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.
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.
Maschke's theorem is commonly formulated as a corollary to the following result: Theorem —
is a direct sum of irreducible representations.
[6][7] The vector space of complex-valued class functions of a group
-invariant inner product structure, described in the article Schur orthogonality relations.
Maschke's theorem was originally proved for the case of representations over
One of the approaches to representations of finite groups is through module theory.
are replaced by modules over its group algebra
(to be precise, there is an isomorphism of categories between
Irreducible representations correspond to simple modules.
In the module-theoretic language, Maschke's theorem asks: is an arbitrary module semisimple?
a field whose characteristic does not divide the order of
[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.
is the field of complex numbers, this shows that the algebra
is a product of several copies of complex matrix algebras, one for each irreducible representation.
is the field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra
is a product of matrix algebras over division rings over
The summands correspond to irreducible representations of
[11] 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.
Let π be any K-linear projection of K[G] onto V. Consider the map
This proves that every submodule is a direct summand, that is, K[G] is semisimple.
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?
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.
The theorem can not apply to the case where G is infinite, or when the field K has characteristics dividing #G. For example,