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REVIEW 2 major objections 5 minor 17 references

Cyclotomic character fields and sets of primes

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read For π={2,q} and p∈π, the principal p-block of any finite group contains a non-trivial character of π'-degree whose values lie in the q-th cyclotomic field.

desk verdict A competent, useful principal-block upgrade of known character results, with an honest correction of the authors' earlier theorem; worth refereeing with minor revision requests. read the letter →

arxiv 2607.16732 v1 pith:62DFXQU6 submitted 2026-07-18 math.RT

Eugenio Giannelli , J. Miquel Martínez , Carolina Vallejo This is my paper
classification math.RT MSC 20C1520C3020C33
keywords characterdegreesprincipalblockstwoprimesfieldsofvaluescyclotomicfinitegroupsirreduciblecharactersLie-type
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 proves that for any finite group G, any odd prime q, and any prime p among 2 and q that divides |G|, the principal p-block of G contains a non-trivial irreducible character whose degree is divisible by neither 2 nor q and whose field of values is contained in the q-th cyclotomic field. This is a principal-block strengthening of earlier results that produced such characters without the block restriction. The proof reduces the statement to finite simple groups via classical block theory, then treats alternating groups with explicit partitions and Lie-type groups with semisimple characters. The authors also show the prime 2 is essential in the statement, and that the claim fails when more than two primes are considered.

What carries the argument

The central object is the principal p-block B_p(G), the block containing the trivial character. The reduction theorem uses standard block theory: the structure of principal blocks in p-solvable groups, a block isomorphism under normal subgroups with p'-quotient, and the third main theorem for blocks, to reduce the claim to simple non-abelian groups. For alternating groups the proof works combinatorially with partitions, hook lengths, and the p-core that determines the block; for Lie-type groups it uses semisimple characters coming from the dual group, lifted through regular embeddings and pushed into the principal block.

What would settle it

Compute the principal 2-block of a small Lie-type group such as PSL(2,8) (with q=3, p=2): the theorem predicts a non-trivial character of degree coprime to 2 and 3 with values in Q(ξ_3). Checking the character table and the block decomposition directly would either confirm the predicted character or produce a counterexample if no such character exists.

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Extended reading notes

Core claim

The central claim is that for π={2,q} with q an odd prime and p∈π, whenever p divides the order of a finite group G, the principal p-block B_p(G) contains a non-trivial irreducible character χ with χ(1) coprime to both 2 and q and Q(χ) ⊆ Q(ξ_q). The proof goes through a reduction theorem that shows it suffices to verify the statement for non-abelian simple groups, where it is established for alternating groups by selecting partitions with controlled p- and q-cores, and for groups of Lie type by producing a semisimple character of p-power order with values in the right cyclotomic field.

Load-bearing premise

The infinite Lie-type case rests on an external proposition about semisimple characters whose hypotheses are not restated; if that proposition does not apply to every relevant simple group of Lie type, the principal-block character for those groups would not be established.

Editorial extensions

If this is right

  • For every prime p, any finite group of order divisible by p has a non-trivial character in its principal p-block of degree coprime to p with field of values contained in the p-th cyclotomic field, generalizing a classical result to principal blocks.
  • Characters with fields of values in cyclotomic extensions are known to control normal p-complements; Theorem A makes such control available for characters inside the principal block, where structure theorems are typically stronger.
  • The hypothesis 2∈π is necessary: for the sporadic simple group J_4, no character of π'-degree with values in Q(ξ_q) exists when π={23,43}, so the set must contain the prime 2.
  • The statement cannot be extended to three primes: for A_5 with π={2,3,5}, no non-trivial character has π'-degree, so the two-prime setting is sharp.
  • A corollary is a principal-block version of a recent result for the pair {2,q}, strengthening the conclusion by forcing the character into the principal block.

Reading between the lines

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

  • If the method could be pushed to arbitrary pairs of primes not containing 2, it would settle a conjecture posed in the paper about principal blocks always containing characters of π'-degree; the authors identify the Lie-type case as the main remaining difficulty.
  • The necessity of the prime 2 suggests a parity mechanism: with 2 present, the cyclotomic field Q(ξ_q) is forced to absorb the character, whereas for odd-prime pairs the field restriction alone is too strong (as in J_4).
  • Since the proof relies on a previously published proposition for Lie-type groups, re-proving that proposition in a self-contained way would remove the only non-verified step on the infinite family of groups.
  • The combinatorial construction for alternating groups corrects an error in a prior paper, which indicates that this area is delicate; a similar audit of the Lie-type semisimple character construction could reveal hidden exceptional cases.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proves Theorem A: for π={2,q} with q an odd prime and p∈π, every finite group G of order divisible by p has a nontrivial irreducible character in the principal p-block whose degree is coprime to both 2 and q and whose field of values lies in Q(ξ_q). This is presented as a principal-block refinement of results of Navarro–Tiep and of the same authors' earlier work, and Corollary B gives the p-block version of the Navarro–Tiep theorem for a single prime. The proof reduces the statement to finite simple nonabelian groups (Theorem 1.4), proves the alternating-group case by explicit partition/hook combinatorics (Theorem 2.6), and handles sporadic and Lie-type groups in Propositions 2.8 and 2.9, relying on the classification and on several external results. The paper also corrects a small error in the proof of [GHSV21, Theorem D].

Significance. If the result is correct, it is a genuine contribution: it shows that for the two-prime set {2,q} the existence of a character with small cyclotomic field and prescribed degree can be combined with membership in the principal block, and it provides a principal-block analogue of a known non-block theorem. The alternating-group section is the main novel combinatorial part and contains an honest correction of a prior oversight. The reduction theorem is coherent, and the finite-group part of the proof is plausible. The principal weakness is that the infinite Lie-type case in Proposition 2.9 is not self-contained: it relies on [GHSV21, Proposition 4.5] without stating its hypotheses or verifying that the character produced there has all the properties needed after descent. The GAP verification for sporadic and exceptional groups is also not reproducible as written. These issues are local but load-bearing, so the manuscript needs revision before the central claim can be accepted.

major comments (2)
  1. [§2, Proposition 2.9 (case ℓ∈π, ℓ≠p)] The entire argument for the remaining infinite families of Lie-type groups rests on [GHSV21, Proposition 4.5], but the hypotheses of that proposition are not stated. Please restate it as a lemma and verify that it applies to every simple group of Lie type with nonexceptional Schur multiplier. In particular, show that for every relevant Lie rank and defining characteristic ℓ, the semisimple character χ_s associated to a p-power-order element s is nontrivial, has π'-degree, and satisfies Q(χ_s)⊆Q(ξ_q) simultaneously. If [GHSV21, Proposition 4.5] has hidden exclusions, the theorem is not established for the affected infinite families.
  2. [§2, Proposition 2.9 (descent to S)] Even accepting [GHSV21, Proposition 4.5], the descent from the reductive group G to S=G/Z(G) is incomplete. The proof asserts that 'the characters χ_s constructed in [GHSV21, Proposition 4.5] contain Z(G) in their kernel' and that χ_s lies in Irr(B_p(S)), but neither assertion is demonstrated. The application of [CE04, Lemma 17.2] depends exactly on the kernel condition, and the block containment in S requires checking that the induced block from G is the principal block of S. Please supply the missing verification or formulate it as a lemma with proof.
minor comments (5)
  1. [§2, Lemma 2.4] In the case pw=p^k the proof says 'We omit the full details of this verification.' Since Lemma 2.4 is used to correct [GHSV21, Theorem D], please include the full verification or a clearer outline.
  2. [§2, Proposition 2.8] The statement 'This can be confirmed using [GAP]' is not reproducible. Please provide the GAP code or a table listing the chosen characters for the sporadic groups and Lie-type groups with exceptional Schur multiplier.
  3. [§1, Theorem 1.4] The deduction 'the only irreducible character of G/E with π'-degree and values in Q(ξ_q) is 1_{G/E}, hence G/E is a group of odd order using [GHSV21, Theorem A]' should be expanded. The exact form of [GHSV21, Theorem A] being invoked is not stated, and the contrapositive used is not immediate to the reader.
  4. [§2, Theorem 2.6] In the line 'a_1 p^{m_1} = n = b_1 p^{k_1}+1', the last term should presumably be b_1 q^{k_1}+1, since the q-adic expansion of n is used. Please correct this typo.
  5. [§2, Proposition 2.9 (case ℓ=p)] The equality 'Irr_{p'}(B_p(S)) = Irr(S)' is formally false; the first set consists of p'-degree characters while the second contains all irreducible characters. The intended statement is likely 'Irr(B_p(S)) = Irr(S)\setminus\{St_S\}' or 'Irr_{p'}(B_p(S)) = Irr_{p'}(S)'. Please rephrase.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the principal-block refinement is derived from standard block theory and CFSG; [GHSV21] supplies only the block-free character input.

full rationale

Walking the derivation chain, Theorem 1.4 reduces Theorem A to the assertion that every non-abelian simple group of order divisible by p has the desired character in its principal p-block. That assertion is not assumed: Section 2 proves it. For alternating groups (Theorem 2.6), [GHSV21, Theorem D] is used only to supply a block-free character χ ∈ Irr_{π'}(A_n) with the required field of values; the paper then proves, by p- and q-core computations (Lemma 2.3), that the same χ lies in B_p(A_n) ∩ B_q(A_n). For Lie-type groups (Prop. 2.9), [GHSV21, Prop. 4.5] is cited for a semisimple character χ_s of π'-degree with Q(χ_s) ⊆ Q(ξ_q) outside the block context; the principal-block membership is then established using [Hi90, Cor. 3.4], [CE04, Prop. 15.6] and [CE04, Lemma 17.2], and the descent to S = G/Z(G). Thus the block condition, the novel content, is not an input to the cited results. The overlapping authorship of [GHSV21] does not make this circular: it is a parameter-free published theorem whose assumptions do not include the target block conclusion. The only flagged concern is that Prop. 2.9 does not restate the hypotheses of [GHSV21, Prop. 4.5]; that is a possible gap in coverage, not a definitional or fitted-input circularity. No fitted parameters, renamed results, or imported uniqueness theorems occur. Score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities; this is a pure existence theorem. The load-bearing inputs are CFSG, standard block theory, the authors' earlier [GHSV21] results, and an unscripted GAP computation for exceptional simple groups.

assumptions (6)
  • standard math Classification of Finite Simple Groups (CFSG)
    Invoked in the proof of Theorem 2.11 to reduce Theorem A to alternating, sporadic, and Lie-type simple groups; not proved in this paper.
  • domain assumption [GHSV21, Theorem D]: existence of pi'-degree characters in A_n with cyclotomic fields
    Theorem 2.6 uses the characters and degree/field properties from the authors' earlier paper, which the present paper partially corrects in Remark 2.5.
  • domain assumption [GHSV21, Proposition 4.5]: semisimple characters of pi'-degree in Lie-type groups
    Proposition 2.9's main case l != p relies on this prior proposition for existence of chi_s with Q(chi_s) subset of Q(xi_q); assumptions are not restated.
  • domain assumption GAP computation for sporadic groups, the Tits group, and Lie-type groups with exceptional Schur multiplier
    Proposition 2.8 asserts the result 'can be confirmed using [GAP]' without a script or computation record; reproducibility rests on GAP's character tables.
  • standard math Feit--Thompson odd order theorem
    Used in the proof of Theorem 1.4 after [GHSV21, Theorem A] to conclude G/E is solvable when the only character is trivial.
  • standard math Fong's theorem, Alperin--Dade theorem, and Brauer's third main theorem
    Standard block-theory results cited from Navarro's book and Alperin/Dade for the reduction and block transfer arguments.

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Pith. "Pith review of Cyclotomic character fields and sets of primes." pith.science (2026). https://pith.science/paper/62DFXQU6

@misc{pith2026260716732,
  author       = {Pith},
  title        = {Pith review of: Cyclotomic character fields and sets of primes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/62DFXQU6}},
  note         = {Machine review of arXiv:2607.16732}
}
abstract

Let $\pi=\{ 2, q \}$ where $q$ is an odd prime. Let $G$ be a finite group of order divisible by a prime $p \in \pi$. We show that the principal $p$-block of $G$ contains a nontrivial irreducible character of degree not divisible by $2$ nor $q$ and with field of values contained in the $q$th cyclotomic extension. This statement simultaneously provides a principal block version of results of Navarro--Tiep and Giannelli--Hung--Schaeffer Fry--Vallejo.

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