[#P2750] Trace-indistinguishable triples in sl2(F5)
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.
1Context
The output is a small atlas of trace failures. Word cutoffs, canonical orbit representatives, and stabilizers can be reused in finite-field invariant computations.
2Definitions
Definition 1 (The space sl_2(F_5) consists of 2 by 2 matrices over F_5 with trace zero). The space sl_2(F_5) consists of 2 by 2 matrices over F_5 with trace zero.
Definition 2 (Simultaneous conjugation sends (A,B,C) to (gAg^{-1},gBg^{-1},gCg^{-1}) using one g in GL_2(F_5)). 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.
3What 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.
1Status
Current status (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.[1]
1Records
Notes and companion material
Original intake status. OPEN as of 2026-08-01. 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.
- 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.
- Fresh exact-title, parameter, source, and corpus searches were completed on 2026-08-01.
Recorded example 1. 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.
How the 2 records connect
ProblemTrace-indistinguishable triples in sl2(F5)
2See also
How to cite
TheoremDB contributors, “Trace-indistinguishable triples in sl2(F5),” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/sl2-f5-triple-trace-fibersThis page as plain text: sl2-f5-triple-trace-fibers.md
This problem includes 2 records joined by 1 typed links, sourced from arxiv.org[1], current as of August 1, 2026.
1References
- Packet source. 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 simultaneous-conjugation trace-invariant setting for tuples of 2 by 2 matrices. ↗preprint · primary source · arXiv:2202.05717v3 · checked 2026-08-01Source use: original summary.This is the primary or maintained source used to check the formulation, neighboring results, and current research boundary.Also cited at Source identified in the candidate’s dated status assessment; exact page or record linked above.Also cited at The contributor formulated the finite-field fiber question after reviewing separating trace sets for matrix tuples.Also cited at Editorial research route recorded 2026-08-01.Supports the dated status review or a neighboring result for “Trace-indistinguishable triples in sl2(F5).”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.
- Vesselin Drensky, “Computing with matrix invariants,” Mathematica Balkanica (N.S.) 21(1-2) (2007), 141-172. arXiv:math/0506614v2. Survey of simultaneous-conjugation invariants for matrix tuples. ↗preprint · secondary source · arXiv:math/0506614v2 · checked 2026-08-01Source use: original summary.This later or complementary source was checked for equivalent formulations, methods, and possible prior answers.Also cited at Source identified in the candidate’s dated status assessment; exact page or record linked above.Also cited at survey of simultaneous-conjugation matrix invariants over characteristic zero.Supports the dated status review or a neighboring result for “Trace-indistinguishable triples in sl2(F5).”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.
CC0 finite classification of the failure of trace words to separate nonsemisimple matrix-tuple orbits.