is a nonzero representation that has no proper nontrivial subrepresentation
Every finite-dimensional unitary representation on a Hilbert space
is the direct sum of irreducible representations.
Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible.
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
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.
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
If there is a proper nontrivial invariant subspace,
Group elements can be represented by matrices, although the term "represented" has a specific and precise meaning in this context.
The last two statements correspond to the requirement that D is a group homomorphism.
A representation is reducible if it contains a nontrivial G-invariant subspace, that is to say, all the matrices
can be put in upper triangular block form by the same invertible matrix
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.
It will only have this form if we choose a suitable basis, which can be obtained by applying the matrix
can be put in block-diagonal form by the same invertible matrix
Each such block is then a group subrepresentation independent from the others.
[2] The (k-dimensional, say) representation can be decomposed into a direct sum of k > 1 matrices: so D(a) is decomposable, and it is customary to label the decomposed matrices by a superscript in brackets, as in D(n)(a) for n = 1, 2, ..., k, although some authors just write the numerical label without parentheses.
It will only have this form if we choose a suitable basis, which can be obtained by applying the matrix
An irreducible representation is by nature an indecomposable one.
have a zero-dimensional, irreducible trivial representation by mapping all group elements to the identity transformation.
Any one-dimensional representation is irreducible since it has no proper nontrivial invariant subspaces.
The irreducible complex representations of a finite group G can be characterized using results from character theory.
is equal to the number of conjugacy classes of
[5] In quantum physics and quantum chemistry, each set of degenerate eigenstates of the Hamiltonian operator comprises a vector space V 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 V. 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.
[6] The irreps of D(K) and D(J), where J is the generator of rotations and K 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.