Representation theory of finite groups

The representation theory of groups is a part of mathematics which examines how groups act on given structures.

Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations.

Representation theory is used in many parts of mathematics, as well as in quantum chemistry and physics. Among other things it is used in algebra to examine the structure of groups. There are also applications in harmonic analysis and number theory. For example, representation theory is used in the modern approach to gain new results about automorphic forms.

The representation of a group in a module instead of a vector space is also called a linear representation.

A representation is called semisimple or completely reducible if it can be written as a direct sum of irreducible representations. This is analogous to the corresponding definition for a semisimple algebra.

For the definition of the direct sum of representations please refer to the section on direct sums of representations.

A representation is called isotypic if it is a direct sum of pairwise isomorphic irreducible representations.

Thus, without loss of generality we can assume that every further considered representation is unitary.

This subrepresentation is also irreducible. That means, the original representation is completely reducible:

As it is sufficient to consider the image of the generating element, we find that

Remark. Note that the direct sum and the tensor products have different degrees and hence are different representations.

In order to understand representations more easily, a decomposition of the representation space into the direct sum of simpler subrepresentations would be desirable. This can be achieved for finite groups as we will see in the following results. More detailed explanations and proofs may be found in [1] and [2].

A subrepresentation and its complement determine a representation uniquely.

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:

Theorem. Every linear representation of a compact group over a field of characteristic zero is a direct sum of irreducible representations.

Note that this decomposition is not unique. However, the number of how many times a subrepresentation isomorphic to a given irreducible representation is occurring in this decomposition is independent of the choice of decomposition.

To achieve a unique decomposition, one has to combine all the irreducible subrepresentations that are isomorphic to each other. That means, the representation space is decomposed into a direct sum of its isotypes. This decomposition is uniquely determined. It is called the canonical decomposition.

In the following, we show how to determine the isotype to the trivial representation:

This proposition enables us to determine the isotype to the trivial subrepresentation of a given representation explicitly.

The theorems above are in general not valid for infinite groups. This will be demonstrated by the following example: let

The moral of the story is that if we consider infinite groups, it is possible that a representation - even one that is not irreducible - can not be decomposed into a direct sum of irreducible subrepresentations.

Even though the character is a map between two groups, it is not in general a group homomorphism, as the following example shows.

In order to show some particularly interesting results about characters, it is rewarding to consider a more general type of functions on groups:

Note that every character is a class function, as the trace of a matrix is preserved under conjugation.

Proofs of the following results of this chapter may be found in [1], [2] and [3].

An inner product can be defined on the set of all class functions on a finite group:

In the following, these bilinear forms will allow us to obtain some important results with respect to the decomposition and irreducibility of representations.

It is possible to derive the following theorem from the results above, along with Schur's lemma and the complete reducibility of representations.

Corollary. Two representations with the same character are isomorphic. This means that every representation is determined by its character.

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.

Using the description of representations via the convolution algebra we achieve an equivalent formulation of these equations:

As was shown in the section on properties of linear representations, we can - by restriction - obtain a representation of a subgroup starting from a representation of a group. Naturally we are interested in the reverse process: Is it possible to obtain the representation of a group starting from a representation of a subgroup? We will see that the induced representation defined below provides us with the necessary concept. Admittedly, this construction is not inverse but rather adjoint to the restriction.

By using the group algebra we obtain an alternative description of the induced representation:

The results introduced in this section will be presented without proof. These may be found in [1] and [2].

George Mackey established a criterion to verify the irreducibility of induced representations. For this we will first need some definitions and some specifications with respect to the notation.

In this section we present some applications of the so far presented theory to normal subgroups and to a special group, the semidirect product of a subgroup with an abelian normal subgroup.

Amongst others, the criterion of Mackey and a conclusion based on the Frobenius reciprocity are needed for the proof of the proposition. Further details may be found in [1].

Induction theorems relate the representation ring of a given finite group G to representation rings of a family X consisting of some subsets H of G. More precisely, for such a collection of subgroups, the induction functor yields a map

Artin's induction theorem is the most elementary theorem in this group of results. It asserts that the following are equivalent:

of 3. For such a partition, a Young tableau is a graphical device depicting a partition. The irreducible representation corresponding to such a partition (or Young tableau) is called a Specht module.

inherits from these constructions the structure of a Hopf algebra which, it turns out, is closely related to symmetric functions.

The theory of representations of compact groups may be, to some degree, extended to locally compact groups. The representation theory unfolds in this context great importance for harmonic analysis and the study of automorphic forms. For proofs, further information and for a more detailed insight which is beyond the scope of this chapter please consult [4] and [5].

Most properties of representations of finite groups can be transferred with appropriate changes to compact groups. For this we need a counterpart to the summation over a finite group:

All the definitions to representations of finite groups that are mentioned in the section ”Properties”, also apply to representations of compact groups. But there are some modifications needed:

By transferring the results of the section decompositions to compact groups, we obtain the following theorems:

Every representation of a compact group is isomorphic to a direct Hilbert sum of irreducible representations.

Note that the isotypes of not equivalent irreducible representations are pairwise orthogonal.

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.

Generally, representations of compact groups are investigated on Hilbert- and Banach spaces. In most cases they are not finite-dimensional. Therefore, it is not useful to refer to characters when speaking about representations of compact 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.

As long as the constructed representations stay finite-dimensional, the characters of the newly constructed representations may be obtained in the same way as for finite groups.

The bilinear form on the representation spaces is defined exactly as it was for finite groups and analogous to finite groups the following results are therefore valid:

Another important result in the representation theory of compact groups is the Peter-Weyl Theorem. 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. Later the modular representation theory of Richard Brauer was developed.