Products of commutators of symplectic involutions

Abstract

Let $I_n$ be the $n\times n$ identity matrix and $J=\left[\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right]$. A matrix A is called symplectic if $A^{T}JA=J$. A symplectic matrix A is a commutator of symplectic involutions if $A=XY{X}^{-1}{Y}^{-1}$, where X and Y are symplectic matrices satisfying $X^2=Y^2=I$. In this article, we give necessary and sufficient condition for a symplectic matrix over the complex number field to be expressed as a product of two commutators of symplectic involutions.

Keywords:

• Involutions
• commutators of involutions
• symplectic matrices

Mathematics Subject Classification (2010):

• 15A23
• 20H25
• 20H20

Acknowledgements

The author thanks the referees for the many helpful comments which greatly improved the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).