# P2750: Trace-indistinguishable triples in sl2(F5)

- ID: `P2750`
- Reference: `sl2-f5-triple-trace-fibers`
- Page: https://theoremdb.org/statements/P2750
- Record maturity: Reviewed problem with recorded work

## Problem

For an ordered triple \(T=(A,B,C)\in\mathfrak{sl}_2(\mathbb F_5)^3\), define its trace profile by \(\tau_T(w)=\operatorname{tr}(w(A,B,C))\) for every word \(w\) in three noncommuting letters. Among fibers of \(T\mapsto\tau_T\), determine the largest number of simultaneous \(GL_2(\mathbb F_5)\)-conjugacy classes in one fiber, and classify every fiber attaining that maximum.

### Problem setup

- **Definition.** The space sl_2(F_5) consists of 2 by 2 matrices over F_5 with trace zero.
- **Remark.** Simultaneous conjugation sends (A,B,C) to (gAg^{-1},gBg^{-1},gCg^{-1}) using one g in GL_2(F_5). Two triples have the same trace profile when the stated traces agree for every finite word, including powers and mixed words.

### What counts as a solution

- Give the maximum number of simultaneous-conjugacy classes in a trace-profile fiber and canonical representatives for every maximizing fiber.
- Prove a finite trace-word cutoff sufficient for this space, or compare all words through an equivalent exact invariant-theoretic certificate, and certify the full orbit enumeration.

## Status

UNKNOWN as of 2026-08-01. Separating trace sets are studied for matrix tuples, but the exact maximum fiber and its orbit classification for traceless triples over F5 was not located. [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Current status and unresolved remainder).** UNKNOWN as of 2026-08-01. Separating trace sets are studied for matrix tuples, but the exact maximum fiber and its orbit classification for traceless triples over F5 was not located.

UNKNOWN as of 2026-08-01. Separating trace sets are studied for matrix tuples, but the exact maximum fiber and its orbit classification for traceless triples over F5 was not located.

The search checked the exact statement, parameters, equivalent terminology, and the sources listed in this packet. Database silence is treated only as bounded status evidence. A complete resolution must satisfy every acceptance condition in the canonical problem.

### Background and intake notes

The output is a small atlas of trace failures. Word cutoffs, canonical orbit representatives, and stabilizers can be reused in finite-field invariant computations.

- Original intake status: UNKNOWN as of 2026-07-27. Separating trace sets are studied for matrix tuples, but the exact maximum fiber and its orbit classification for traceless triples over F5 was not located.
- A 2026-07-27 search checked Lopatin and Reimers, arXiv:2202.05717, Drensky, arXiv:math/0506614, and exact parameter searches. No matching F5 triple-fiber table was found.
- The field F5 avoids the characteristic-two trace collapse while keeping the full set at 125^3 triples. Three matrices are the first tuple size where genuinely mixed trace data proliferate.
- Use Cayley-Hamilton and trace identities to derive a finite word-length cutoff, then enumerate simultaneous conjugacy orbits and hash their certified profiles.
- Trap: trace words generally see semisimplification rather than every nonclosed orbit. Finding one separating word for sampled pairs does not prove a fiber complete.

- Recorded example: Let N=E_12. The triples (0,0,0) and (N,0,0) are not conjugate, but they have the same trace profile because N^2=0 and tr(N)=0.

### Open directions

- **Route 1** (reported): Give the maximum number of simultaneous-conjugacy classes in a trace-profile fiber and canonical representatives for every maximizing fiber. Prove a finite trace-word cutoff sufficient for this space, or compare all words through an equivalent exact invariant-theoretic certificate, and certify the full orbit enumeration. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `sl2-f5-triple-trace-fibers`, 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>Jonathan Elmer, “The separating variety for 2x2 matrix invariants,” Linear and Multilinear Algebra 72(3) (2024), 389-411. DOI 10.1080/03081087.2022.2158300; arXiv:2202.05717v3. The contributor formulated the finite-field fiber question after reviewing separating trace sets for matrix tuples. https://arxiv.org/abs/2202.05717
   - Also cited at The simultaneous-conjugation trace-invariant setting for tuples of 2 by 2 matrices
   - Also cited at Editorial research route recorded 2026-08-01
   - preprint; reference source; arXiv:2202.05717, checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Trace-indistinguishable triples in sl2(F5), the reviewed source scope is The contributor formulated the finite-field fiber question after reviewing separating trace sets for matrix tuples.. The packet makes no inference beyond that cited scope.
   - Source named by the research packet.
2. <a id="reference-2"></a>Vesselin Drensky, “Computing with matrix invariants,” Mathematica Balkanica (N.S.) 21(1-2) (2007), 141-172. arXiv:math/0506614v2. survey of simultaneous-conjugation matrix invariants over characteristic zero https://arxiv.org/abs/math/0506614
   - preprint; reference source; arXiv:math/0506614, checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Trace-indistinguishable triples in sl2(F5), this source supplies adjacent invariant-theory methods and does not settle the finite-field triple-trace target.