Maschke's theorem

The set of Maschke (after Heinrich Maschke , 1899 ) is a key message from the mathematical sub-region of the representation theory of finite groups . It states that representations are composed of irreducible representations , except in the special case of modular representations .

Let it be a finite group and a body . The essence of the theory of linear representations of depends fundamentally on whether the characteristic of is a factor of the order of or not. In the first case one speaks of modular representations. The difference lies essentially in the statement made in Maschke's theorem. ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle G}$

Not modular case

It applies ; this is especially the case when characteristic has 0, for example for . ${\displaystyle \operatorname {char} K\nmid \#G}$ ${\displaystyle K}$ ${\displaystyle K=\mathbb {C} }$

Then Maschke's sentence says:

Every -linear representation of is a direct sum of irreducible representations. ${\displaystyle K}$ ${\displaystyle G}$

Equivalent formulations are:

• Each representation is semi-simple .
• Every -invariant subspace of a representation has an -invariant complement , i. H. . ${\displaystyle G}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle G}$ ${\displaystyle W}$ ${\displaystyle V=U\oplus W}$

Modular presentations

If, on the other hand , the following applies: The group ring is not completely reducible, i. H. the regular representation is not completely reducible. ${\displaystyle \operatorname {char} K\mid \#G}$ ${\displaystyle K[G]}$ ${\displaystyle G\rightarrow GL_{K}(K[G])}$

Not every sub-module of has a complement. ${\displaystyle K[G]}$ ${\displaystyle K[G]}$

See also

Representation theory of finite groups

literature

Kurt Meyberg, Algebra II , Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , chapter 9.3 "Maschke's theorem"