|Algebraic structure → Group theory|
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 Hermitian[Sesquilinear form or clarification needed] vector space 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), 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.
- 1 History
- 2 Overview
- 3 Examples of irreducible representations
- 4 Applications in theoretical physics and chemistry
- 5 Lie groups
- 6 See also
- 7 References
- 8 Further reading
- 9 External links
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.
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 (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 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 (all representations can form a subrepresentation with the trivial -invariant subspaces, e.g. the whole vector space , and ). If there is a proper non-trivial invariant subspace, 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 a, b, c... denote elements of a group G with group product signified without any symbol, so ab is the group product of a and b and is also an element of G, and let representations be indicated by D. 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:
and similarly for all other group elements. The last two staments correspond to the requirement that D is a group homomorphism.
Decomposable and indecomposable representations
which diagonalizes every matrix in the representation into the same pattern of diagonal blocks – each of the blocks are representations of the group independent of each other. The representations D(a) and D′(a) are said to be equivalent representations. The 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.
The dimension of D(a) is the sum of the dimensions of the blocks:
Examples of irreducible representations
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
- and the orthogonal subspace given by
- The former irrep is one-dimensional and isomorphic to the trivial representation of . The latter is dimensional and is known as the standard representation of .
- 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 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.
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.
- Representation theory of Lie algebras
- Representation theory of SU(2)
- Representation theory of SL2(R)
- Representation theory of the Galilean group
- Representation theory of diffeomorphism groups
- Representation theory of the Poincaré group
- Theorem of the highest weight
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