Finding linear representations of a group

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This article discusses various methods to find linear representations of a group over various fields.

Representations in terms of quotient-subgroup matching

A representation of a group $G$ of degree $n$ over a field $k$ is a homomorphism from $G$ to the general linear group $GL(n,k)$. In particular, it is an isomorphism between a quotient of $G$ and a subgroup of $GL(n,k)$. In order to tackle the general question of finding representations of degree $n$ over $k$, it is important to know the (conjugacy classes of) subgroups of $GL(n,k)$.

Classifying over the reals and complex numbers

Over the reals, classifying finite subgroups of the general linear group is equivalent to classifying finite subgroups of the orthogonal group, because any finite subgroup of the general linear group is conjugate to a finite subgroup of the orthogonal group. The classification is easy for small values of $n$:

• For $n = 1$, the only finite subgroups are the trivial group and a cyclic group of order two. In fact, the orthogonal group itself is cyclic of order two.
• For $n = 2$, the only finite subgroups are cyclic groups and dihedral groups. Further information: Classification of finite subgroups of O(2,R)
• For $n = 3$, the finite subgroups include cyclic groups, dihedral groups, and finitely many exceptions. These exceptions include the von Dyck groups and the triangle groups with parameters $(3,3,2)$, $(4,3,2)$ and $(5,3,2)$. In particular, they include the alternating groups of degrees four and five, the symmetric group of degree four, the cube group, which is a direct product of the symmetric group of degree four and a cyclic group of order two, and a group of order $120$. Further information: Classification of finite subgroups of SO(3,R), Classification of finite subgroups of O(3,R)

Thus, given a finite group $G$, determining all the representations of degree less than or equal to $3$ is equivalent to determining all the quotient groups of $G$ that are isomorphic to one of the subgroups listed above.

Over the complex numbers, classifying finite subgroups of the general linear group is equivalent to classifying finite subgroups of the unitary group, because any finite subgroup of the general linear group is conjugate to a finite subgroup of the unitary group. The classification here is somewhat more tricky:

• For $n = 1$, the only finite subgroups are the cyclic subgroups, because $U(1,\mathbb{C})$ is the circle group.
• For $n = 2$, the finite subgroups are completely classified. The finite subgroups of $SU(2,\mathbb{C})$ central extensions of the cyclic group of order two by the finite subgroups of $SO(3,\R)$. A little more work gives the list of finite subgroups of $U(2,\mathbb{C})$. Further information: Classification of finite subgroups of SU(2,C), Classification of finite subgroups of U(2,C).

Thus, given a finite group $G$, determining all the representations of degree less than or equal to $2$ is equivalent to determining all the quotient groups of $G$ that are isomorphic to one of the subgroups listed above.

Finding natural actions

Permutation actions

One way to construct linear representations is to use permutation representations that may arise from the way the group is defined. Permutation representations on sets of size more than one are never irreducible by themselves, because the one-dimensional subspace comprising vectors with all coordinates equal is an invariant subspace. However, subtracting this trivial representation may give an irreducible representation or a representation that is easily decomposed into irreducible constituents.

For instance, the symmetric group of degree $n$ acts naturally on a $n$-dimensional vector space by permuting the basis elements. The action on the $(n-1)$-dimensional subspace of vectors whose coordinates add up to zero is irreducible, and this representation is termed the standard representation. The restriction of this representation to the alternating group also gives an irreducible representation.

Actions on coset spaces and other group-theoretic constructs

Some group actions can be constructed using subgroups, by the fact that a group acts on left coset space by left multiplication. In fact, every transitive group action arises in this way, by the fundamental theorem of group actions. Thus, if there is a subgroup of index $d$, we get a homomorphism from the group to the symmetric group of degree $d$, giving a representation of degree $d - 1$ if we look at the subspace of vectors whose coordinates add up to zero.

The subgroups approach is comprehensive if we have complete information about the subgroup structure, but in the absence of this information, we may use more indirect methods:

• The action of the group by conjugation on the set of elements in any conjugacy class gives a transitive group action. This corresponds to the action on the coset space of the centralizer of any element of the conjugacy class.
• The action of the group by conjugation on the collection of conjugate subgroups to a given subgroup is a transitive group action. This corresponds to the action on the coset space of the normalizer subgroup.
• Actions on the collection of all subsets of a given size by left multiplication, right multiplication, or conjugation, may be of interest. Such actions are not transitive, but their orbits may be of interest.

Using old representations to obtain new ones

Sums and products

We can take a direct sum of linear representations to obtain new linear representations from old ones, but this is not a good method for obtaining irreducible linear representations. The tensor product of linear representations is another method to obtain new linear representations from old ones.

The tensor product of two linear representations is usually not linear (unless one of them has degree one). However, it may decompose as a sum of new irreducible linear representations.

Automorphism groups

Given a linear representation $\rho:G \to GL(n,k)$ and an automorphism $\sigma$ of $G$, we can define a linear representation $\rho \circ \sigma: G \to GL(n,k)$. If $\sigma$ is a class-preserving automorphism, then this new representation is conjugate to the old one, because character determines representation in characteristic zero. Otherwise, we may get a new representation.

In fact, it suffices to construct an automorphism of $G/\operatorname{Ker}(\rho)$ to get a new representation, or more generally, an automorphism of any intermediate quotient.

Induction

Induction from subgroups is a very powerful tool for obtaining new representations. These representations are usually not irreducible but they may have new irreducible constituents.

Induction from the principal character

Induction from the principal character is equivalent to taking the permutation representation on the coset space of the subgroup. Thus, the permutation representations approach can be viewed as a special case of induction from subgroups.

Powerful theorems about induction

Artin's induction theorem states that every linear representation over a field of characteristic zero is a rational linear combination of representations induced from cyclic subgroups. Brauer's induction theorem states that every linear representation is an integral linear combination of representations induced from elementary subgroups. In fact, a stronger version of the statement is that every linear representation is an integral linear combination of representations induced from degree one representations on elementary subgroups.