On commutators of unipotent matrices of index 2

Abstract

A commutator of unipotent matrices of index 2 is a matrix of the form $XY{X}^{-1}{Y}^{-1}$, 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 $F$ is a field with $\left|F\right|\ge 4$, then it is shown that every $n\times n$ matrix over $F$ with determinant 1 is a product of at most four commutators of unipotent matrices of index 2. Consequently, every $n\times n$ matrix over $F$ with determinant 1 is a product of at most eight unipotent matrices of index 2. Conditions on $F$ 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 $F$ such as its characteristic or its set of perfect squares.