In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Maschke's theorem, named after
Heinrich Maschke, is a theorem in
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
theory that concerns the decomposition of representations of a
finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
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 representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
s, since when the theorem applies, any representation is a
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of irreducible pieces (constituents). Moreover, it follows from the
Jordan–Hölder theorem
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many na ...
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
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
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
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
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
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
to the following result:
Then the corollary is
The
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
of complex-valued
class functions of a
group has a natural
-invariant
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
structure, described in the article
Schur orthogonality relations. Maschke's theorem was originally
proved for the case of representations over
by constructing
as the
orthogonal complement
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W^\perp of all vectors in V that are orthogonal to every vector in W. I ...
of
under this inner product.
Module-theoretic
One of the approaches to representations of finite groups is through
module theory
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since t ...
. ''Representations'' of a group
are replaced by ''modules'' over its
group algebra