# 马施克定理

Heinrich Maschke

## 定理

${\displaystyle V}$${\displaystyle K}$上的有限维线性空间${\displaystyle (V,\rho )}$有限群${\displaystyle G}$表示, ${\displaystyle U_{0}}$${\displaystyle V}$${\displaystyle G}$不变子空间, ${\displaystyle K}$特征不能整除${\displaystyle G}$

## 证明

${\displaystyle U_{0}}$${\displaystyle V}$的子空间，所以存在${\displaystyle U_{0}}$${\displaystyle V}$中的补空间${\displaystyle W_{0}}$，及投影${\displaystyle P_{0}}$, ${\displaystyle Q_{0}}$，使得

${\displaystyle U_{0}=P_{0}V}$

${\displaystyle W_{0}=Q_{0}V}$

${\displaystyle P_{0}^{2}-P_{0}=Q_{0}^{2}-Q_{0}=P_{0}Q_{0}=Q_{0}P_{0}=0}$

${\displaystyle P_{0}+Q_{0}=1}$

${\displaystyle P=N^{-1}\sum _{g\in G}gP_{0}g^{-1}}$

${\displaystyle Q=N^{-1}\sum _{g\in G}gQ_{0}g^{-1}}$

${\displaystyle P+Q=1}$

${\displaystyle P^{2}=P}$

${\displaystyle Q^{2}=Q}$

${\displaystyle PQ=QP=0}$

${\displaystyle V=U\oplus W}$

${\displaystyle P}$的定义 ${\displaystyle U={\textrm {Im}}P\subseteq U_{0}}$

${\displaystyle U=U_{0}}$

${\displaystyle V=U_{0}\oplus W}$

${\displaystyle W}$${\displaystyle G}$不变子空间。