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 $PSL_2(F_p)$, the symmetric groups $S_n$ and the alternating groups $A_n$. In modern terminology, a (complex) representation of a finite group $G$ is a group homomorphism

$$\pi:G\to GL(V),$$

where $V$ is a complex vector space. The representation is called irreducible if $V$ has no subspaces other than $0$ or $V$ that are invariant under $\pi(G)$. Two representations $\pi$ and $\pi^\prime$ of $G$ are called equivalent if there exists a linear isomorphism $A:V\to V^\prime$ such that $A$ is $G$-equivariant, i.e.

$$A\circ \pi(g)=\pi^\prime(g)\circ A\ (\forall g\in G).$$

In general, linear maps $A$ that satisfy this equation are called $G$-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 $V$ such that $\pi(G)\subset U(V)$.

(2)

Consequently, every representation of $G$ is equivalent to a direct sum of irreducible representations.

(3)

(Schur's lemma) When $\pi$ and $\pi^\prime$ are irreducible and inequivalent, the only intertwiner $A$ between $V$ and $V^\prime$ is $A=0$. On the other hand, when $\pi$ and $\pi^\prime$ are equivalent, every nonzero intertwiner $A$ is an isomorphism. In particular, when $\pi=\pi^\prime$ every intertwiner is scalar.

(4)

By (3), the decomposition in (2) is essentially unique.

(5)

The left regular representation of $G$ is defined by the left multiplication of $G$ on itself. More precisely, we take $V$ as the space of complex valued functions on $G$, and define the left regular representation $\lambda$ by $\lambda(g)f(x):=f(g^{-1}x)$. Similarly, we define the right regular representation $\rho$ on $V$ by the right multiplication of $G$ on itself. Then $V$ is a $G\times G$ representation via $(\lambda,\rho)$. As such, $V$ is equivalent to the direct sum

$$V\simeq \sum_{\pi\in\hat G} {\rm End}(V_\pi),$$

where $\pi$ runs over the set $\hat G$ of all equivalence classes of irreducible representations of $G$.

(6)

(Plancherel formula) If $f\in V$, we define its Fourier transform $\hat f$ as the function on $\hat G$ such that ${\hat f}(\pi):=\frac{1}{\vert G\vert}\sum_{g\in G}f(g)\pi(g)\in {\rm End}(V_\pi)$. We give $V$ a Hilbert space structure by the Hermitian inner product $(f_1,f_2):=\frac{1}{\vert G\vert} \sum_{g\in G}f_1(g)\overline{f_2(g)}$. Then

$$\Vert f\Vert^2=\sum_{\pi\in \hat G}{\rm dim}(V_\pi) \Vert {\hat f}(\pi)\Vert^2.$$

Here $\Vert {\hat f}(\pi)\Vert$ denotes the Hilbert-Schmidt norm of $\hat f(\pi)\in {\rm End}(V_\pi)$.