Determinants preserving maps on the spaces of symmetric matrices and skew-symmetric matrices

Abstract

Denote ${\Sigma }_n$ and $Q_n$ the set of all $n\times n$ symmetric and skew-symmetric matrices over a field $\mathbb{F}$, respectively, where $char(\mathbb{F})\ne 2$ and $|\mathbb{F}|\ge n^2+1$. A characterization of $\varphi ,\psi :{\Sigma }_n\to {\Sigma }_n$, for which at least one of them is surjective, satisfying $$\det (\varphi (x)+\psi (y))=\det (x+y)(x,y\in {\Sigma }_n)$$ is given. Furthermore, if $n$ is even and $\varphi ,\psi :Q_n\to Q_n$, for which $\psi$ is surjective and $\psi (0)=0$, satisfy $$\det (\varphi (x)+\psi (y))=\det (x+y)(x,y\in Q_n),$$ then $\varphi =\psi$ and $\psi$ must be a bijective linear map preserving the determinant.

9 pages