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 to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication.

The bulk of this article describes linear representation theory; see the last section for generalizations.

The various theories are quite different in detail, though some basic definitions and concepts are similar.

The most important divisions are: Representation theory also depends heavily on the type of vector space on which the group acts.

One must also consider the type of field over which the vector space is defined.

The characteristic of the field is also significant; many theorems for finite groups depend on the characteristic of the field not dividing the order of the group.

It is common practice to refer to V itself as the representation when the homomorphism is clear from the context.

invertible matrices on the field K. Consider the complex number u = e2πi / 3 which has the property u3 = 1.

, isomorphic to the previous one, is σ given by: The group C3 may also be faithfully represented on

be the space of homogeneous degree-3 polynomials over the complex numbers in variables

A subspace W of V that is invariant under the group action is called a subrepresentation.

Under the assumption that the characteristic of the field K does not divide the size of the group, representations of finite groups can be decomposed into a direct sum of irreducible subrepresentations (see Maschke's theorem).

This holds in particular for any representation of a finite group over the complex numbers, since the characteristic of the complex numbers is zero, which never divides the size of a group.

A set-theoretic representation (also known as a group action or permutation representation) of a group G on a set X is given by a function ρ : G → XX, the set of functions from X to X, such that for all g1, g2 in G and all x in X: where

Every group G can be viewed as a category with a single object; morphisms in this category are just the elements of G. Given an arbitrary category C, a representation of G in C is a functor from G to C. Such a functor selects an object X in C and a group homomorphism from G to Aut(X), the automorphism group of X.

In the case where C is VectK, the category of vector spaces over a field K, this definition is equivalent to a linear representation.

When C is Ab, the category of abelian groups, the objects obtained are called G-modules.

Representations in Top are homomorphisms from G to the homeomorphism group of a topological space X.