Maschke's theorem

Maschke's theorem is commonly formulated as a corollary to the following result:

Theorem. If V is a complex representation of a finite group G with a subrepresentation W, then there is another subrepresentation U of V such that V=W⊕U.[4][5]

Then the corollary is

Corollary (Maschke's theorem). Every representation of a finite group G over a field F with characteristic not dividing the order of G is a direct sum of irreducible representations.[6][7]

The vector space of complex-valued class functions of a group G has a natural G-invariant inner product structure, described in the article Schur orthogonality relations. Maschke's theorem was originally proved for the case of representations over ${\displaystyle \mathbb {C} }$ by constructing U as the orthogonal complement of W under this inner product.

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[G] (to be precise, there is an isomorphism of categories between K[G]-Mod and Rep_G, the category of representations of G). 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:

Maschke's Theorem. Let G be a finite group and K a field whose characteristic does not divide the order of G. Then K[G], the group algebra of G, is semisimple.[8][9]

The importance of this result stems from the well developed theory of semisimple rings, in particular, the Artin–Wedderburn theorem (sometimes referred to as Wedderburn's Structure Theorem). When K is the field of complex numbers, this shows that the algebra K[G] is a product of several copies of complex matrix algebras, one for each irreducible representation.[10] 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[G] is a product of matrix algebras over division rings over K. The summands correspond to irreducible representations of G over K.[11]

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

Maschke's theorem. If G is a group and F is a field with characteristic not dividing the order of G, then the category of representations of G over F is semi-simple.

Let U be a subspace of V complement of W. Let ${\displaystyle p_{0}:V\to W}$ be the projection function, i.e., ${\displaystyle p_{0}(w+u)=w}$ for any ${\displaystyle u\in U,w\in W}$.

Define ${\displaystyle p(x)={\frac {1}{\#G}}\sum _{g\in G}g\cdot p_{0}\cdot g^{-1}(x)}$, where ${\displaystyle g\cdot p_{0}\cdot g^{-1}}$ is an abbreviation of ${\displaystyle \rho _{W}{g}\cdot p_{0}\cdot \rho _{V}{g^{-1}}}$, with ${\displaystyle \rho _{W}{g},\rho _{V}{g^{-1}}}$ being the representation of G on W and V. Then, ${\displaystyle ker\ p}$ is preserved by G under representation ${\displaystyle \rho _{V}}$: for any ${\displaystyle w'\in ker\ p,h\in G}$,

${\displaystyle {\begin{aligned}p(hw')&=h\cdot h^{-1}{\frac {1}{\#G}}\sum _{g\in G}g\cdot p_{0}\cdot g^{-1}(hw')\\&=h\cdot {\frac {1}{\#G}}\sum _{g\in G}(h^{-1}\cdot g)\cdot p_{0}\cdot (g^{-1}h)w'\\&=h\cdot {\frac {1}{\#G}}\sum _{g\in G}g\cdot p_{0}\cdot g^{-1}w'\\&=h\cdot p(w')\\&=0\end{aligned}}}$

so ${\displaystyle w'\in ker\ p}$ implies that ${\displaystyle hw'\in ker\ p}$. So the restriction of ${\displaystyle \rho _{V}}$ on ${\displaystyle ker\ p}$ is also a representation.

By the definition of ${\displaystyle p}$, for any ${\displaystyle w\in W}$, ${\displaystyle p(w)=w}$, so ${\displaystyle W\cap ker\ p=\{0\}}$, and for any ${\displaystyle v\in V}$, ${\displaystyle p(p(v))=p(v)}$. Thus, ${\displaystyle p(v-p(v))=0}$, and ${\displaystyle v-p(v)\in ker\ p}$. Therefore, ${\displaystyle V=W\oplus ker\ p}$.

Let V be a K[G]-submodule. We will prove that V is a direct summand. Let π be any K-linear projection of K[G] onto V. Consider the map

${\displaystyle {\begin{cases}\phi :K[G]\to V\\\phi (x)={\frac {1}{\#G}}\sum _{s\in G}s\cdot \pi (s^{-1}\cdot x)\end{cases}}}$

Then φ is again a projection: it is clearly K-linear, maps K[G] to V, and induces the identity on V (therefore, maps K[G] onto V). Moreover we have

${\displaystyle {\begin{aligned}\phi (t\cdot x)&={\frac {1}{\#G}}\sum _{s\in G}s\cdot \pi (s^{-1}\cdot t\cdot x)\\&={\frac {1}{\#G}}\sum _{u\in G}t\cdot u\cdot \pi (u^{-1}\cdot x)\\&=t\cdot \phi (x),\end{aligned}}}$

so φ is in fact K[G]-linear. By the splitting lemma, ${\displaystyle K[G]=V\oplus \ker \phi }$. This proves that every submodule is a direct summand, that is, K[G] is semisimple.