
Next: Locally compact groups Up: Representation Theory of Algebraic Previous: Representation Theory of Algebraic
At the end of the nineteenth century Frobenius started the systematic study of complex representations of finite groups. He defined the notion of a group representation, and studied the representations of important groups such as
, the symmetric groups
and the alternating groups
. In modern terminology, a (complex) representation of a finite group
is a group homomorphism
where
is a complex vector space. The representation is called irreducible if
has no subspaces other than
or
that are invariant under
. Two representations
and
of
are called equivalent if there exists a linear isomorphism
such that
is
-equivariant, i.e.
In general, linear maps
that satisfy this equation are called
-maps or intertwiners. The investigation of representations of general finite groups was soon picked up by Burnside, Schur, and many others. The main results of their work can be formulated as follows:
- (1)
- Every representation of a finite group is unitarizable. This means that there exists a Hilbert space structure on
such that
.
- (2)
- Consequently, every representation of
is equivalent to a direct sum of irreducible representations.
- (3)
- (Schur's lemma) When
and
are irreducible and inequivalent, the only intertwiner
between
and
is
. On the other hand, when
and
are equivalent, every nonzero intertwiner
is an isomorphism. In particular, when
every intertwiner is scalar.
- (4)
- By (3), the decomposition in (2) is essentially unique.
- (5)
-
The left regular representation of
is defined by the left multiplication of
on itself. More precisely, we take
as the space of complex valued functions on
, and define the left regular representation
by
. Similarly, we define the right regular representation
on
by the right multiplication of
on itself. Then
is a
representation via
. As such,
is equivalent to the direct sum
where
runs over the set
of all equivalence classes of irreducible representations of
.
- (6)
-
(Plancherel formula) If
, we define its Fourier transform
as the function on
such that
. We give
a Hilbert space structure by the Hermitian inner product
. Then
Here
denotes the Hilbert-Schmidt norm of
.

Next: Locally compact groups Up: Representation Theory of Algebraic Previous: Representation Theory of Algebraic