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## Historical context

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 .

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