Isometries of symmetric gauge functions

Abstract

Denote by F the field R of real numbers, the field C of complex numbers or the skew-field H of real quaternions. Let F n be the linear space of all column vectors with n coordinates over F A norm on F n is a symmetric gauge function if for any , for all permutations (i 1,…,i n) of (l,…,n), and for all possible choices of . The purpose of this note is to characterize the isometries of symmetric gauge functions on F n . In particular, we show that G is the isometry group of a symmetric gauge function on F n if and only if G is a compact subgroup of GL n (F) containing P n (F), the group of generalized permutation matrices over F. To prove the main theorem, we establish some results concerning the groups that contain P n (F), and characterize all finite subgroups of GL n (R) that contain P n (R), which are of independent interest. While most of these results could be deduced from the advanced theories of Lie groups and reflection groups, elementary proofs are given to them in this paper.