Let be a finite group. The group algebra is the complex algebra with basis , with multiplication defined by linearly extending the group law of . The description of as an algebra is called representation theory. Relatedly, we want to understand the structure of -modules. We often call -modules (complex) representations of , and simple -modules irreducible representations.
Lemma 1 (Schur’s lemma) If and are simple -modules, then every nonzero homomorphism is an isomorphism. Moreover every homomorphism is a scalar multiple of the identity.
Proof: Suppose that is a nonzero homomorphism. Then is a submodule, so , and similarly , so is an isomorphism. Now suppose . Since every eigenspace of is a submodule, by simplicity every eigenspace is either or . From the spectral theorem we deduce that is a scalar multiple of the identity.
It turns out that we can ask for a unitary structure in -modules for free.
Lemma 2 (Weyl’s averaging tricking) Every -module has a -invariant inner product. Moreover if is simple then this inner product is unique up to scaling.
Proof: Let be a -module and let be any inner product on . Define a new inner product by
Clearly has the desired properties. Now suppose that is simple, and that is another -invariant inner product. Let be the adjoint of the formal identity
In other words let be the unique function satisfying
for all . Then is a homomorphism, so by Schur’s lemma it must be a multiple of the identity, so must be a multiple of .
Lemma 3 Let be a finite-dimensional -module with -invariant inner product , and for each simple -module let be the sum of all submodules of isomorphic to (the isotypic component of corresponding to ). Then for some , and is the orthogonal direct sum of the submodules .
Proof: Since the orthogonal complement of a submodule is a submodule, it follows by induction on dimension that is an orthogonal direct sum of simple submodules, so , where runs over simple -modules and each for some . Moreover by Schur’s lemma any two nonisomorphic simple submodules must be orthogonal, as the orthogonal projection from one to the other is a homomorphism, so every submodule of isomorphic to must be contained in , so .
The above lemma is particularly interesting when applied to the -module itself, the regular representation. We give a -invariant inner product by declaring the basis to be orthonormal. From the above lemma we know then that , where runs over simple -modules, and for some . Note then by Schur’s lemma that . On the other hand every homomorphism is determined uniquely by the destination of the unit in , so . We deduce that as -modules
In particular there are only finitely many simple -modules, and their dimensions obey
One can also prove in a more informative way, as follows. Fix an invariant inner product on , and consider any homomorphism . The adjoint is also a homomorphism, so
Conversely for any we may define a homomorphism by
We deduce therefore that is the subspace of spanned by the elements , where . Moreover using Schur’s lemma one can show that the images of and are orthogonal whenever and are orthogonal in , so by letting range over a basis of we thus see that is the orthogonal direct sum of copies of .
In any case we now understand the structure of as a -module, and we are only a short step away from understanding its structure as an algebra. Consider the obvious map
where as always the sum runs over a complete set of irreducible representations up to isomorphism. We claim this map is an isomorphism. Since we already know the dimensions agree, it suffices to prove injectivity. Thus suppose maps to zero, i.e., that acts trivially on each simple -module . Then acts trivially on , so . Thus we have proved the following theorem.
Theorem 4 As complex algebras, .
Finally, it is useful to understand how to project onto isotypic components. Given , we can compute the trace of as an operator on in two different ways. On the one hand, by looking at the basis ,
On the other hand from the decomposition we have
As a consequence, for every we have
where is the operator defined by
These identities are most easily verified first for , then extending to all of by linearity. Now if for then acts as zero on , so , so . On the other hand one can verify directly that is a homomorphism, so by Schur’s lemma the image of must be contained in . We deduce therefore from that is the orthogonal projection onto .
The function is usually called the character of . From the relations and for one can deduce the well known orthogonality relations for characters. In fact the distinction between and is hardly more than notational. Often we identify functions with elements , in which case the operation of convolution corresponds to multiplication in the group algebra. The operator then is just convolution with . So, in brief, to project onto the -isotypic component you convolve with the character of and multiply by .
We have kept to almost the bare minimum in the above discussion: the complex numbers and finite groups . There are a number of directions we could try to move in. We could replace with a different field, say one which is not algebraically closed, or one which has positive characteristic. Alternatively we could replace with an infinite group, say with a locally compact topology. We mention two such generalisations.
Theorem 5 (Artin–Wedderburn) Every semisimple ring is isomorphic to a product of matrix rings, where the are integers and the are division rings. In particular every semisimple -algebra is isomorphic to a product .
When defining unitary representations for compact groups we demand that the map be continuous, where is given the strong operator topology.
Theorem 6 (Peter–Weyl) Let be a compact group and its normalised Haar measure. Let be the set of all irreducible unitary representations of up to isomorphism. Then is countable, every is finite-dimensional, and the algebra of square-integrable functions with the operation of convolution decomposes as a Hilbert algebra as
where is the space together with the Hilbert–Schmidt inner product.