Let and be two vector spaces over the same field , and let be a mapping between the two vector spaces. If fulfills
then is said to be a linear transformation between the two vector spaces. The group of all such linear transformations when is called the general linear group of and denoted .
If is finite dimensional with dimensionality , then any element of is isomorphic to a matrix in . Choosing a basis for , the effect of a linear representation is given by its effect on the basis vectors:
Let be a linear transformation on . If is a subspace which is unaltered by , i.e.
then is said to be an invariant subspace (under ).
Let and let the linear transformation pick out the -component of any vector :
Then the subspace is an invariant subspace.
Linear group representations[edit | edit source]
A linear representation of a group is a mapping from the group to linear transformations on a vector space , , such that group multiplication is preserved:
Note that to the left the multiplication is in the group , while to the right the multiplication the combination of successive linear transformations in .
Let be any group and represent all of its elements by the unit element 1 (of ). This is allways a representation (check it!), and it is for obvious reasons called the trivial representation.
Irreducible representations[edit | edit source]
If is a linear representation of on , we say that a subspace is an invariant subspace under the representation if
for all group elements .
If and are the only invariant subspaces of (under ), then the representation is said to be irreducible.
The irreducible representation can be thought of as the building blocks of which one can construct general representations of the group.
Our previous example, where all group elements were represented by the unit element 1, is an irreducible representation. Since any vector multiplied by unity equals itself, each unique vector defines its own subspace under this representation.
Since we are concerned with finite groups, i.e. groups with only a finite number of members, it suffices also to choose finite dimensional vector spaces . If we will choose a basis for the vector space , we can further regard all representations as matrix representations:
where is any field and is the dimension of . We will mostly be concerned with the fields of the real () and complex () numbers, in which case the entries of the representation matrices will be real or complex, respectively.
The components of the representation matrix are obtained from the effect of the representation on the basis vectors , where is the dimension of the vectors space :
These representation matrices have to obey
which is nothing other than ordinary matrix multiplication.
Proof: Since is a representation, the two calculations
must yield the same.
Let and be irreducible representations of the group on and . Assume that is a linear transformation such that
Then is either invertible or identically zero.
Proof: is a subspace of which is invariant under :
Since is irreducible this means that either
- , in which case , or
- , in which case is onto.
We also have that is an invariant subspace of under , since belongs to the kernel if does:
Therefore, since is irreducible either
- , in which case is one-to-one, or
- , in which case .
Therefore, is either zero or invertible.