PERMUTATION INVARIANT NORMS

A norm or seminorm on parallel to . parallel to on R(n) is permutation invariant if parallel to Px parallel to = parallel to x parallel to for all n x n permutation matrices P and for all x is an element of R(n). We present a systematic study of permutation invariant seminorms. Their relations with other types of norms on matrices are discussed. In addition, we consider a special class of permutation invariant seminorms, the c-radii, defined and denoted by R(c)(x) = max{\c(t)Px\ : P a permutation} for any given c is an element of R(n). It is shown that R(c) are the building blocks of all permutation invariant seminorms. If the entries of c are not all equal and their sum is not 0, then R(c) is a norm on R(n). For such R(c), we study their norm properties and characterize their isometry groups.