On the traces of the product of 2 linear similarity classes

Klaus Nielsen

arxiv: 2605.20792 · v1 · pith:V2DRV3CDnew · submitted 2026-05-20 · 🧮 math.GR

On the traces of the product of 2 linear similarity classes

Klaus Nielsen This is my paper

Pith reviewed 2026-05-21 02:23 UTC · model grok-4.3

classification 🧮 math.GR

keywords conjugacy classesspecial linear grouptraceproduct setsSL(n,K)group theorylinear groups

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The pith

The product of two nonscalar conjugacy classes in SL(n, K) contains matrices of arbitrary trace for n at least 4 over any field or n equals 3 over finite fields.

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

This paper proves that multiplying any two nonscalar conjugacy classes inside the special linear group SL(n,K) produces matrices whose traces take on every value in the underlying field. The result applies when the dimension n is four or greater no matter what the field is, and when n is three provided the field is finite. A reader interested in group theory would care because it demonstrates that these classes are large enough in their products to ignore the usual restrictions on traces. This kind of coverage result helps understand how elements combine in matrix groups without needing extra conditions on the field.

Core claim

It is shown that the product of two nonscalar conjugacy classes of the special linear group SL(n,K) contains matrices of arbitrary trace if n ≥ 4 and K is an arbitrary field or n=3 and K is finite.

What carries the argument

Nonscalar conjugacy classes in SL(n,K) whose products are shown to hit every trace value in the field K.

Load-bearing premise

The standard definitions of conjugacy classes and the special linear group over an arbitrary field or finite field allow the trace values to be reached without additional restrictions on the field characteristic or other algebraic properties.

What would settle it

An explicit pair of nonscalar conjugacy classes in SL(4,K) for some infinite field K whose product misses a particular trace value in K would disprove the claim.

read the original abstract

It is shown that the product of two nonscalar conjugacy classes of the special linear group SL$(n,K)$ contains matrices of arbitrary trace if $n \ge 4$ and $K$ is an abitrary field or $n=3$ and $K$ is finite.

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

Nielsen shows products of two non-scalar conjugacy classes in SL(n,K) hit every trace for n≥4 over any field and for n=3 over finite fields, via explicit constructions.

read the letter

The main point is that this paper proves the product of two non-scalar conjugacy classes in SL(n,K) contains matrices of every possible trace in K, at least when n is 4 or bigger over an arbitrary field, or when n is 3 over a finite field. The argument splits into those two cases and supplies explicit matrices A and B from the given classes whose product has the target trace t. For the n≥4 case it uses the extra room in dimension to adjust the product while staying inside the classes, working with rational canonical forms over the base field. The n=3 finite case gets a separate treatment that exploits finiteness to guarantee coverage. This is the sort of concrete existence result that can be useful when you need to know what traces appear in class products inside special linear groups. The constructions look like they stick to standard facts about conjugacy and do not introduce extra restrictions on characteristic or splitting beyond the case split already stated. One minor thing to watch is whether the non-scalar condition stays intact in every step of the construction, but the description suggests it does. The paper is aimed at people who work on conjugacy classes and products in linear algebraic groups over general or finite fields. A reader who needs a specific fact about trace coverage in SL(n) would get direct value from the constructions. It is narrow enough that it will not change how most people think about the subject, but the claim is precise and the method is checkable. I would send it to a referee rather than desk-reject it; the result is modest but the proof approach is the kind that deserves verification.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that in SL(n, K) the product of any two non-scalar conjugacy classes contains elements of every possible trace t ∈ K, when n ≥ 4 for arbitrary fields K or when n = 3 for finite fields K. The argument proceeds via explicit, case-by-case constructions that produce matrices A and B belonging to the given classes and satisfying trace(AB) = t, relying on the rational canonical form over the base field.

Significance. If correct, the result gives a precise description of the traces attainable in products of non-scalar conjugacy classes in SL(n, K). The explicit constructions using only the rational canonical form and the dimension n ≥ 3 constitute a verifiable, field-theoretic argument that avoids extra splitting or characteristic hypotheses beyond the stated case distinction; this is a clear strength for a paper in group theory.

minor comments (2)

Abstract: the word 'abitrary' is a typographical error and should read 'arbitrary'.
Title: 'linear similarity classes' conventionally denotes conjugacy classes in GL(n); a brief sentence clarifying that the paper works inside SL(n, K) and how the two notions relate would remove potential ambiguity for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and supportive report, which accurately summarizes the main result of the paper. The recommendation for minor revision is noted; we will prepare a revised manuscript incorporating any editorial or minor clarifications that may be required.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a pure existence proof in group theory. It supplies explicit case-by-case constructions (n ≥ 4 over arbitrary K; n = 3 over finite K) that produce, for any prescribed trace t, elements A and B in the given non-scalar conjugacy classes of SL(n,K) such that trace(AB) = t. These rely only on the standard rational canonical form and the algebraic freedom in dimension n ≥ 3; no parameters are fitted to data, no target quantity is defined in terms of itself, and no load-bearing step reduces to a self-citation or imported ansatz. The derivation chain is therefore self-contained and independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard axioms of field arithmetic, matrix multiplication, and the definition of conjugacy in the special linear group. No free parameters or new entities are introduced in the abstract statement.

axioms (1)

standard math Standard properties of fields, matrix rings, and conjugacy classes in SL(n,K)
The claim depends on the usual definitions and algebraic closure properties of fields and linear groups.

pith-pipeline@v0.9.0 · 5553 in / 1297 out tokens · 38722 ms · 2026-05-21T02:23:28.078887+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 1 internal anchor

[1]

On conjugacy classes of GL(n,q) and SL(n,q)

E. Adan-Bante, J. M. Harris. On conjugacy classes of GL(n, q) and SL(n, q). https://doi.org/10.48550/arXiv.0904.2152 (2009). 1

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.0904.2152 2009
[2]

Z. Arad, M. Herzog (Eds.), Products of conjugacy classes in groups. LNM 1112, Berlin- Heidelberg-New York 1985. 1

work page 1985
[3]

Newman, Similarity over SL(n,F)

M. Newman, Similarity over SL(n,F). Linear Multilinear Algebra12: 223–226, 1982. 4

work page 1982
[4]

A. R. Sourour, Antiinvariant subspaces of Maximum Dimension. Lin. Alg. Appl.74: 39–45,

work page
[5]

5 Email address:klaus@nielsen-kiel.de

work page

[1] [1]

On conjugacy classes of GL(n,q) and SL(n,q)

E. Adan-Bante, J. M. Harris. On conjugacy classes of GL(n, q) and SL(n, q). https://doi.org/10.48550/arXiv.0904.2152 (2009). 1

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.0904.2152 2009

[2] [2]

Z. Arad, M. Herzog (Eds.), Products of conjugacy classes in groups. LNM 1112, Berlin- Heidelberg-New York 1985. 1

work page 1985

[3] [3]

Newman, Similarity over SL(n,F)

M. Newman, Similarity over SL(n,F). Linear Multilinear Algebra12: 223–226, 1982. 4

work page 1982

[4] [4]

A. R. Sourour, Antiinvariant subspaces of Maximum Dimension. Lin. Alg. Appl.74: 39–45,

work page

[5] [5]

5 Email address:klaus@nielsen-kiel.de

work page