On the Product of Coninvolutory Affine Transformations

A complex matrix is called \emph{coninvolutory} if $T\overline{T}=I$. In this paper, we study decompositions of affine transformations in $\mathrm{Aff}(n,\mathbb{C})=\mathrm{GL}(n,\mathbb{C})\ltimes \mathbb{C}^n$ into products of coninvolutions. We prove that an affine transformation $g$ is a product of two coninvolutions in $\mathrm{Aff}(n,\mathbb{C})$ if and only if its linear part $L(g)$ is $c$-reversible; that is, $L(g)$ is conjugate to $\overline{L(g)}^{-1}$ in $\mathrm{GL}(n,\mathbb{C})$. Equivalently, $g$ is conjugate to $\overline{g}^{-1}$ in $\mathrm{Aff}(n,\mathbb{C})$. We further characterize elements that are products of three coninvolutions via consimilarity and show that every $g=(A,v)\in \mathrm{Aff}(n,\mathbb{C})$ with $|\det(A)|=1$ can be expressed as a product of at most four coninvolutions.