Irreducible representation

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation formula_1 or irrep of an algebraic structure formula_2 is a nonzero representation that has no proper nontrivial subrepresentation formula_3, with formula_4 closed under the action of formula_5.
Every finite-dimensional unitary representation on a Hilbert space formula_6 is the direct sum of irreducible representations. As irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), the converse may not hold, such as the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices, which is indecomposable but reducible.
History.
Group representation theory was generalized by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act on a vector space over a field formula_7 of arbitrary characteristic, rather than a vector space over the field of real numbers or over the field of complex numbers. The structure analogous to an irreducible representation in the resulting theory is a simple module.
Overview.
Let formula_8 be a representation i.e. a homomorphism formula_9 of a group formula_10 where formula_6 is a vector space over a field formula_12. If we pick a basis formula_13 for formula_6, formula_8 can be thought of as a function (a homomorphism) from a group into a set of invertible matrices and in this context is called a matrix representation. However, it simplifies things greatly if we think of the space formula_6 without a basis.
A linear subspace formula_17 is called formula_10-invariant if formula_19 for all formula_20 and all formula_21. The co-restriction of formula_8 to the general linear group of a formula_10-invariant subspace formula_17 is known as a subrepresentation. A representation formula_9 is said to be irreducible if it has only trivial subrepresentations (all representations can form a subrepresentation with the trivial formula_10-invariant subspaces, e.g. the whole vector space formula_6, and {0}). If there is a proper nontrivial invariant subspace, formula_8 is said to be reducible.
Notation and terminology of group representations.
Group elements can be represented by matrices, although the term "represented" has a specific and precise meaning in this context. A representation of a group is a mapping from the group elements to the general linear group of matrices. As notation, let denote elements of a group with group product signified without any symbol, so is the group product of and and is also an element of , and let representations be indicated by . The representation of "a" is written
By definition of group representations, the representation of a group product is translated into matrix multiplication of the representations:
If is the identity element of the group (so that , etc.), then is an identity matrix, or identically a block matrix of identity matrices, since we must have
and similarly for all other group elements. The last two statements correspond to the requirement that is a group homomorphism.
Reducible and irreducible representations.
A representation is reducible if it contains a nontrivial G-invariant subspace, that is to say, all the matrices formula_32 can be put in upper triangular block form by the same invertible matrix formula_33. In other words, if there is a similarity transformation:
which maps every matrix in the representation into the same pattern upper triangular blocks. Every ordered sequence minor block is a group subrepresentation. That is to say, if the representation is of dimension k, then we have:
formula_35
where formula_36 is a nontrivial subrepresentation. If we are able to find a matrix formula_37 that makes formula_38 as well, then formula_32 is not only reducible but also decomposable.
Notice: Even if a representation is reducible, its matrix representation may still not be the upper triangular block form. It will only have this form if we choose a suitable basis, which can be obtained by applying the matrix formula_40 above to the standard basis.
Decomposable and indecomposable representations.
A representation is decomposable if all the matrices formula_32 can be put in block-diagonal form by the same invertible matrix formula_33. In other words, if there is a similarity transformation:
which diagonalizes every matrix in the representation into the same pattern of diagonal blocks. Each such block is then a group subrepresentation independent from the others. The representations and are said to be equivalent representations. The representation can be decomposed into a direct sum of matrices:
so is decomposable, and it is customary to label the decomposed matrices by a superscript in brackets, as in for , although some authors just write the numerical label without parentheses.
The dimension of is the sum of the dimensions of the blocks:
If this is not possible, i.e. , then the representation is indecomposable.
Notice: Even if a representation is decomposable, its matrix representation may not be the diagonal block form. It will only have this form if we choose a suitable basis, which can be obtained by applying the matrix formula_40 above to the standard basis.
Connection between irreducible representation and decomposable representation.
An irreducible representation is by nature an indecomposable one. However, the converse may fail. 
But under some conditions, we do have an indecomposable representation being an irreducible representation. 
Examples of irreducible representations.
Trivial representation.
All groups formula_10 have a one-dimensional, irreducible trivial representation.
One-dimensional representation.
Any one-dimensional representation is irreducible by virtue since it has no proper nontrivial subspaces.
Irreducible complex representations.
The irreducible complex representations of a finite group G can be characterized using results from character theory. In particular, all such representations decompose as a direct sum of irreps, and the number of irreps of formula_10 is equal to the number of conjugacy classes of formula_10.
Applications in theoretical physics and chemistry.
In quantum physics and quantum chemistry, each set of degenerate eigenstates of the Hamiltonian operator comprises a vector space for a representation of the symmetry group of the Hamiltonian, a "multiplet", best studied through reduction to its irreducible parts. Identifying the irreducible representations therefore allows one to label the states, predict how they will split under perturbations; or transition to other states in . Thus, in quantum mechanics, irreducible representations of the symmetry group of the system partially or completely label the energy levels of the system, allowing the selection rules to be determined.
Lie groups.
Lorentz group.
The irreps of and , where is the generator of rotations and the generator of boosts, can be used to build to spin representations of the Lorentz group, because they are related to the spin matrices of quantum mechanics. This allows them to derive relativistic wave equations.