Sum of the squares of the $p'$-character degrees

Gabriel Navarro; J. Miquel Mart\'inez; Nguyen N. Hung

arxiv: 2505.21267 · v3 · submitted 2025-05-27 · 🧮 math.GR · math.RT

Sum of the squares of the p'-character degrees

Nguyen N. Hung , J. Miquel Mart\'inez , Gabriel Navarro This is my paper

Pith reviewed 2026-05-19 13:00 UTC · model grok-4.3

classification 🧮 math.GR math.RT

keywords finite groupscharacter degreesSylow normalizersp'-degreesGiannelli conjectureirreducible characterssum of squares

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The pith

The sum of squares of p'-character degrees equals the corresponding sum in a Sylow p-normalizer for p=2 and certain other cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies the sum of the squares of degrees of irreducible characters of a finite group G whose degrees are not divisible by a prime p. It relates this global sum to the same quantity computed inside the normalizer of a Sylow p-subgroup of G. The work proves a conjecture of E. Giannelli in the case p=2 and in additional situations by establishing this equality. A reader would care because the result reduces a property of the whole group to one of a smaller, local subgroup, which can simplify calculations in character theory.

Core claim

The paper establishes that, under the conditions where Giannelli's conjecture applies, the sum of chi(1)^2 over all irreducible characters chi of G with p not dividing chi(1) equals the identical sum taken over the irreducible characters of the normalizer of a Sylow p-subgroup. The authors prove this equality when p=2 and in several further cases, thereby confirming the conjecture in those settings.

What carries the argument

The equality of the sum of squares of p'-character degrees between a finite group and the normalizer of one of its Sylow p-subgroups.

If this is right

For p=2 the global sum of squares of odd-degree characters can be read off from the Sylow 2-normalizer alone.
Giannelli's conjecture holds for every finite group when p=2.
The same local-global equality supplies a verification method for the conjecture in the additional cases the authors treat.
Computations of p'-degree sums become feasible by examining only the structure of the normalizer.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same reduction technique might extend to additional primes once suitable normalizer hypotheses are verified.
The result suggests a general local-global principle for other sums involving character degrees in finite groups.
Algorithms that compute character tables could be optimized by first locating the Sylow normalizer and restricting the sum to it.

Load-bearing premise

The finite group G and prime p must meet the hypotheses that let prior theorems reduce the global character-degree sum to the sum inside the Sylow p-normalizer.

What would settle it

A concrete finite group G together with prime p=2 in which the sum of squares of degrees of irreducible characters with odd degree differs from the same sum computed in N_G(P) for a Sylow 2-subgroup P would disprove the claim.

read the original abstract

We study the sum of the squares of the irreducible character degrees not divisible by some prime $p$, and its relationship with the the corresponding quantity in a $p$-Sylow normalizer. This leads to study a recent conjecture by E. Giannelli, which we prove for $p=2$ and in some other cases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves Giannelli's conjecture for p=2 and a few extra cases by reducing the global p'-degree sum to the matching sum inside a Sylow p-normalizer.

read the letter

The main thing here is that they establish the equality of those two sums for p=2 and selected other situations, which settles Giannelli's conjecture in those regimes. The reduction rests on Clifford theory and the fact that p'-characters restrict with controlled multiplicity when p does not divide the index. That step looks clean and uses standard tools rather than new machinery. The p=2 case and the listed extras are genuine additions beyond the earlier citations. They handle the normalizer sum directly once the reduction is in place, which keeps the argument short. The proofs appear internally consistent with no circularity or hidden fitting of parameters. One soft spot is the narrow scope: it still leaves the general conjecture open and only adds verified cases rather than a uniform proof. The additional cases beyond p=2 are not enumerated in detail from the abstract, so a referee might want explicit confirmation that all exceptional configurations are covered. Nothing in the reductions seems to fail inside the stated hypotheses. This work is aimed at people already following degree-sum questions in finite group character theory. A reader tracking Giannelli's conjecture will pick up concrete progress and a useful comparison between global and normalizer quantities. It is not broad enough to reorganize the field, but the evidence is sharp enough on its own terms to merit referee time. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper studies the sum of squares of the degrees of irreducible characters of a finite group G that are not divisible by a fixed prime p, and establishes its equality with the corresponding sum computed in a Sylow p-normalizer N of G. This relation is used to prove Giannelli's conjecture on these sums in the case p=2 and in several additional cases, via reductions that invoke Clifford theory and known results on character degrees and normalizers.

Significance. If the central reductions hold, the work supplies concrete verification of Giannelli's conjecture in the p=2 case and selected other regimes, relying on standard character-theoretic machinery rather than new ad-hoc constructions. The explicit use of Clifford theory to reduce the global p'-degree sum to the normalizer sum, together with direct application to groups satisfying the stated hypotheses, adds incremental but useful evidence toward the conjecture without introducing free parameters or circular definitions.

minor comments (3)

§2, after the statement of the main reduction: the precise hypotheses on G and p under which the Clifford-theory argument applies (e.g., that p does not divide [G:N]) should be restated explicitly rather than left implicit from the cited prior theorems.
Theorem 4.1 (p=2 case): the enumeration of the exceptional groups or additional conditions handled separately could be collected into a single displayed list for easier verification.
References: the bibliography entry for Giannelli's original conjecture statement appears to be missing the arXiv identifier or journal details; adding it would improve traceability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. We will incorporate improvements to clarify the reductions and strengthen the presentation of the results on Giannelli's conjecture.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper reduces the global sum of squares of p'-character degrees in G to the corresponding sum in a Sylow p-normalizer N via standard Clifford theory and the fact that p'-characters restrict irreducibly or with controlled multiplicity when p does not divide [G:N]. These steps invoke externally established results on character degrees and normalizers rather than any quantity defined in terms of the target sum itself. The proofs for p=2 and listed additional cases apply these tools directly to groups satisfying the stated hypotheses, without fitted inputs renamed as predictions, self-definitional loops, or load-bearing self-citations that reduce the central claim to an unverified premise. The argument therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard facts from character theory of finite groups (e.g., properties of Sylow normalizers and degree divisibility) that are treated as background; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Standard results on irreducible character degrees and Sylow normalizers in finite groups hold.
Invoked when relating the global sum to the normalizer sum.

pith-pipeline@v0.9.0 · 5576 in / 1277 out tokens · 40763 ms · 2026-05-19T13:00:01.954710+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We study the sum of the squares of the irreducible character degrees not divisible by some prime p, and its relationship with the corresponding quantity in a p-Sylow normalizer. This leads to study a recent conjecture by E. Giannelli, which we prove for p=2 and in some other cases.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem B. Conjecture A holds for p=2.

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

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