REVIEW 2 major objections 3 minor 1 cited by
The continuity of $p$-rationality of characters and the principal block
T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read The paper proves Hung's continuity conjecture for p=2 inside the principal 2-block, and proposes a local criterion for the level-1 gap.
desk verdict A genuine principal-block refinement of Hung's p=2 theorem plus a new local criterion for the rationality gap; the Lie-type checks are compressed but the argument holds up. 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 key objects are the $p$-rationality level of a character, the exponent $\alpha$ such that $c(\chi)_p=p^\alpha$, and the principal block $B_0(G)$. The proofs use the Galois automorphism $\sigma_\alpha$, which fixes $p$-power order roots and sends $p'$-roots to their $1+p^\alpha$-th power, and the semi-inertia subgroup of a character under the Galois action. For the $p=2$ verification, the load-bearing structure is the decomposition of $\mathrm{Irr}(B_0(G))$ into Lusztig series $\mathcal{E}(G,t)$ indexed by 2-elements $t$ in the dual group, together with Jordan decomposition, which forces odd-degree characters to correspond to 2-central $t$, and Proposition 4.2, which says odd-degree unipotent characters are rational valued.
What would settle it
Compute directly the unipotent character tables of $E_7(q)$ or $E_8(q)$ with $q$ odd and check whether any odd-degree unipotent character has non-rational values; such a character would refute Proposition 4.2 and with it the proof of Theorem A.
Extended reading notes
Core claim
The central result is Theorem A: Conjecture 1.1 holds for $p=2$. That is, if $B_0(G)$ contains an irreducible character $\chi$ of odd degree with $c(\chi)_2=2^a$, then for every $2\le b\le a$ there is an odd-degree $\psi\in \mathrm{Irr}(B_0(G))$ with $c(\psi)_2=2^b$. The route is a general reduction, Theorem 3.3, showing that the conjecture follows for any $p$ once an explicit statement about almost simple groups (Conjecture 3.1) is verified; the authors verify it for $p=2$ in Theorem 4.3, using the description of the principal 2-block of a group of Lie type as a union of Lusztig series and the fact that odd-degree unipotent characters are rational. The paper also formulates Conjecture B, a necessary-and-sufficient local condition on $N_G(P)/\Phi(P)$ acting on $P/\Phi(P)$ for the absence of characters of $p'$-degree with $p$-rationality level 1, and proves it for groups with a normal Sylow $p$-subgroup.
Load-bearing premise
The proof of the $p=2$ simple-group check relies on the claim, justified in the text only by a quick inspection, that every irrational unipotent character of a finite group of Lie type in odd characteristic has even degree; if an odd-degree unipotent character were irrational, the chain leading to Theorem A would break.
Editorial extensions
If this is right
- Hung's continuity conjecture holds for $p=2$ for every finite group, so the principal 2-block cannot hide gaps in the ladder of 2-rationality levels.
- Any counterexample to Conjecture 1.1 for an odd prime must occur in an almost simple group, reducing the full conjecture to the verification of Conjecture 3.1.
- The results are consistent with the Alperin–McKay–Navarro conjecture: both the continuity in the principal block and the level-1 gap criterion are consequences of that conjecture.
- Conjecture B gives a computable local criterion for the level-1 gap, and Theorem 5.5 confirms it for groups with a normal Sylow $p$-subgroup, covering $p$-solvable, sporadic, symmetric, alternating, and defining-characteristic Lie-type groups.
Reading between the lines
- The dependence on $P/\Phi(P)$ in Conjecture B suggests the level-1 gap is governed by the minimal number of generators of a Sylow subgroup; one could test this by computing, in families not yet covered, whether the condition can be simplified to an inequality involving that number.
- The $p=2$ proof for Lie-type groups points to a possible odd-prime analogue: when $p$ divides $q-\varepsilon$, the role of 2-central elements might be played by elements of order $|t|_p$, and the semisimple characters in the corresponding Lusztig series would supply the required levels.
- The paper's $D_{24}$ example shows that level-1 phenomena are not visible in the principal block; this suggests that any general principal-block version of the gap must pass to $G/O_{p'}(G)$ or to the defect group, as Conjecture 5.7 does.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper studies the p-part of character conductors of finite groups. Its main result, Theorem A, proves the principal-block form of Hung's continuity conjecture for p=2: if the principal 2-block B0(G) contains a p′-degree irreducible character whose 2-rationality level is α≥2, then B0(G) contains p′-degree irreducible characters of every level β with 2≤β≤α. The proof combines a reduction theorem (Theorem 3.3) that, assuming a conjectural statement for almost simple groups (Conjecture 3.1), reduces the problem to almost simple groups, with a verification for p=2 in Theorem 4.3 using Lusztig series and Jordan decomposition. The paper also formulates Conjecture B, a local group-theoretic criterion for the absence of p′-degree characters of p-rationality level 1, proves it for groups with a normal Sylow p-subgroup (Theorem 5.5), and derives applications to p-solvable, sporadic, symmetric, and alternating groups via the McKay–Navarro conjecture.
Significance. The main theorem is a genuine refinement of Hung's continuity result, since it localizes the phenomenon to the principal block and the reduction is effective for p=2. The proof strategy is transparent: external results of Hung, Malle, Navarro–Tiep, and others are used as hypotheses rather than as circular inputs, and I see no circularity in the main argument. Theorem 3.3 is carefully structured and is a reusable reduction statement. Section 5 gives a substantive normal-Sylow proof of a new local conjecture, and the use of semi-inertia subgroups and coprime actions is elegant. If the compressed points in Section 4 are filled in, the paper will be a solid contribution to the area and relevant to the Alperin–McKay–Navarro program.
major comments (2)
- [§4, Theorem 4.3 (final paragraph)] The treatment of G=SL_{2n}(εq) is too compressed to be checked as written. After quoting [8, Tab. 4.5.1] and [3, Thm 21.14], the proof ends with 'we can argue exactly as in the previous case', but the previous case used the explicit cyclic group Z of order (q−ε)_2 and then formed the powers t^k. For PGL_{2n}(εq) this step is not automatic from connectedness of centralizers: one must say that if t is the given 2-central element of order 2^α, then each t^{2^{α−β}} is again 2-central and its centralizer is among the connected Levi subgroups, so [9, Thm 5.6] supplies rationality level β and the unique semisimple character in E(G,t^{2^{α−β}}) is A-invariant. As written, the chain of characters realizing levels 2≤β≤α is not actually constructed for this infinite family. Please expand this paragraph explicitly.
- [§4, Proposition 4.2] The proof of the claim that every unipotent character of odd degree is rational is reduced to 'a quick inspection shows that all irrational unipotent characters have even degree'. This is load-bearing, because Theorem 4.3 uses it to discard all unipotent characters from the p=2 argument. The finite check over the exceptional families should be made explicit, for example by citing the relevant character-degree and character-field tables in [6] and explaining why every row with irrational field has even degree. Without this, the statement is an assertion rather than a verified step.
minor comments (3)
- [§2, Lemma 2.7] The sentence 'By hypothesis, we can assume that P⊴G' is not justified by Conjecture 2.1 as stated. Conjecture 2.1 gives a bijection between Irr_{p′}(G) and Irr_{p′}(N_G(P)); the proof should transfer the given character to N_G(P), construct characters there, and transfer back. As written the lemma is incomplete, although the intended reduction is clear.
- [§5, proof of Theorem 5.5] The symbol G is used for both the finite group and the Galois group Gal(Q_p/Q), which makes the proof hard to follow (for example, 'Write G = Gal(Q_p/Q)' while G is also the group under study). Please use a different notation such as Γ or 𝒢 for the Galois group.
- [§4, Theorem 4.3 (E6 paragraph)] The odd-degree condition for the semisimple characters χ_t is implicit: it follows because t is 2-central, so a Sylow 2-subgroup is contained in C_{G^*}(t) and the degree is odd. Stating this explicitly would make the argument easier to verify and would prepare the same fact for the SL_{2n} case.
Circularity Check
No significant circularity: Theorem A is derived from external simple-group results and prior theorems, not from its own conclusion.
full rationale
The derivation chain is: Conjecture 1.1 is reduced to Conjecture 3.1 in Theorem 3.3; Theorem 4.3 proves Conjecture 3.1 for p = 2 using independent results [13], [20], [9], [3], [5], and [8]. The target statement is never used as an input. The E6 and SL_{2n} arguments compute rationality levels via [9, Thm 5.6] from the order of 2-central elements and propagate A-invariance to powers t^k; this is a computation, not a restatement. Two passages warrant explicit flagging but are not circular: (1) Proposition 4.2's 'a quick inspection shows that all irrational unipotent characters have even degree' is an unstated case check; (2) Theorem 4.3's SL_{2n} case closes with 'But then we can argue exactly as in the previous case', omitting the cyclic-power argument that was explicit for E6. Both are proof gaps, not self-referential inputs. The same-author citation [13] is load-bearing for Conjecture 3.1(a), but it is an independent published proof of the Navarro–Tiep Galois conjecture for p = 2, so it counts as real evidence under the rules and does not raise the circularity score.
Assumptions & free parameters
assumptions (5)
- standard math Classification of finite simple groups (CFSG)
- standard math Deligne-Lusztig theory and Jordan decomposition of characters
- domain assumption Navarro-Tiep [20, Thms B and 5.7] on exponents and p-rationality levels
- domain assumption Malle [13] settles the Navarro-Tiep Galois conjecture for p=2
- standard math Brauer's lemma on character tables and Clifford-Gallagher theory
Cite this review
Pith. "Pith review of The continuity of $p$-rationality of characters and the principal block." pith.science (2026). https://pith.science/paper/ROIS7EUS
@misc{pith2026241216128,
author = {Pith},
title = {Pith review of: The continuity of $p$-rationality of characters and the principal block},
year = {2026},
howpublished = {\url{https://pith.science/paper/ROIS7EUS}},
note = {Machine review of arXiv:2412.16128}
}
abstract
We study rationality properties of irreducible characters of finite groups. We show that the continuity of $2$-rationality is a phenomenon that can be detected in the principal $2$-block, thus refining a recent result of N. N. Hung. We also propose a conjectural group theoretical criterion for the continuity gap at level $1$ for all primes
Forward citations
Cited by 1 Pith paper
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The Isaacs--Navarro Galois conjecture
The Isaacs-Navarro Galois conjecture is proved: for every finite group G and prime l, there is an H0-equivariant bijection between the l'-degree characters of G and those of the normalizer of a Sylow l-subgroup.
Reference graph
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