On commutators of unipotent matrices of index 2

TL;DR

This paper investigates how matrices with determinant 1 over certain fields can be expressed as products of a limited number of commutators of unipotent matrices of index 2, revealing links to field properties.

Contribution

It establishes new bounds on the number of unipotent commutator factors needed to factorize matrices with determinant 1, depending on field characteristics and matrix size.

Findings

Every determinant 1 matrix over suitable fields is a product of at most four unipotent commutators.

Under certain conditions, the number of factors reduces to three or two.

The results connect matrix factorization properties with field characteristics like characteristic and perfect squares.

Abstract

A commutator of unipotent matrices of index 2 is a matrix of the form $XYX^{-1}Y^{-1}$, where $X$ and $Y$ are unipotent matrices of index 2, that is, $X\ne I_n$, $Y\ne I_n$, and $(X-I_n)^2=(Y-I_n)^2=0_n$. If $n>2$ and $\mathbb F$ is a field with $|\mathbb F|\geq 4$, then it is shown that every $n\times n$ matrix over $\mathbb F$ with determinant 1 is a product of at most four commutators of unipotent matrices of index 2. Consequently, every $n\times n$ matrix over $\mathbb F$ with determinant 1 is a product of at most eight unipotent matrices of index 2. Conditions on $\mathbb F$ are given that improve the upper bound on the commutator factors from four to three or two. The situation for $n=2$ is also considered. This study reveals a connection between factorability into commutators of unipotent matrices and properties of $\mathbb F$ such as its characteristic or its set of perfect squares.