Linear representations of finite groups

Linear transformationsEdit

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:


Invariant subspacesEdit

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 representationsEdit

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 representationsEdit

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.

Matrix representationsEdit

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.

Schur's lemmaEdit

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.