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

Then the corollary is

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

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

Module-theoretic

One of the approaches to representations of finite groups is through module theory. Representations of a group [math]\displaystyle{ G }[/math] are replaced by modules over its group algebra [math]\displaystyle{ K[G] }[/math] (to be precise, there is an isomorphism of categories between [math]\displaystyle{ K[G]\text{-Mod} }[/math] and [math]\displaystyle{ \operatorname{Rep}_{G} }[/math], the category of representations of [math]\displaystyle{ G }[/math]). 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 [math]\displaystyle{ G }[/math] be a finite group and [math]\displaystyle{ K }[/math] a field whose characteristic does not divide the order of [math]\displaystyle{ G }[/math]. Then [math]\displaystyle{ K[G] }[/math], the group algebra of [math]\displaystyle{ G }[/math], 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 [math]\displaystyle{ K }[/math] is the field of complex numbers, this shows that the algebra [math]\displaystyle{ K[G] }[/math] is a product of several copies of complex matrix algebras, one for each irreducible representation. If the field [math]\displaystyle{ K }[/math] has characteristic zero, but is not algebraically closed, for example if [math]\displaystyle{ K }[/math] is the field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra [math]\displaystyle{ K[G] }[/math] is a product of matrix algebras over division rings over [math]\displaystyle{ K }[/math]. The summands correspond to irreducible representations of [math]\displaystyle{ G }[/math] over [math]\displaystyle{ K }[/math].

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 [math]\displaystyle{ p_0 : V \to W }[/math] be the projection function, i.e., [math]\displaystyle{ p_0(w + u) = w }[/math] for any [math]\displaystyle{ u \in U, w \in W }[/math].

Define [math]\displaystyle{ p(x) = \frac{1}{\#G} \sum_{g \in G} g \cdot p_0 \cdot g^{-1} (x) }[/math], where [math]\displaystyle{ g \cdot p_0 \cdot g^{-1} }[/math] is an abbreviation of [math]\displaystyle{ \rho_W{g} \cdot p_0 \cdot \rho_V{g^{-1}} }[/math], with [math]\displaystyle{ \rho_W{g}, \rho_V{g^{-1}} }[/math] being the representation of G on W and V. Then, [math]\displaystyle{ \ker p }[/math] is preserved by G under representation [math]\displaystyle{ \rho_V }[/math]: for any [math]\displaystyle{ w' \in \ker p, h \in G }[/math], [math]\displaystyle{ \begin{align} 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{align} }[/math]

so [math]\displaystyle{ w' \in \ker p }[/math] implies that [math]\displaystyle{ hw' \in \ker p }[/math]. So the restriction of [math]\displaystyle{ \rho_V }[/math] on [math]\displaystyle{ \ker p }[/math] is also a representation.

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

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 [math]\displaystyle{ \begin{cases} \varphi:K[G]\to V \\ \varphi:x \mapsto \frac{1}{\#G}\sum_{s \in G} s\cdot \pi(s^{-1} \cdot x) \end{cases} }[/math]

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

[math]\displaystyle{ \begin{align} \varphi(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\varphi(x), \end{align} }[/math]

so φ is in fact K[G]-linear. By the splitting lemma, [math]\displaystyle{ K[G]=V \oplus \ker \varphi }[/math]. 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 [math]\displaystyle{ x = \sum\lambda_g g\in K[G] }[/math] define [math]\displaystyle{ \epsilon(x) = \sum\lambda_g }[/math]. Let [math]\displaystyle{ I=\ker\epsilon }[/math]. Then I is a K[G]-submodule. We will prove that for every nontrivial submodule V of K[G], [math]\displaystyle{ I \cap V \neq 0 }[/math]. Let V be given, and let [math]\displaystyle{ v=\sum\mu_gg }[/math] be any nonzero element of V. If [math]\displaystyle{ \epsilon(v)=0 }[/math], the claim is immediate. Otherwise, let [math]\displaystyle{ s = \sum 1 g }[/math]. Then [math]\displaystyle{ \epsilon(s) = \#G \cdot 1 = 0 }[/math] so [math]\displaystyle{ s \in I }[/math] and [math]\displaystyle{ sv = \left(\sum1g\right)\!\left(\sum\mu_gg\right) = \sum\epsilon(v)g = \epsilon(v)s }[/math]

so that [math]\displaystyle{ sv }[/math] 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 [math]\displaystyle{ \mathbb{Z} }[/math] and the representation [math]\displaystyle{ \rho: \mathbb{Z} \to \mathrm{GL}_2(\Complex) }[/math] defined by [math]\displaystyle{ \rho(n) = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}^n = \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} }[/math]. Let [math]\displaystyle{ W = \Complex \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix} }[/math], a 1-dimensional subspace of [math]\displaystyle{ \Complex^2 }[/math] spanned by [math]\displaystyle{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} }[/math]. Then the restriction of [math]\displaystyle{ \rho }[/math] on W is a trivial subrepresentation of [math]\displaystyle{ \mathbb{Z} }[/math]. However, there's no U such that both W, U are subrepresentations of [math]\displaystyle{ \mathbb{Z} }[/math] and [math]\displaystyle{ \Complex^2 = W \oplus U }[/math]: any such U needs to be 1-dimensional, but any 1-dimensional subspace preserved by [math]\displaystyle{ \rho }[/math] has to be spanned by an eigenvector for [math]\displaystyle{ \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} }[/math], and the only eigenvector for that is [math]\displaystyle{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} }[/math].
• Consider a prime p, and the group [math]\displaystyle{ \mathbb{Z}/p\mathbb{Z} }[/math], field [math]\displaystyle{ K = \mathbb{F}_p }[/math], and the representation [math]\displaystyle{ \rho: \mathbb{Z}/p\mathbb{Z} \to \mathrm{GL}_2(\mathbb{F}_p) }[/math] defined by [math]\displaystyle{ \rho(n) = \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} }[/math]. Simple calculations show that there is only one eigenvector for [math]\displaystyle{ \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} }[/math] here, so by the same argument, the 1-dimensional subrepresentation of [math]\displaystyle{ \mathbb{Z}/p\mathbb{Z} }[/math] is unique, and [math]\displaystyle{ \mathbb{Z}/p\mathbb{Z} }[/math] 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.