The isometries and the \(G\)-invariance of certain seminorms. (Q1414151)

Let \({\mathbb F}\) be either \({\mathbb R}\) or \({\mathbb C}\), and let \(G\) be a subgroup of the general linear group \(\text{GL}_n({\mathbb F})\). Given an \(m\times n\) matrix \(A\) over~\({\mathbb F}\), let \(\| \cdot\| _{A,\infty} \) be the seminorm on \({\mathbb F}^n\) defined by \(\| x\| _{A,\infty}=\| Ax\| _\infty\) for all \(x\in {\mathbb F}^n\). The author characterizes the linear isometries for the seminorm \(\| \cdot\| _{A,\infty}\) and shows that the group \({\mathcal G}_{A,\infty}\) of isometries for the norm \(\| \cdot\| _{A,\infty}\) can be embedded in the group \(\text{GP}_n({\mathbb F})\) of general permutation matrices. Further, conditions on~\(A\) for which \(\| \cdot\| _{A,\infty}\) is \(G\)@-invariant are studied. As a special case the matrices \(A\) for which \(\| \cdot\| _{A,\infty}\) is absolute or a symmetric gauge function are described. The results answer an open question from \textit{R. Hemasinha} [Linear Multilinear Algebra 35, 135--151 (1993; Zbl 0790.15025)] of characterizing matrices \(A\in {\mathbb R}^{n\times m}\) for which \(\| \cdot\| _{A,\infty}\) on~\({\mathbb R}^n\) are sign invariant.