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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
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Previous: Representation Theory of Algebraic