Conversely, given a positive-definite function, one can define a unitary representation of G in a natural way.

Indeed, one has the following: given a smooth representation of G, we define a new action on V:

These give the irreducible representations of G with K not in their kernel.

The trivial representation of G is weakly contained in the left regular representation.

First, the representations of G are semisimple (completely reducible).

An irreducible representation of G is completely determined by its character.

Given a group G, representation theory then asks what representations of G exist.

These are the continuous finite-dimensional linear representations of G on complex vector spaces.

To each irreducible representation of G corresponds a node in .

We say that ρ is a real representation of G if the matrices are real.