Maschke's theorem

In mathematics, Maschke's theorem, ^{[1] [2]} named after Heinrich Maschke, ^[3] 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 — ${\displaystyle V}$ is a representation of a finite group ${\displaystyle G}$ over a field ${\displaystyle \mathbb {F} }$ with characteristic not dividing the order of ${\displaystyle G}$. If ${\displaystyle V}$ has a subrepresentation ${\displaystyle W}$, then it has another subrepresentation ${\displaystyle U}$ such that ${\displaystyle V=W\oplus U}$. ^{[4] [5]}

Then the corollary is

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

The vector space of complex-valued class functions of a group ${\displaystyle G}$ has a natural ${\displaystyle G}$-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 ${\displaystyle U}$ as the orthogonal complement of ${\displaystyle W}$ under this inner product.

Module-theoretic

One of the approaches to representations of finite groups is through module theory. Representations of a group ${\displaystyle G}$ are replaced by modules over its group algebra  ${\displaystyle K[G]}$ (to be precise, there is an isomorphism of categories between ${\displaystyle K[G]{\text{-Mod}}}$ and ${\displaystyle \operatorname {Rep} _{G}}$, the category of representations of ${\displaystyle G}$). 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 ${\displaystyle G}$ be a finite group and ${\displaystyle K}$ a field whose characteristic does not divide the order of ${\displaystyle G}$. Then ${\displaystyle K[G]}$, the group algebra of ${\displaystyle G}$, is semisimple. ^{[8] [9]}

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 ${\displaystyle K}$ is the field of complex numbers, this shows that the algebra ${\displaystyle K[G]}$ is a product of several copies of complex matrix algebras, one for each irreducible representation. ^[10] If the field ${\displaystyle K}$ has characteristic zero, but is not algebraically closed, for example if ${\displaystyle K}$ is the field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra ${\displaystyle K[G]}$ is a product of matrix algebras over division rings over ${\displaystyle K}$. The summands correspond to irreducible representations of ${\displaystyle G}$ over ${\displaystyle K}$. ^[11]

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

Define ${\textstyle p(x)={\frac {1}{\#G}}\sum _{g\in G}g\cdot p_{0}\cdot g^{-1}(x)}$, 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 andV. 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, ${\displaystyle V=W\oplus \ker p}$.

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

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

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

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

so φ is in fact K[G]-linear. By the splitting lemma, ${\displaystyle K[G]=V\oplus \ker \varphi }$. 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. ^[12]

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

$${\displaystyle sv=\left(\sum 1g\right)\!\left(\sum \mu _{g}g\right)=\sum \epsilon (v)g=\epsilon (v)s}$$

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

Notes

1. Maschke, Heinrich (1898-07-22). "Ueber den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen" [On the arithmetical character of the coefficients of the substitutions of finite linear substitution groups]. Math. Ann. (in German). 50 (4): 492–498. doi:10.1007/BF01444297. JFM   29.0114.03. MR   1511011.
2. Maschke, Heinrich (1899-07-27). "Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind" [Proof of the theorem that those finite linear substitution groups, in which some everywhere vanishing coefficients appear, are intransitive]. Math. Ann. (in German). 52 (2–3): 363–368. doi:10.1007/BF01476165. JFM   30.0131.01. MR   1511061.
3. O'Connor, John J.; Robertson, Edmund F., "Heinrich Maschke", MacTutor History of Mathematics Archive , University of St Andrews
4. Fulton & Harris 1991, Proposition 1.5.
5. Serre 1977, Theorem 1.
6. Fulton & Harris 1991, Corollary 1.6.
7. Serre 1977, Theorem 2.
8. It follows that every module over ${\displaystyle K[G]}$ is a semisimple module.
9. The converse statement also holds: if the characteristic of the field divides the order of the group (the modular case), then the group algebra is not semisimple.
10. The number of the summands can be computed, and turns out to be equal to the number of the conjugacy classes of the group.
11. One must be careful, since a representation may decompose differently over different fields: a representation may be irreducible over the real numbers but not over the complex numbers.
12. Serre 1977, Exercise 6.1.

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. MR   1878556. Zbl   0984.00001.
• 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. MR   0450380. Zbl   0355.20006.
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN   978-0-387-97495-8. MR   1153249. OCLC   246650103.