On the traces of the product of 2 linear similarity classes

TL;DR

The paper demonstrates that the product of two non-scalar conjugacy classes in SL(n,K) can produce matrices with any trace when n ≥ 4 over any field, or when n=3 over finite fields.

Contribution

It establishes new conditions under which the product of conjugacy classes in SL(n,K) can generate matrices with arbitrary trace, extending previous understanding.

Findings

For n ≥ 4, the product contains matrices of any trace over any field.

For n=3, the result holds when K is finite.

The work generalizes known results to broader classes of fields and dimensions.

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.