[#P2902] Positivity of the p-element centralizer generalized character

Problem. Let \(G\) be a finite group and \(p\) a prime. Define a class function \(\Psi_{G,p}:G\to\mathbb Z\) by \(\Psi_{G,p}(g)=0\) when \(p\) divides the order of \(g\), and, when the order of \(g\) is coprime to \(p\), let \(\Psi_{G,p}(g)\) be the number of elements of the centralizer \(C_G(g)\) whose order is a power of \(p\). Is \(\Psi_{G,p}\) an ordinary complex character of \(G\) for every finite \(G\) and every prime \(p\)?

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Definitions and notation

1Context

Each finite group yields exact character multiplicities, so systematic computations can eliminate families and expose minimal counterexamples. Structural reductions and failed induction inequalities are also highly reusable.

2Definitions

Definition 1 (A p-element). A p-element is an element whose order is a power of p, including the identity of order 1; an element is p-regular when its order is coprime to p.

Definition 2 (An ordinary complex character). An ordinary complex character is the trace character of a finite-dimensional complex representation, equivalently a nonnegative integer combination of irreducible complex characters.

3What counts as a solution

Prove that every irreducible-character multiplicity in Psi_{G,p} is a nonnegative integer for all finite G and p, or give a finite group G and prime p for which some multiplicity is negative.
A counterexample must identify G by a reproducible presentation or database ID and include its character table, conjugacy-class data, centralizer p-element counts, and the resulting negative inner product.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. The source has zero answers. Robinson's open-access 2026 Journal of Algebra paper studies this exact class function, proves many families, and retains universal character positivity as a conjecture. Prove that every irreducible-character multiplicity in Psi_{G,p} is a nonnegative integer for all finite G and p, or give a finite group G and prime p for which some multiplicity is negative.^[1]

1Packet records

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Record	Kind	Assessment
UNKNOWN as of 2026-07-31. The source has zero answers. Robinson's open-access 2026 Journal of Algebra paper studies this exact class function, proves many families, and retains universal character positivity as a conjecture. Prove that every irreducible-character multiplicity in Psi_{G,p} is a nonnegative integer for all finite G and p, or give a finite group G and prime p for which some multiplicity is negative.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 UNKNOWN as of 2026-07-31. The source has zero answers. Robinson's open-access 2026 Journal of Algebra paper studies this exact class function, proves many families, and retains universal character positivity as a conjecture. Prove that every irreducible-character multiplicity in Psi_{G,p} is a nonnegative integer for all finite G and p, or give a finite group G and prime p for which some multiplicity is negative.^[1] Relevance to this problem For p element centralizer generalized character, pins the dated research frontier: UNKNOWN as of 2026-07-31. The source has zero answers. Robinson's open-access 2026 Journal of Algebra paper studies this exact class function, proves many families, and retains universal character positivity as a conjecture. Prove that every irreducible-character multiplicity in Psi_{G,p} is a. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details UNKNOWN as of 2026-07-31. The source has zero answers. Robinson's open-access 2026 Journal of Algebra paper studies this exact class function, proves many families, and retains universal character positivity as a conjecture. A complete resolution must satisfy this condition: Prove that every irreducible-character multiplicity in Psi_{G,p} is a nonnegative integer for all finite G and p, or give a finite group G and prime p for which some multiplicity is negative. Open full record →
Prove that every irreducible-character multiplicity in Psi_{G,p} is a nonnegative integer for all finite G and p, or give a finite group G and prime p for which some multiplicity is negative.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove that every irreducible-character multiplicity in Psi_{G,p} is a nonnegative integer for all finite G and p, or give a finite group G and prime p for which some multiplicity is negative.^[1] Relevance to this problem For Positivity of the p-element centralizer generalized character, record p-element-centralizer-generalized-character-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove that every irreducible-character multiplicity in Psi_{G,p} is a nonnegative integer for all finite G and p, or give a finite group G and prime p for which some multiplicity is negative. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove that every irreducible-character multiplicity in Psi_{G,p} is a nonnegative integer for all finite G and p, or give a finite group G and prime p for which some multiplicity is negative. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. UNKNOWN as of 2026-07-31. The source has zero answers. Robinson's open-access 2026 Journal of Algebra paper studies this exact class function, proves many families, and retains universal character positivity as a conjecture.

On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for MathOverflow question 478499; comments provide computations and special cases only.
Robinson, Journal of Algebra 690 (2026), 37-74, defines the same Psi_{1,p,G}, conjectures that it is always a character and even projective, and proves the projective claim for PSL(2,q) and SL(2,q) for every relevant p and q.
The source comments report checks for all groups of order at most 200, symmetric and alternating groups through degree 15, and several low-rank Lie-type groups; these are reusable finite baselines rather than a general proof.
A TheoremDB search for Psi_{1,p,G}, p-element centralizer counts, truncated conjugation modules, and projective character positivity found no duplicate.

Recorded example 1. If p does not divide |G|, then Psi_{G,p} is the trivial character. If G is a p-group, it is the regular character.

Computational notes

The checked source reports no counterexample among all groups of order at most 200 and several larger standard families, but the exact scripts and digests were not attached to the MO post.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemPositivity of the p-element centralizer generalized character

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Largest girth from two generators of SL(2,101)finite groups
Common divisors of disjoint coset indicesfinite groups
A multiplicative lower bound for numbers of finite groupsfinite groups

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“Positivity of the p-element centralizer generalized character.” TheoremDB. P2902. Problem statement; statement text SHA-256 a6c4251f5d3dbd14f27031dbbe5aafd737e8c52c84c9327bbb96bfab9efab4cd. https://theoremdb.org/statement/?ref=P2902

BibTeX

@misc{theoremdb-problem-a6c4251f5d3dbd14f27031dbbe5aafd737e8c52c84c9327bbb96bfab9efab4cd,
  title = {{Positivity of the p-element centralizer generalized character}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 a6c4251f5d3dbd14f27031dbbe5aafd737e8c52c84c9327bbb96bfab9efab4cd},
  url = {https://theoremdb.org/statement/?ref=P2902}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. MathOverflow: Is this generalized character always a character?. Question 478499 and all visible comments, checked through the Stack Exchange API on 2026-07-27, including the November 2025 paper update. Question 478499 and all visible comments, checked through the Stack Exchange API on 2026-07-27, including the November 2025 paper update. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The source has zero answers. Robinson's open-access 2026 Journal of Algebra paper studies this exact class function, proves many families, and retains universal character positivity as a conjecture.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Positivity of the p-element centralizer generalized character: UNKNOWN as of 2026-07-27. The source has zero answers. Robinson's open-access 2026 Journal of Algebra paper studies this exact class function, proves many families, and retains universal character positivity as a conjecture.Source named by the research packet.
Geoffrey R. Robinson, “A generalized character related to the p-local structure and representation theory of a finite group”. Journal of Algebra 690 (2026), 37-74. DOI 10.1016/j.jalgebra.2025.11.001. Status evidence identified in the source record and checked at the linked publication. ↗preprint · primary source · arXiv:2505.03976, checked 2026-07-31 · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The source has zero answers. Robinson's open-access 2026 Journal of Algebra paper studies this exact class function, proves many families, and retains universal character positivity as a conjecture.Also cited at Full journal article relevant to Positivity of the p-element centralizer generalized character.Source used to assess the problem's recorded status.For Positivity of the p-element centralizer generalized character: UNKNOWN as of 2026-07-27. The source has zero answers. Robinson's open-access 2026 Journal of Algebra paper studies this exact class function, proves many families, and retains universal character positivity as a conjecture.

This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.

[#P2902] Positivity of the p-element centralizer generalized character

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