Group Representation

A 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

(3)

Two representations are considered equivalent if they are similar. For example, performing similarity transformations of the above matrices by

(4)

gives the following equivalent representation of ,

(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 a vector space structure is preserved. For instance, a unitary representation is a group homomorphism into the group of unitary transformations which preserve a Hermitian inner product on .

In favorable situations, such as a finite group, an arbitrary representation will break up into irreducible representations, i.e., where the are irreducible. For many groups, the irreducible representations have been classified.