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 $char(\mathbb{F})\neq 2$ and $\lvert \mathbb{F} \rvert \geq n^2+1$. A characterization of $\phi,\psi:\Sigma_n \rightarrow \Sigma_n$, for which at least one of them is surjective, satisfying $$\det(\phi(x)+\psi(y))=\det(x+y)\qquad(x,y\in \Sigma_n)$$ is given. Furthermore, if $n$ is even and $\phi,\psi:Q_n \rightarrow Q_n$, for which $\psi$ is surjective and $\psi(0)=0$, satisfy $$\det(\phi(x)+\psi(y))=\det(x+y)\qquad(x,y\in Q_n),$$ then $\phi=\psi$ and $\psi$ must be a bijective linear map preserving the determinant.