group majorization

‎‎In this note, we present an equivalent condition for linear preservers of group majorization induced by closed subgroup $G$ of $O(\mathbb{R}^n)$. Moreover, a new concept of majorization  is defined on $\mathbb{R}^3$ as acu-majorization and this is extended for $3 \times m$ matrices. Then we characterize all its linear preservers on $\mathbb{R}^3$ and $M_{3,m}$.

T. Ando characterized linear preservers of majorization in  [Linear Algebra Appl. 118 (1989) 163-248]. In this note, we present a method to state a simple proof of Ando's theorem. By using this method, we state an equivalent condition for matrix representations of linear preservers of $G$-majorization on matrices, where  $G$ is a finite subgroup of orthogonal  group $O(\mathbb{R}^n)$.Moreover, we introduce reflective majorization and characterize all its linear preservers.