information:highlighting, structural:expansion
Finite groups

Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to geometry and crystallography. Representations of finite groups exhibit many of the features of the general theory and point the way to other branches and topics in representation theory.

Over a field of characteristic zero, the representation of a finite group G has a number of convenient properties. First, the representations of G are semisimple (completely reducible). This is a consequence of Maschke's theorem, which states that any subrepresentation V of a G -representation W has a G -invariant complement. One proof is to choose any projection π from W to V and replace it by its average π defined by

π is equivariant, and its kernel is the required complement.

The finite-dimensional G -representations can be understood using character theory: the character of a representation φ: G → GL(V) is the class function χ: G → F defined by

where T r {\displaystyle \mathrm {Tr} } is the trace. An irreducible representation of G is completely determined by its character.

Maschke's theorem holds more generally for fields of positive characteristic p, such as the finite fields, as long as the prime p is coprime to the order of G. When p and | G | have a common factor, there are G -representations that are not semisimple, which are studied in a subbranch called modular representation theory.

Averaging techniques also show that if F is the real or complex numbers, then any G -representation preserves an inner product ⟨ ⋅, ⋅ ⟩ {\displaystyle \langle \cdot,\cdot \rangle } on V in the sense that

for all g in G and v, w in W. Hence any G -representation is unitary.

Unitary representations are automatically semisimple, since Maschke's result can be proven by taking the orthogonal complement of a subrepresentation. When studying representations of groups that are not finite, the unitary representations provide a good generalization of the real and complex representations of a finite group.

Results such as Maschke's theorem and the unitary property that rely on averaging can be generalized to more general groups by replacing the average with an integral, provided that a suitable notion of integral can be defined. This can be done for compact topological groups (including compact Lie groups), using Haar measure, and the resulting theory is known as abstract harmonic analysis.

Over arbitrary fields, another class of finite groups that have a good representation theory are the finite groups of Lie type. Important examples are linear algebraic groups over finite fields. The representation theory of linear algebraic groups and Lie groups extends these examples to infinite-dimensional groups, the latter being intimately related to Lie algebra representations. The importance of character theory for finite groups has an analogue in the theory of weights for representations of Lie groups and Lie algebras.

Representations of a finite group G are also linked directly to algebra representations via the group algebra F, which is a vector space over F with the elements of G as a basis, equipped with the multiplication operation defined by the group operation, linearity, and the requirement that the group operation and scalar multiplication commute.