Irreducible Representation

An irreducible representation of a group is a nonzero group representation that has no nontrivial invariant subspaces. For example, the orthogonal group has an irreducible representation on .

Every finite-dimensional complex group representation of a finite group or a connected semisimple Lie group breaks up into a direct sum of irreducible representations. But in general, this is not the case, e.g., has a group representation on by

(1)

i.e., . But the subspace is fixed, hence is not irreducible, but there is no complementary invariant subspace.

The irreducible representation has a number of remarkable properties, as formalized in the group orthogonality theorem. For a finite group with complex irreducible realizations by unitary transformations, let the group order be , and the dimension of the th group representation (the order of each constituent matrix) be (a positive integer). Let any group element be denoted , and let the th row and th column of the matrix corresponding to the group element in the th irreducible representation be . The following properties can be derived from the group orthogonality theorem,

(2)

1. The dimensionality theorem:

(3)

where each must be a positive integer and is the group character (trace) of the group representation.

2. The sum of the squared absolute values of the group characters in any irreducible representation equals ,

(4)

3. orthogonality of different representations

(5)

4. In a given group representation, reducible or irreducible, the group characters of all matrices belonging to group elements in the same conjugacy class are identical (group characters of different group representations can coincide on a particular conjugacy class).

5. The number of irreducible representations of a group is equal to the number of conjugacy classes in the group. This number is the size of the character table, while the dimensions of the individual group representations may differ.

6. A one-dimensional group representation with all 1s (totally symmetric) will always exist for any group.

7. For a matrix group representation, the determinant gives a one-dimensional group representation. Its group character, the trace, is generally not a one-dimensional group representation.

8. The multiplicity of the irreducible representation with group character in a reducible group representation with group character is

(6)

where is the group order of the group . Since group characters are constant on conjugacy classes, this may be written

(7)

where the sum is over the conjugacy classes and is the number of elements in .

Irreducible representations can be indicated using mulliken symbols.