A group representation of a group 
is a group action of 
on a vector
space 
by invertible linear maps. For example, the
group of two elements 
has a representation 
by 
and 
. A representation is a group
homomorphism 
.
Most groups have many different representations, possibly on different vector spaces. For example, the symmetric
group 
has a representation on 
by
 |
(1)
|
where 
is the permutation symbol of the permutation 
. It also has a representation on
by
 |
(2)
|
A representation gives a matrix for each element, and so another representation of 
is given by the matrices
![[1 0; 0 1],[0 1; 1 0],[-1 0; -1 1],[1 -1; 0 -1],[0 -1; 1 -1],[-1 1; -1 0].](/images/equations/GroupRepresentation/NumberedEquation3.svg) |
(3)
|
Two group representations are equivalent if there is one common invertible linear map 
between their vector spaces such
that 
for every 
.
In finite dimensions, this
means that their matrices are related by a common similarity
transformation. For example, performing similarity
transformations of the above matrices by
![[1 19; 0 1]](/images/equations/GroupRepresentation/NumberedEquation4.svg) |
(4)
|
gives the following equivalent representation of 
,
![[1 0; 0 1],[-19 -360; 1 19],[18 323; -1 -18],[1 37; 0 -1],[-19 -343; 1 18],[18 343; -1 -19].](/images/equations/GroupRepresentation/NumberedEquation5.svg) |
(5)
|
Any representation 
of 
can be restricted
to a representation of any subgroup 
, in which case, it is denoted 
. More surprisingly, any representation 
on 
can be extended to a representation of 
, on a larger vector space 
, called the induced
representation.
Representations have applications to many branches of mathematics, aside from applications to physics and chemistry. The name of the theory depends on the group 
and on the vector
space 
.
Different approaches are required depending on whether 
is a finite group, an infinite discrete group, or a Lie
group. Another important ingredient is the field of
scalars for 
. The vector space 
can be infinite dimensional
such as a Hilbert space. Also, special kinds of
representations may require that additional structure on a vector
space is preserved. For instance, a unitary
representation is a group homomorphism 
into the group
of unitary transformations which preserve
a Hermitian inner product on 
.
For a finite group over a field whose characteristic does not divide the
group order, an arbitrary group representation breaks
up into irreducible representations,
i.e., 
where the 
are irreducible. For many groups, the irreducible
representations have been classified.
