# P2902: Positivity of the p-element centralizer generalized character

- ID: `P2902`
- Reference: `p-element-centralizer-generalized-character`
- Page: https://theoremdb.org/statements/P2902
- Record maturity: Reviewed problem with recorded work

## 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 \(p\)?

### Definitions

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

### What 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.

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

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.

- Original intake status: 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.
- 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: 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### 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.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `p-element-centralizer-generalized-character`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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. https://mathoverflow.net/questions/478499/is-this-generalized-character-always-a-character
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - forum; reference source; checked 2026-07-31
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>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. Full journal article relevant to Positivity of the p-element centralizer generalized character. https://arxiv.org/abs/2505.03976
   - preprint; reference source; arXiv:2505.03976, checked 2026-07-31; checked 2026-07-31
   - Source use: citation_only
   - 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.