On commutators of unipotent matrices of index 2

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.