For example, representation theory is used in the modern approach to gain new results about automorphic forms.
In this article we will restrict ourselves to the study of finite-dimensional representation spaces, except for the last chapter.
A group acting on a finite set is sometimes considered sufficient for the definition of the permutation representation.
However, since we want to construct examples for linear representations - where groups act on vector spaces instead of on arbitrary finite sets - we have to proceed in a different way.
A closer inspection of the convolution of two basis elements as shown in the equation above reveals that the multiplication in
the definition above provides the equation: For an example, see the main page on this topic: Dual representation.
are the imaginary unit and the primitive cube root of unity respectively): Then As it is sufficient to consider the image of the generating element, we find that Let
We reexamine the example provided for the direct sum: The outer tensor product Using the standard basis of
The following theorem will be presented in a more general way, as it provides a very beautiful result about representations of compact – and therefore also of finite – groups: Or in the language of
-linear map, because This proposition enables us to determine the isotype to the trivial subrepresentation of a given representation explicitly.
Note that every character is a class function, as the trace of a matrix is preserved under conjugation.
An inner product can be defined on the set of all class functions on a finite group: Orthonormal property.
, they form an orthonormal basis for the vector space of all class functions with respect to the inner product defined above, i.e. One might verify that the irreducible characters generate
by showing that there exists no nonzero class function which is orthogonal to all the irreducible characters.
In the following, these bilinear forms will allow us to obtain some important results with respect to the decomposition and irreducibility of representations.
This formula is a "necessary and sufficient" condition for the problem of classifying the irreducible representations of a group up to isomorphism.
It provides us with the means to check whether we found all the isomorphism classes of irreducible representations of a group.
Thus, we achieve the following result: Amongst others, the criterion of Mackey and a conclusion based on the Frobenius reciprocity are needed for the proof of the proposition.
Likewise, the induction on class functions defines a homomorphism of abelian groups
is finitely generated as a group, the first point can be rephrased as follows: Serre (1977) gives two proofs of this theorem.
The irreducible representation corresponding to such a partition (or Young tableau) is called a Specht module.
The direct sum of all these representation rings inherits from these constructions the structure of a Hopf algebra which, it turns out, is closely related to symmetric functions.
The representation theory unfolds in this context great importance for harmonic analysis and the study of automorphic forms.
If the representation is finite-dimensional, it is possible to determine the direct sum of the trivial subrepresentation just as in the case of finite groups.
Nevertheless, in most cases it is possible to restrict the study to the case of finite dimensions: Since irreducible representations of compact groups are finite-dimensional and unitary (see results from the first subsection), we can define irreducible characters in the same way as it was done for finite groups.
However, because a compact group has in general infinitely many conjugacy classes, this does not provide any useful information.
However, the induced representation can be defined more generally, so that the definition is valid independent of the index of the subgroup
then there exists a canonical isomorphism The Frobenius reciprocity transfers, together with the modified definitions of the inner product and of the bilinear form, to compact groups.
It is usually presented and proven in harmonic analysis, as it represents one of its central and fundamental statements.
We can reformulate this theorem to obtain a generalization of the Fourier series for functions on compact groups: The general features of the representation theory of a finite group G, over the complex numbers, were discovered by Ferdinand Georg Frobenius in the years before 1900.