In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper subrepresentation closed under the action of. Every finite-dimensional unitary representation on a Hilbert space is the direct sum of irreducible representations. As irreducible representations are always indecomposable, these terms are often confused; however, in general there are many reducible but indecomposable representations, such as the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices.
Let be a representation i.e. a homomorphism of a group where is a vector space over a field. If we pick a basis for, can be thought of as a function 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 without a basis. A linear subspace is called -invariant if for all and all. The restriction of to a -invariant subspace is known as a subrepresentation. A representation is said to be irreducible if it has only trivial subrepresentations. If there is a proper non-trivial invariant subspace, is said to be reducible.
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, 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 staments correspond to the requirement that is a group homomorphism.
Decomposable and indecomposable representations
A representation is decomposable if all the matrices can be put in block-diagonal form by the same invertible matrix. 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 representations 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.
Examples of irreducible representations
Trivial representation
All groups have a one-dimensional, irreducible trivial representation. More generally, any one-dimensional representation is irreducible by virtue of having 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 is equal to the number of conjugacy classes of.
The irreducible complex representations of are exactly given by the maps, where is an th root of unity.
Let be an -dimensional complex representation of with basis. Then decomposes as a direct sum of the irreps
Let be a group. The regular representation of is the free complex vector space on the basis with the group action, denoted All irreducible representations of appear in the decomposition of as a direct sum of irreps.
Example of an irreducible representation over
Let be a group and be a finite dimensional irreducible representation of G over. By the theory of group actions, the set of fixed points of is non empty, that is, there exists some such that for all. This forces every irreducible representation of a group over to be one dimensional.
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 rulesto 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.
