{"id":"628e58e2-66ca-4963-a2a8-0ce84857095a","arxiv_id":"2607.16732","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every finite group with p in {2,q} dividing its order has a non-trivial principal-p-block irreducible character of pi'-degree with values in Q(xi_q).","lead":"The paper proves a new existence theorem in finite group character theory: whenever a finite group's order is divisible by 2 or a chosen odd prime q, its principal p-block contains a non-trivial irreducible character whose degree is coprime to 2 and q and whose values lie in the qth cyclotomic field. This extends earlier results of Navarro--Tiep and Giannelli--Hung--Schaeffer Fry--Vallejo into the principal-block setting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lie-type case in Prop. 2.9 rests entirely on [GHSV21, Prop. 4.5] without restating its hypotheses; if that proposition excludes any simple Lie-type group, the principal-p-block character is not proved for that family.","rationale":"The reader's weakest assumption correctly identifies the reliance on [GHSV21, Proposition 4.5] as the least secure point. The rest of the proof — the reduction to simple groups, the alternating-group combinatorics, and the p=2 Brauer-character argument — is coherent and no internal inconsistency was found. However, the Lie-type case is infinite and the paper's Proposition 2.9 does not establish that the external proposition covers all relevant groups. This is a genuine but local and potentially fixable gap: if the hypotheses of Prop. 4.5 match exactly, the theorem stands; if not, the affected Lie-type families need a separate argument. The reader's CONDITIONAL verdict is appropriate, so no change in verdict is recommended.","tokens_in":9722,"tokens_out":18178,"duration_ms":164441,"concrete_test":"Obtain the full statement of [GHSV21, Proposition 4.5] and check its hypotheses against every simple group of Lie type not covered by Proposition 2.8. In particular, recompute the smallest nontrivial cases, e.g. PSL(3,3) with p=2,q=3 and PSL(2,8) with p=3,q=2, using GAP/Chevie or the cited construction: verify that the semisimple character χ_s is nontrivial, has degree coprime to 2q, satisfies Q(χ_s) ⊆ Q(ξ_q), lies in the principal p-block, and vanishes on Z(G) so that it descends to S. If any of these checks fails, Proposition 2.9 needs a new argument or a revised hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2.9 is the only step covering simple groups of Lie type without exceptional Schur multiplier. In the case ℓ ∈ π, ℓ ≠ p, the proof invokes [GHSV21, Proposition 4.5] to obtain a semisimple character χ_s attached to a p-power-order semisimple element s with Q(χ_s) ⊆ Q(ξ_q). The paper does not restate the hypotheses of that proposition or verify that it applies to every simple Lie-type group not already handled by Prop. 2.8. In particular, it is not demonstrated that the character produced by Prop. 4.5 is nontrivial, has π'-degree, lies in the principal p-block after descent to S = G/Z(G), and has Z(G) in its kernel. The descent via [CE04, Lemma 17.2] depends on the latter assertion, which is simply stated. If Prop. 4.5 has hidden exclusions — for example, restrictions on the Lie rank, the characteristic ℓ, or the relationship between q and the group — then the main theorem is not established for the affected infinite families. This is load-bearing because Theorem A is reduced in Theorem 1.4 to exactly these simple groups, and no alternative argument is supplied.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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].","tokens_in":10041,"tokens_out":21109,"duration_ms":177799,"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":[{"comment":"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.","section":"§2, Proposition 2.9 (case ℓ∈π, ℓ≠p)"},{"comment":"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.","section":"§2, Proposition 2.9 (descent to S)"}],"minor_comments":[{"comment":"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.","section":"§2, Lemma 2.4"},{"comment":"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.","section":"§2, Proposition 2.8"},{"comment":"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.","section":"§1, Theorem 1.4"},{"comment":"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.","section":"§2, Theorem 2.6"},{"comment":"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.","section":"§2, Proposition 2.9 (case ℓ=p)"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the black-box use of [GHSV21, Proposition 4.5] for infinitely many Lie-type groups. Since the same author group is involved, it is essential that the editor ensure the cited proposition is in the public record, has no hidden exclusions, and that the authors either reproduce the relevant statement or provide a proof of the exact consequence they need. The GAP verification should be made available for checking. I do not see evidence of circularity; the theorem is derived from CFSG and published block theory."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a solid, workmanlike paper. The genuinely new content is Theorem A: for π={2,q}, every finite group of order divisible by p∈π has a nontrivial π′-degree irreducible character in the principal p-block with field of values in Q(ξ_q). That principal-block version was not known, and it does not follow trivially from the earlier [GHSV21] results. The reduction theorem (Theorem 1.4) is coherent and the alternating-group core computations in Theorem 2.6 are careful. The paper also fixes a real error in [GHSV21, Theorem D] via Lemma 2.4 and Remark 2.5, with a concrete counterexample (p=3, n=14, λ=(11,3)) — that alone is a useful service to the literature.\n\nThe soft spots are real but minor. Lemma 2.4 says a verification is omitted; a referee should ask for it to be included. Proposition 2.8 is a GAP computation with no code or output supplied; for sporadic groups and exceptional Schur multipliers that is borderline acceptable, but the paper should at least say what exactly was checked. The stress-test concern about Proposition 2.9 is fair as a clarity issue: the paper relies on [GHSV21, Prop. 4.5] without restating its hypotheses or explicitly verifying that the resulting semisimple character has Z(G) in its kernel and lies in the principal p-block after descent. The authors assert the kernel property, and the descent uses [CE04, Lemma 17.2]. This is not a discovered counterexample or a hole I can find; it is an omitted set of details. Since the same authors wrote the cited proposition and are correcting that paper in Remark 2.5, I would be surprised if the hypothesis check failed, but a referee should make them spell it out.\n\nThe citation pattern looks fine: the heavy lifting is from [GHSV21], but the paper says so plainly and the new block-membership arguments are their own. The paper does not overclaim; the discussion in Section 3 correctly notes limits, including the necessity of 2∈π and failure for three primes.\n\nWho gets value from this: people working on principal blocks and character fields. It is not a breakthrough, but it is a correct, citable refinement. I would accept it for peer review and ask for minor revisions: fill the omitted verification, give GAP details, and restate Prop. 4.5's hypotheses in Prop. 2.9. My own take is that the central argument holds up, and the condition in the reader's report is about presentation, not substance.","headline":"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.","tokens_in":10490,"tokens_out":2154,"would_cite":true,"duration_ms":22113,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20C15","20C30","20C33"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["character degrees","principal blocks","two primes","fields of values","cyclotomic fields","finite groups","irreducible characters","Lie-type groups"],"falsifier":"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.","tokens_in":9633,"feed_emoji":"🎭","tokens_out":13609,"duration_ms":107392,"temperature":0.7,"pith_summary":"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.","feed_headline":"Every principal block has a character of degree coprime to 2 and q","feed_subtitle":"Its field of values is also contained in the q-th cyclotomic extension, sharpening a classical theorem to principal blocks.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Principal p-blocks: cyclotomic characters with 2,q-free degree","Cyclotomic character fields in principal p-blocks: degrees coprime to 2 and q","2,q-coprime degree with cyclotomic values in every principal p-block","Principal p-blocks: characters with cyclotomic values, 2,q-free degree"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Principal p-blocks: cyclotomic characters with 2,q-free degree","Cyclotomic character fields in principal p-blocks: degrees coprime to 2 and q","2,q-coprime degree with cyclotomic values in every principal p-block","Principal p-blocks: characters with cyclotomic values, 2,q-free degree"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00188,"raw_usage":{"total_tokens":7152,"prompt_tokens":621,"completion_tokens":6531,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":365,"completion_tokens_details":{"reasoning_tokens":6441}},"tokens_in":365,"tokens_out":6531,"duration_ms":41319,"temperature":1.0,"reasoning_tokens":6441,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T20:06:09.352090+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}