Maschke's theorem


In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make general conclusions about representations of a finite group G without actually computing them. It reduces the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces. Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. In particular, a representation of a finite group over a field of characteristic zero is determined up to isomorphism by its character.

Formulations

Maschke's theorem addresses the question: when is a general representation built from irreducible subrepresentations using the direct sum operation? This question are formulated differently for different perspectives on group representation theory.

Group-theoretic

Maschke's theorem is commonly formulated as a corollary to the following result:
Then the corollary is
The vector space of complex-valued class functions of a group has a natural -invariant inner product structure, described in the article Schur orthogonality relations. Maschke's theorem was originally proved for the case of representations over by constructing as the orthogonal complement of under this inner product.

Module-theoretic

One of the approaches to representations of finite groups is through module theory. Representations of a group G are replaced by modules over its group algebra K. Irreducible representations correspond to simple modules. In the module-theoretic language, Maschke's theorem asks: is an arbitrary module semisimple? In this context, the theorem can be reformulated as follows:
The importance of this result stems from the well developed theory of semisimple rings, in particular, the Artin–Wedderburn theorem. When K is the field of complex numbers, this shows that the algebra K is a product of several copies of complex matrix algebras, one for each irreducible representation. If the field K has characteristic zero, but is not algebraically closed, for example, K is a field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra K is a product of matrix algebras over division rings over K. The summands correspond to irreducible representations of G over K.

Category-theoretic

Reformulated in the language of semi-simple categories, Maschke's theorem states

Proofs

Module-theoretic

Let V be a K-submodule. We will prove that V is a direct summand. Let π be any K-linear projection of K onto V. Consider the map
Then φ is again a projection: it is clearly K-linear, maps K onto V, and induces the identity on V. Moreover we have
so φ is in fact K-linear. By the splitting lemma,. This proves that every submodule is a direct summand, that is, K is semisimple.

Converse statement

The above proof depends on the fact that #G is invertible in K. This might lead one to ask if the converse of Maschke's theorem also holds: if the characteristic of K divides the order of G, does it follow that K is not semisimple? The answer is yes.
Proof. For define. Let. Then I is a K-submodule. We will prove that for every nontrivial submodule V of K,. Let V be given, and let be any nonzero element of V. If, the claim is immediate. Otherwise, let. Then so and
so that is an element of both I and V. This proves V is not a direct complement of I for all V, so K is not semisimple.