Irreducible representation

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation ${\displaystyle (\rho ,V)}$ or irrep of an algebraic structure ${\displaystyle A}$ is a nonzero representation that has no proper subrepresentation ${\displaystyle (\rho |_{W},W),W\subset V}$ closed under the action of ${\displaystyle \{\rho (a):a\in A\}}$.

Every finite-dimensional unitary representation on a Hermitian[Sesquilinear form or clarification needed] vector space ${\displaystyle V}$ 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.

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 ${\displaystyle K}$  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.[citation needed]

Overview

Let ${\displaystyle \rho }$  be a representation i.e. a homomorphism ${\displaystyle \rho :G\to GL(V)}$  of a group ${\displaystyle G}$  where ${\displaystyle V}$  is a vector space over a field ${\displaystyle F}$ . If we pick a basis ${\displaystyle B}$  for ${\displaystyle V}$ , ${\displaystyle \rho }$  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 ${\displaystyle V}$  without a basis.

A linear subspace ${\displaystyle W\subset V}$  is called ${\displaystyle G}$ -invariant if ${\displaystyle \rho (g)w\in W}$  for all ${\displaystyle g\in G}$  and all ${\displaystyle w\in W}$ . The restriction of ${\displaystyle \rho }$  to a ${\displaystyle G}$ -invariant subspace ${\displaystyle W\subset V}$  is known as a subrepresentation. A representation ${\displaystyle \rho :G\to GL(V)}$  is said to be irreducible if it has only trivial subrepresentations (all representations can form a subrepresentation with the trivial ${\displaystyle G}$ -invariant subspaces, e.g. the whole vector space ${\displaystyle V}$ , and {0}). If there is a proper non-trivial invariant subspace, ${\displaystyle \rho }$  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

${\displaystyle D(a)={\begin{pmatrix}D(a)_{11}&D(a)_{12}&\cdots &D(a)_{1n}\\D(a)_{21}&D(a)_{22}&\cdots &D(a)_{2n}\\\vdots &\vdots &\ddots &\vdots \\D(a)_{n1}&D(a)_{n2}&\cdots &D(a)_{nn}\\\end{pmatrix}}}$

By definition of group representations, the representation of a group product is translated into matrix multiplication of the representations:

${\displaystyle D(ab)=D(a)D(b)}$

If e is the identity element of the group (so that ae = ea = a, etc.), then D(e) is an identity matrix, or identically a block matrix of identity matrices, since we must have

${\displaystyle D(ea)=D(ae)=D(a)D(e)=D(e)D(a)=D(a)}$

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

A representation is decomposable if a similar matrix P can be found for the similarity transformation:[1]

${\displaystyle D'(a)\equiv P^{-1}D(a)P}$

which diagonalizes every matrix in the representation into the same pattern of diagonal blocks – each of the blocks are representation of the group independent of each other. The representations D(a) and D'(a) are said to be equivalent representations.[2] The representation can be decomposed into a direct sum of k matrices:

${\displaystyle D'(a)=P^{-1}D(a)P={\begin{pmatrix}D^{(1)}(a)&0&\cdots &0\\0&D^{(2)}(a)&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &D^{(k)}(a)\\\end{pmatrix}}=D^{(1)}(a)\oplus D^{(2)}(a)\oplus \cdots \oplus D^{(k)}(a)}$

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:

${\displaystyle \mathrm {dim} [D(a)]=\mathrm {dim} [D^{(1)}(a)]+\mathrm {dim} [D^{(2)}(a)]+\ldots +\mathrm {dim} [D^{(k)}(a)]}$

If this is not possible, i.e. k=1, then the representation is indecomposable.[1][3]

Examples of Irreducible Representations

Trivial Representation

All groups ${\displaystyle G}$  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 ${\displaystyle G}$  is equal to the number of conjugacy classes of ${\displaystyle G}$ .[4]

• The irreducible complex representations of ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$  are exactly given by the maps ${\displaystyle 1\mapsto \gamma }$ , where ${\displaystyle \gamma }$  is an ${\displaystyle n}$ th root of unity.
• Let ${\displaystyle V}$  be an ${\displaystyle n}$ -dimensional complex representation of ${\displaystyle S_{n}}$  with basis ${\displaystyle \{v_{i}\}_{i=1}^{n}}$ . Then ${\displaystyle V}$  decomposes as a direct sum of the irreps ${\displaystyle V_{triv}=\mathbb {C} (\sum _{i=1}^{n}v_{i})}$  and the orthogonal subspace given by:
${\displaystyle V_{std}=\{\sum _{i=1}^{n}a_{i}v_{i}:a_{i}\in \mathbb {C} ,\sum _{i=1}^{n}a_{i}=0\}}$
The former irrep is one-dimensional and isomorphic to the trivial representation of ${\displaystyle S_{n}}$ . The latter is ${\displaystyle n-1}$  dimensional and is known as the standard representation of ${\displaystyle S_{n}}$ .[4]
• Let ${\displaystyle G}$  be a group. The regular representation of ${\displaystyle G}$  is the free complex vector space on the basis ${\displaystyle \{e_{g}\}_{g\in G}}$  with the group action ${\displaystyle g\cdot e_{g'}=e_{gg'}}$ , denoted ${\displaystyle \mathbb {C} G}$ . All irreducible representations of ${\displaystyle G}$  appear in the decomposition of ${\displaystyle \mathbb {C} G}$  as a direct sum of irreps.

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.[5]

Lie groups

Lorentz group

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.[6]

References

1. ^ a b E.P. Wigner (1959). Group theory and its application to the quantum mechanics of atomic spectra. Pure and applied physics. Academic press. p. 73.
2. ^ W.K. Tung (1985). Group Theory in Physics. World Scientific. p. 32. ISBN 978-997-1966-560.
3. ^ W.K. Tung (1985). Group Theory in Physics. World Scientific. p. 33. ISBN 978-997-1966-560.
4. ^ a b Serre, Jean-Pierre (1977). Linear Representations of Finite Groups. Springer-Verlag. ISBN 978-0387901909.
5. ^ "A Dictionary of Chemistry, Answers.com" (6th ed.). Oxford Dictionary of Chemistry.
6. ^ T. Jaroszewicz; P.S Kurzepa (1992). "Geometry of spacetime propagation of spinning particles". Annals of Physics. 216 (2): 226–267. Bibcode:1992AnPhy.216..226J. doi:10.1016/0003-4916(92)90176-M.