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. If (V, ρ) is a finite-dimensional representation of a finite group G over a field of characteristic zero, and U is an invariant subspace of V, then the theorem claims that U admits an invariant direct complement W; in other words, the representation (V, ρ) is completely reducible. More generally, the theorem holds for fields of positive characteristic p, such as the finite fields, if the prime p does not divide the order of G.

Reformulation and the meaning

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]. Irreducible representations correspond to simple modules. Maschke's theorem addresses the question: is a general (finite-dimensional) representation built from irreducible subrepresentations using the direct sum operation? In the module-theoretic language, 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.^[4][5]

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.^[6] 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.^[7]

Returning to representation theory, Maschke's theorem and its module-theoretic version allow one to make general conclusions about representations of a finite group G without actually computing them. They reduce 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.

Proof

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

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

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.

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. It follows that every module over K[G] is a semisimple module.
5. 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.
6. The number of the summands can be computed, and turns out to be equal to the number of the conjugacy classes of the group.
7. 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.

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.