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 allow 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. If V is a complex representation of a 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 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 $\mathbb {C}$ 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]-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]

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

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 $\varphi :K[G]\to V$ given by $\varphi (x)={\frac {1}{\#G}}\sum _{{s\in G}}s\cdot \pi (s^{{-1}}\cdot x).$ Then φ is again a projection: it is clearly K-linear, maps K[G] onto V, and induces the identity on V. Moreover we have

{\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, $K[G]=V\oplus \ker \varphi$ . 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 $x=\sum \lambda _{g}g\in K[G]$ define $\epsilon (x)=\sum \lambda _{g}$ . Let $I=\ker \epsilon$ . Then I is a K[G]-submodule. We will prove that for every nontrivial submodule V of K[G], $I\cap V\neq 0$ . Let V be given, and let $v=\sum \mu _{g}g$ be any nonzero element of V. If $\epsilon (v)=0$ , the claim is immediate. Otherwise, let $s=\sum 1g$ . Then $\epsilon (s)=\#G\cdot 1=0$ so $s\in I$ and $sv=\left(\sum 1g\right)\left(\sum \mu _{g}g\right)=\sum \epsilon (v)g=\epsilon (v)s$ so that $sv$ is an element of both I and V. This proves that V is not a direct complement of I for all V, so K[G] is not semisimple.

Notes

↑ 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.
↑ 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.
↑ O'Connor, John J.; Robertson, Edmund F., "Heinrich Maschke", MacTutor History of Mathematics archive, University of St Andrews .
↑ Fulton & Harris, Proposition 1.5.
↑ Serre, Theorem 1.
↑ Fulton & Harris, Corollary 1.6.
↑ Serre, Theorem 2.
↑ It follows that every module over K[G] is a semisimple module.
↑ 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.
↑ The number of the summands can be computed, and turns out to be equal to the number of the conjugacy classes of the group.
↑ 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.
↑ Serre, 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. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.

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[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 .

[FOOTNOTEFultonHarrisProposition_1.5-4] Fulton & Harris, Proposition 1.5.

[FOOTNOTESerreTheorem_1-5] Serre, Theorem 1.

[FOOTNOTEFultonHarrisCorollary_1.6-6] Fulton & Harris, Corollary 1.6.

[FOOTNOTESerreTheorem_2-7] Serre, Theorem 2.

[8] It follows that every module over 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.

[FOOTNOTESerreExercise_6.1-12] Serre, Exercise 6.1.