Isometries of the special orthogonal group (Q389604)

Let \(c=(c_1,\dots,c_n)\in\mathbb R^n\) satisfy \(c_1\geq\dots\geq c_n\geq 0\) and \(c_1>0\). The \(c\)-spectral norm of \(A\in\mathbb R^{n\times n}\) is defined by \(\|A\|_c=c_1\sigma_1(A)+\dots+c_n\sigma_n(A)\), where \(\sigma_1(A)\geq\dots\geq\sigma_n(A)\) are the singular values of~\(A\). The authors give a complete characterization of all isometries on the special orthogonal group~\(SO(n)\) in~\(\mathbb R^{n\times n}\) with respect to~\(\|\cdot\|_c\). As an application, they characterize all spectrally multiplicative mappings on~\(SO(n)\).