordinary character table of the Monster group

Kay Magaard; Robert A.Wilson; Thomas Breuer

REVIEW 2 major objections 4 minor 1 cited by

Some steps in the verification of the ordinary character table of the Monster group

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A fully documented computer session reconstructs the Monster group's 194-class character table and checks it against the stored Atlas table.

desk verdict A transparent, reproducible supplement that honestly documents the computational steps behind [BMW24], with a real but disclosed oracle dependence that limits its standalone verification claim. read the letter →

arxiv 2412.09313 v1 pith:AM2MGB4M submitted 2024-12-12 math.RT

Thomas Breuer , Kay Magaard , Robert A.Wilson This is my paper · ORCID

classification math.RT MSC 20C1520C4020D08

keywords MonstergroupordinarycharactertableverificationconjugacyclassespowermapsirreduciblecharactersGAPsessionMAGMAcomputation

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 spells out, as a reproducible GAP/MAGMA session, the computations used in the companion verification of the ordinary character table of the Monster group. Its aim is to show that a 194-class character table of the Monster, with correct centralizer orders, element orders, and power maps, and with 194 irreducible characters, can be built from previously verified subgroup tables plus the assumed degree-196883 irreducible character. It also documents the construction of three subgroup tables that the verification needs, and it ends with the explicit check that the reconstructed table is permutation-equivalent to the Monster table stored in the Atlas/GAP library. A sympathetic reader would care because the Monster is the largest sporadic simple group and its character table is a basic resource in group theory; this paper pins down exactly which steps and which input tables the verification rests on.

What carries the argument

The carrying mechanism is a partial 'head' record for the Monster table: a list of class centralizer orders and element orders together with partial class fusions from subgroup character tables into the Monster. Two utility functions extend this head by root classes, whose p-th power lies in an already known class, and by centralizer orders read off from permutation-character values. Around this head, the argument proceeds in stages: restriction of the 196883-character to known subgroups, computation of the nine permutation-character constituents, completion of the 194-class head, determination of all power maps by transfer-diagram consistency and quadratic-field arguments, and finally induction from subgroup irreducibles reduced by the LLL algorithm to obtain the full 194-character table.

What would settle it

Recompute the character table of any one of the three appendix subgroups by an independent method, say by constructing 2^(1+24)+.Co1 from a different matrix representation and comparing with the library table using TransformingPermutationsCharacterTables. A failure there, or a failure of the final check TransformingPermutationsCharacterTables(m, atlas_m) when re-run with the published session, would refute the paper's claim.

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Extended reading notes

Core claim

The central claim is that the Monster's ordinary character table can be reconstructed and independently checked by a deterministic computer session, rather than inherited as a black box. Starting from the assumed existence of an irreducible character of degree 196883 and of exactly two involution classes, the paper computes the restriction of that character to 2.B and 3.Fi24, builds the nine transitive constituents of the permutation character, and then constructs the 194 conjugacy classes by adding root classes of prime-order elements and using permutation-character values and Sylow arguments. Power maps are fixed by consistency with subgroup fusions and by Galois-field reasoning for the ambiguous pairs of classes. Once the character values are known on all classes, the irreducible characters are obtained by inducing from subgroups, reducing with the LLL algorithm, and checking the Atlas irreducibles against the resulting lattice; the final command verifies that the two tables are permutation-equivalent. The paper also recomputes the character tables of 2^(1+24)+.Co1, 3^(1+12)+:6.Suz.2, and 5^(1+6)+.4.J2.2 in appendices, so those input tables do not have to be taken on faith.

Load-bearing premise

The reconstruction inherits its correctness from previously computed subgroup character tables, notably 2.B, 3.Fi24, and Th, loaded from the GAP library; if any of those input tables is wrong, the derived Monster table can be wrong, and the paper only re-derives three smaller tables in its appendices.

Editorial extensions

If this is right

The ordinary character table of the Monster stored in the Atlas/GAP library is backed by a reproducible computation, not by the original table's provenance alone.
The three subgroup tables constructed in the appendices are independently certified, so the verification does not rely on those tables as unexamined inputs.
The two candidate tables for the 3B normalizer are resolved: only one is compatible with the degree-196883 character, so the other candidate is excluded by the restriction test.
The remaining ambiguities in the power maps, for Galois-conjugate pairs of element orders 39, 59, 71, 78, and 119, are settled, giving a complete and consistent power-map structure for the Monster.
The final table has 194 irreducible characters of norm 1 whose degree-squared sum equals the group order, which is the standard completeness check for a character table.

Reading between the lines

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

Beyond the paper, the same head-and-fusions protocol could be applied to other large groups whose character tables are stored in libraries, converting a table's authority from 'known from the literature' to 'recomputable on demand.'
Because the verification uses the library irreducibles as an oracle and then checks lattice membership, it certifies that the stored Monster table is consistent with the subgroup input tables; re-running the session with independently recomputed 2.B and 3.Fi24 tables would strengthen that certification further.
A natural testable extension is to repeat the computation from a different construction of the Monster or from different generating sets; if the final table changes, the verification would expose a hidden dependence on the chosen subgroup data.

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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

The paper does exactly what its title says: it lays out the GAP and MAGMA session used to verify the ordinary character table of the Monster, as a companion to the main verification in [BMW24]. The new content is not the character table itself—that has been known since the Atlas—but the explicit, step-by-step protocol, including the independent MAGMA recomputation of the tables of 2^(1+24)+.Co1, 3^(1+12)+:6.Suz.2, and 5^(1+6)+.4.J2.2. That is a genuinely useful contribution to computational verification: it makes the computations reproducible, and it catches any library-table errors in those three subgroups.

The soft spot is the one the stress-test identifies. Section 7 uses the Atlas character table of M as an "oracle" to supply the 194 candidate irreducibles, then checks each candidate lies in the Z-lattice generated by induced characters and has norm 1. That is a sound certificate for each individual character—a norm-1 virtual character is irreducible—but it is not a completeness proof. The list of candidates comes from the Atlas, so if the Atlas table were wrong in a way that passed the lattice test, the final equivalence check would not catch it. The paper is upfront about this, saying plainly it is not computing irreducibles from scratch. That is an honest limitation, not a hidden one.

The other concern, the reliance on library tables for 2.B, 3.Fi24, and Th without recomputing them, is real but is also contextual: those tables come from prior verified work ([BMW20], [BMO17], [Brea]), and this paper is a supplement, not a self-contained proof. The MAGMA dependency itself is not a flaw; the session is reproducible if you have the software.

The reader's conditional verdict is fair in spirit, though I would not treat the oracle issue as a reason to withhold acceptance. This paper is a companion piece, and it never claims to be the full independent verification. The value is in the detailed protocol and the subgroup re-computations, both of which check out internally: the sum of class lengths equals the group order, the power maps become consistent, and the final table is permutation-equivalent to the library table.

Who should read it: anyone working on computational verification of group character tables, and anyone who wants to see how the Monster table can be reconstructed from subgroup data without invoking huge black-box computations. It deserves a serious referee—not because it proves something new, but because it provides a checkable record of an important verification. My recommendation is to send it to peer review, with a referee who understands the scope and the oracle caveat.

Referee Report

2 major / 4 minor

Summary. The paper presents a detailed GAP/MAGMA session protocol that constructs the character table head of the Monster group M: 194 conjugacy classes with centralizer orders and power maps (Sections 4–5), the degree 196883 character χ (Sections 2 and 6), and a verification of the 194 irreducible characters (Section 7). The irreducible characters are obtained by taking the characters from the Atlas table of M as an oracle, permuting them to match the computed table head, and testing each candidate for membership in the Z-lattice generated by characters induced from known subgroups (2.B, 2^{1+24}.Co1, 3^{1+12}:6.Suz.2, 3.Fi24, and cyclic subgroups). The final check is that the resulting table is permutation-equivalent to the Atlas table. Three appendices document the recomputation of the character tables of 2^{1+24}.Co1, 3^{1+12}:6.Suz.2, and 5^{1+6}:4.J2.2.

Significance. The paper is a valuable companion to the authors' verification of the Monster character table, providing an unusually transparent, step-by-step record of a large computational verification. Its strengths are reproducibility — every GAP/MAGMA command and output is shown — and internal consistency checks, such as the sum of class lengths equalling |M| and the selection of the correct 3B normalizer table in Section 6. If accepted, the protocol provides a permanent machine-checkable record of the computations and a template for verifying other large character tables. The main limitation is the reliance on previously verified library tables and on the Atlas table as a candidate source; this is explicitly acknowledged in the text, but the logical status of the verification deserves sharper formulation.

major comments (2)

[§7, pp. 32–36] The verification semantics should be stated more sharply. The lattice membership test plus norm 1 is a sound certificate that each candidate character that passes it is an irreducible character of M, assuming the correctness of the table head (classes, centralizer orders, power maps) and of the induced characters. Since the final table has 194 distinct irreducible characters and the group has 194 conjugacy classes, the final set is automatically complete, regardless of the oracle. The oracle is used only to generate candidate characters; the final equivalence check against the Atlas table is a consistency check, not the certificate. The current phrasing, 'we use the characters printed in the Atlas as an oracle' and 'the final check is equivalence to the Atlas table', invites a circularity objection that does not actually apply to the certificate. Please spell out this logical structure explicitly, and distinguish the certificate (lattice membership) from the consistency check (permutation equivalence).
[§4.3, §5, §7] The paper should explicitly list all external inputs whose correctness is assumed, beyond the three tables recomputed in the appendices. In particular, the GAP library tables for 2.B, 3.Fi24, and Th are loaded without recomputation, and the MAGMA computations in Sections 9–10 are described but not indepently verified here. The text cites [BMW20], [BMO17], and [Brea] for these, which is appropriate, but a consolidated statement of assumptions at the start of Section 4 or Section 7 would make the scope of the verification unambiguous. This is load-bearing because an error in any of these input tables would propagate through the class fusions, power maps, and the lattice used in Section 7.

minor comments (4)

[§7, p. 32] The phrase 'the indirection of χ by the 2nd power map' should read 'the composition of χ with the 2nd power map'.
[§7, p. 36] Typo: 'reuce them with the known irreducibles' should be 'reduce them with the known irreducibles'.
[§2, p. 3] The character degree appears as '57377a' in one sentence and '57477a' in the surrounding lists; please check the intended value and use it consistently.
[References] Reference [Brea] gives 'arXiv:1604.00754.' with a trailing period; the arXiv identifier should be typeset without the period.

Circularity Check

2 steps flagged · score 6.0 of 10

Section 7 imports the irreducible characters from the Atlas character table of M and then verifies m by equivalence to that same Atlas table, so the irreducible part of the verification is circular.

self definitional [Section 7, first paragraph (p. 32)]
"We will not compute the irreducibles of M from scratch but verify the irreducibles from the Atlas character table of M, in the sense that we use the characters printed in the Atlas as an “oracle”."

The candidate irreducible characters of the computed table m are taken from the Atlas table atlas_m, permuted to m's columns. Completeness, i.e., that these 194 rows are all irreducible characters of M, therefore rests on the very table the paper aims to verify. The lattice-membership and norm-1 test is a sound certificate for each candidate that passes it, because a norm-1 element of the lattice spanned by induced characters is an irreducible character of M. But that test does not supply the candidate list and cannot detect a missing irreducible; both come from the Atlas oracle.
self definitional [Section 7, final GAP output (p. 36)]
"SetIrr( m, List( knownirr, x -> ClassFunction( m, x ) ) ); gap> ResetFilterObj( m, HasAutomorphismsOfTable ); gap> TransformingPermutationsCharacterTables( m, atlas_m ) <> fail; true"

After the oracle loop, knownirr has length 194 and consists of the permuted Atlas irreducible characters (the oracle has exactly 194 characters, and the loop is what raises the count to 194). SetIrr(m, knownirr) therefore installs the Atlas rows as the rows of m up to the column permutation pi. The subsequent TransformingPermutationsCharacterTables(m, atlas_m) check then compares m with the table that supplied its irreducible rows. It is a consistency check between the independently computed class head and power maps and the Atlas table, not an independent verification of the ordinary character table; any Atlas error that is compatible with the head, power maps, and the induced lattice would pass.

full rationale

The paper contains substantial independent work outside Section 7: the table head (194 conjugacy classes, centralizer orders, and power maps) is assembled from subgroup character tables and permutation characters, and the sum of the resulting class lengths is checked to equal |M|; the degree-196883 character is computed from restrictions to subgroups; and the appendices recompute three subgroup tables from explicit representations. These parts do not reduce to the target table. The circularity is confined to the irreducible-character part of the verification: the candidate irreducibles and the completeness of the 194-row list are imported from the Atlas character table of M, and the final equivalence to that same Atlas table is therefore partly by construction. The lattice-membership/norm-1 test gives a genuine mathematical certificate for each imported candidate, so the computation does prove compatibility of the Atlas irreducibles with independently computed class data; it does not independently verify the irreducible character table. Score 6 reflects this partial, construction-level circularity rather than a fully circular derivation.

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

No fitted or ad hoc continuous parameters appear. Discrete choices for Galois-conjugate class labels are conventional labeling decisions, not free parameters. No new entities are introduced. The load-bearing assumptions are prior existence facts, correctness of library tables, correctness of Atlas data, correctness of MAGMA, and the use of the Atlas table as an oracle.

assumptions (5)

domain assumption Existence of the Monster group M with an irreducible degree 196883 character and exactly two involution classes.
Stated at the start of Section 2; these facts are taken from prior Monster existence results and are not reproved in this paper.
domain assumption Correctness of the character tables of 2.B, 3.Fi24, Th and other subgroups stored in the GAP Character Table Library.
Relied on throughout Sections 2-5; the paper cites [BMW20], [Brea], and [BMO17] for these tables and only recomputes three subgroup tables in the appendices.
domain assumption Correctness of generator sets and representations from the ATLAS of Group Representations.
Used in Sections 9 and 10 to build permutation groups for MAGMA computations; the paper checks some standard-generator conditions but not full faithfulness of the representations.
ad hoc to paper Correctness of the Atlas character table of M, used as an oracle in Section 7.
Section 7 states: 'we use the characters printed in the Atlas as an oracle'. The final equivalence check against this same table makes reliance on the target result an input assumption.
domain assumption Correctness of MAGMA computations and the GAP/MAGMA software used.
Sections 8, 9, and 10 depend on MAGMA to compute character tables from generators; these heavy computations are not independently verified in this paper.

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Pith. "Pith review of Some steps in the verification of the ordinary character table of the Monster group." pith.science (2026). https://pith.science/paper/AM2MGB4M

@misc{pith2026241209313,
  author       = {Pith},
  title        = {Pith review of: Some steps in the verification of the ordinary character table of the Monster group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AM2MGB4M}},
  note         = {Machine review of arXiv:2412.09313}
}

read the original abstract

We show the details of certain computations that are used in the paper "Verification of the conjugacy classes and ordinary character table of the Monster".

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Verification of the conjugacy classes and ordinary character table of the Monster
math.GR 2024-12 conditional novelty 5.0 of 10

Under standard existence assumptions, the authors verify that the Monster has exactly 194 conjugacy classes and reproduce its Atlas character table.

Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages · cited by 1 Pith paper

[1]

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W. Bosma, J. Cannon, and C. Playoust, The M agma algebra system. I . T he user language , J. Symbolic Comput. 24 (1997), no. 3 4, 235 265. 1484478

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[2]

Bhargava, R

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Breuer, K

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, Verification of the conjugacy classes and ordinary character table of the M onster , submitted, 2024

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T. Breuer, Constructing the ordinary character tables of some A tlas groups using character theoretic methods. , https://export.arxiv.org/abs/1604.00754 arXiv:1604.00754

arXiv
[7]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Bosma, J

W. Bosma, J. Cannon, and C. Playoust, The M agma algebra system. I . T he user language , J. Symbolic Comput. 24 (1997), no. 3 4, 235 265. 1484478

work page 1997

[2] [2]

Bhargava, R

M. Bhargava, R. Guralnick, G. Hiss, K. Lux, and P. H. Tiep (eds.), Finite simple groups: thirty years of the A tlas and beyond , Contemporary Mathematics, vol. 694, Providence, RI, American Mathematical Society, 2017. 3682583

work page 2017

[3] [3]

Breuer, G

T. Breuer, G. Malle, and E. A. O'Brien, Reliability and reproducibility of A tlas information , in Bhargava et al. Atlas2017 , p. 21 31. 3682588

work page

[4] [4]

Breuer, K

T. Breuer, K. Magaard, and R. A. Wilson, Verification of the ordinary character table of the B aby M onster , J. Algebra 561 (2020), 111 130. 4135540

work page 2020

[5] [5]

, Verification of the conjugacy classes and ordinary character table of the M onster , submitted, 2024

work page 2024

[6] [6]

Breuer, Constructing the ordinary character tables of some A tlas groups using character theoretic methods

T. Breuer, Constructing the ordinary character tables of some A tlas groups using character theoretic methods. , https://export.arxiv.org/abs/1604.00754 arXiv:1604.00754

arXiv

[7] [7]

, Permutation C haracters in GAP , https://www.math.rwth-aachen.de/ Thomas.Breuer/ctbllib/doc2/manual.pdf https://www.math.rwth 45 aachen.de/ 126 Thomas.Breuer/ ctbllib/ doc2/ manual.pdf

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R. W. Barraclough and R. A. Wilson, The character table of a maximal subgroup of the M onster , LMS J. Comput. Math. 10 (2007), 161 175. 2308856

work page 2007