Maschke's theorem [1] is a theorem in group representation theory which 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 doesn't divide the order of G.

It is named after Heinrich Maschke.[2]

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 KG. 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:

Let G be a finite group and K a field whose characteristic does not divide the order of G. Then KG, the group algebra of G, is a semisimple algebra.[3][4]

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 KG is a product of several copies of complex matrix algebras, one for each irreducible representation.[5] 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 KG is a product of matrix algebras over division rings over K. The summands correspond to irreducible representations of G over K.[6]

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.

Notes

1. ^ Heinrich Maschke (July 1899) "Beweiss des Satzes, dass diejenigen endlichen linearen Substitutionesgruppen, 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], Mathematische Annalen, vol. 52, no. 3, pages 363-368.
2. ^ O'Connor, John J.; Robertson, Edmund F., "Maschke's theorem", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Maschke.html .
3. ^ It follows that every module over KG is a semisimple module.
4. ^ 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.
5. ^ The number of the summands can be computed, and turns out to be equal to the number of the conjugacy classes of the group.
6. ^ 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), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, MR1878556, ISBN 978-0-387-95385-4
* Serre, Jean-Pierre (1971). Linear representations of finite groups. Graduate texts in mathematics, 42. Springer-Verlag. ISBN 0-387-90190-6.