Maschke's theorem

Short description: Concerns the decomposition of representations of a finite group into irreducible pieces

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 (constituents). 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 (finite-dimensional) representation built from irreducible subrepresentations using the direct sum operation? This question (and its answer) 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:

Theorem — $V$ is a representation of a finite group $G$ over a field $\mathbb{𝔽}$ with characteristic not dividing the order of $G$. If $V$ has a subrepresentation $W$, then it has another subrepresentation $U$ such that $V=W\oplus U$.

Then the corollary is

Corollary (Maschke's theorem) — Every representation of a finite group $G$ over a field $\mathbb{𝔽}$ with characteristic not dividing the order of $G$ is a direct sum of irreducible representations.

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 $\mathbb{ℂ}$ by constructing $U$ as the orthogonal complement of $W$ 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 $K[G]$ (to be precise, there is an isomorphism of categories between $K[G]\text{-Mod}$ and ${\mathrm{Rep}}_G$, 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.

The importance of this result stems from the well developed theory of semisimple rings, in particular, their classification as given by the Wedderburn–Artin 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. If the field $K$ has characteristic zero, but is not algebraically closed, for example if $K$ is the 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 $K$.

Category-theoretic

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.

Proofs

Group-theoretic

Let U be a subspace of V complement of W. Let $p_0:V\to W$ be the projection function, i.e., $p_0(w+u)=w$ for any $u\in U,w\in W$.

Define $p(x)=\frac{1}{\mathrm{\#}G}\sum _{g\in G}g\cdot p_0\cdot g^{-1}(x)$, where $g\cdot p_0\cdot g^{-1}$ is an abbreviation of ${\rho }_{W}g\cdot p_0\cdot {\rho }_V{g}^{-1}$, with ${\rho }_{W}g,{\rho }_V{g}^{-1}$ being the representation of G on W and V. Then, $\ker p$ is preserved by G under representation ${\rho }_V$: for any $w^{\prime }\in \ker p,h\in G$, $$\begin{array}{rl}p(h{w}^{\prime })& =h\cdot h^{-1}\frac{1}{\mathrm{\#}G}\sum _{g\in G}g\cdot p_0\cdot g^{-1}(h{w}^{\prime })\\ & =h\cdot \frac{1}{\mathrm{\#}G}\sum _{g\in G}(h^{-1}\cdot g)\cdot p_0\cdot (g^{-1}h)w^{\prime }\\ & =h\cdot \frac{1}{\mathrm{\#}G}\sum _{g\in G}g\cdot p_0\cdot g^{-1}{w}^{\prime }\\ & =h\cdot p(w^{\prime })\\ & =0\end{array}$$

so $w^{\prime }\in \ker p$ implies that $h{w}^{\prime }\in \ker p$. So the restriction of ${\rho }_V$ on $\ker p$ is also a representation.

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

Module-theoretic

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

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

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

so φ is in fact K[G]-linear. By the splitting lemma, $K[G]=V\oplus \ker \phi$. This proves that every submodule is a direct summand, that is, K[G] 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[G] is not semisimple? The answer is yes.

Proof. For $x=\sum {\lambda }_{g}g\in K[G]$ define $ϵ(x)=\sum {\lambda }_g$. Let $I=\ker ϵ$. Then I is a K[G]-submodule. We will prove that for every nontrivial submodule V of K[G], $I\cap V\ne 0$. Let V be given, and let $v=\sum {\mu }_{g}g$ be any nonzero element of V. If $ϵ(v)=0$, the claim is immediate. Otherwise, let $s=\sum 1g$. Then $ϵ(s)=\mathrm{\#}G\cdot 1=0$ so $s\in I$ and $$sv=(\sum 1g)(\sum {\mu }_{g}g)=\sum ϵ(v)g=ϵ(v)s$$

so that $sv$ is a nonzero element of both I and V. This proves V is not a direct complement of I for all V, so K[G] is not semisimple.

Non-examples

The theorem can not apply to the case where G is infinite, or when the field K has characteristics dividing #G. For example,

• Consider the infinite group $\mathbb{\mathbb{Z}}$ and the representation $\rho :\mathbb{\mathbb{Z}}\to {\mathrm{G}\mathrm{L}}_2(\mathbb{ℂ})$ defined by $\rho (n)={\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]}^n=\left[\begin{array}{cc}1& n\\ 0& 1\end{array}\right]$. Let $W=\mathbb{ℂ}\cdot \left[\begin{array}{c}1\\ 0\end{array}\right]$, a 1-dimensional subspace of ${\mathbb{ℂ}}^2$ spanned by $\left[\begin{array}{c}1\\ 0\end{array}\right]$. Then the restriction of $\rho$ on W is a trivial subrepresentation of $\mathbb{\mathbb{Z}}$. However, there's no U such that both W, U are subrepresentations of $\mathbb{\mathbb{Z}}$ and ${\mathbb{ℂ}}^2=W\oplus U$: any such U needs to be 1-dimensional, but any 1-dimensional subspace preserved by $\rho$ has to be spanned by an eigenvector for $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$, and the only eigenvector for that is $\left[\begin{array}{c}1\\ 0\end{array}\right]$.
• Consider a prime p, and the group $\mathbb{\mathbb{Z}}/p\mathbb{\mathbb{Z}}$, field $K={\mathbb{𝔽}}_p$, and the representation $\rho :\mathbb{\mathbb{Z}}/p\mathbb{\mathbb{Z}}\to {\mathrm{G}\mathrm{L}}_2({\mathbb{𝔽}}_p)$ defined by $\rho (n)=\left[\begin{array}{cc}1& n\\ 0& 1\end{array}\right]$. Simple calculations show that there is only one eigenvector for $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ here, so by the same argument, the 1-dimensional subrepresentation of $\mathbb{\mathbb{Z}}/p\mathbb{\mathbb{Z}}$ is unique, and $\mathbb{\mathbb{Z}}/p\mathbb{\mathbb{Z}}$ cannot be decomposed into the direct sum of two 1-dimensional subrepresentations.

References

• Lang, Serge (2002-01-08). Algebra. Graduate Texts in Mathematics, 211 (Revised 3rd ed.). New York: Springer-Verlag. ISBN 978-0-387-95385-4. 
• Serre, Jean-Pierre (1977-09-01). Linear Representations of Finite Groups. Graduate Texts in Mathematics, 42. New York–Heidelberg: Springer-Verlag. ISBN 978-0-387-90190-9. https://archive.org/details/linearrepresenta1977serr. 
• Fulton, William; Harris, Joe (1991) (in en-gb). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. OCLC 246650103. https://link.springer.com/10.1007/978-1-4612-0979-9.