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

Abstract

Denote $Σ_n$ 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 $ϕ,ψ:Σ_n \rightarrow Σ_n$, for which at least one of them is surjective, satisfying $$\det(ϕ(x)+ψ(y))=\det(x+y)\qquad(x,y\in Σ_n)$$ is given. Furthermore, if $n$ is even and $ϕ,ψ:Q_n \rightarrow Q_n$, for which $ψ$ is surjective and $ψ(0)=0$, satisfy $$\det(ϕ(x)+ψ(y))=\det(x+y)\qquad(x,y\in Q_n),$$ then $ϕ=ψ$ and $ψ$ must be a bijective linear map preserving the determinant.