Irreducible Representation

An irreducible representation of a group is a group representation that has no nontrivial invariant subspaces. For example, the orthogonal group 👁 O(n)
has an irreducible representation on 👁 R^n
.

Any representation of a finite or semisimple Lie group breaks up into a direct sum of irreducible representations. But in general, this is not the case, e.g., 👁 (R,+)
has a representation on 👁 R^2
by

i.e., 👁 phi(a)(x,y)=(x+ay,y)
. But the subspace 👁 y=0
is fixed, hence 👁 phi
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. Let the group order of a group be 👁 h
, and the dimension of the 👁 i
th representation (the order of each constituent matrix) be 👁 l_i
(a positive integer). Let any operation be denoted 👁 R
, and let the 👁 m
th row and 👁 n
th column of the matrix corresponding to a matrix 👁 R
in the 👁 i
th irreducible representation be 👁 Gamma_i(R)_(mn)
. The following properties can be derived from the group orthogonality theorem,

1. The dimensionality theorem:

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

2. The sum of the squares of the group characters in any irreducible representation 👁 i
equals 👁 h
,

3. Orthogonality of different representations

4. In a given representation, reducible or irreducible, the group characters of all matrices belonging to operations in the same class are identical (but differ from those in other representations).

5. The number of irreducible representations of a group is equal to the number of conjugacy classes in the group. This number is the dimension of the 👁 Gamma
matrix (although some may have zero elements).

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

7. A one-dimensional representation for a group with elements expressed as matrices can be found by taking the group characters of the matrices.

8. The number 👁 a_i
of irreducible representations 👁 chi_i
present in a reducible representation 👁 c
is given by

where 👁 h
is the group order of the group and the sum must be taken over all elements in each class. Written explicitly,

where 👁 chi_i^'
is the group character of a single entry in the character table and 👁 n_R
is the number of elements in the corresponding conjugacy class.

Irreducible representations can be indicated using Mulliken symbols.

See also

Character Table, Finite Group, Group, Group Character, Group Orthogonality Theorem, Group Representation, Itô's Theorem, Lie Algebra Representation, Mulliken Symbols, Orthogonal Group Representations, Semisimple Lie Group Unitary Transformation, Vector Space, Wedderburn's Theorem

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Irreducible Representation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IrreducibleRepresentation.html

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