Английская Википедия:Irreducible representation

Материал из Онлайн справочника
Перейти к навигацииПерейти к поиску

Шаблон:Short description Шаблон:Group theory sidebarШаблон:Use American English

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

Every finite-dimensional unitary representation on a Hilbert space <math>V</math> 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.

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 <math>K</math> 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.Шаблон:Citation needed

Overview

Шаблон:Further

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

A linear subspace <math>W\subset V</math> is called <math>G</math>-invariant if <math>\rho(g)w\in W</math> for all <math>g\in G</math> and all <math> w\in W</math>. The co-restriction of <math>\rho</math> to the general linear group of a <math>G</math>-invariant subspace <math>W\subset V</math> is known as a subrepresentation. A representation <math>\rho: G \to GL(V)</math> is said to be irreducible if it has only trivial subrepresentations (all representations can form a subrepresentation with the trivial <math>G</math>-invariant subspaces, e.g. the whole vector space <math>V</math>, and {0}). If there is a proper nontrivial invariant subspace, <math>\rho</math> 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 Шаблон:Math denote elements of a group Шаблон:Math with group product signified without any symbol, so Шаблон:Math is the group product of Шаблон:Math and Шаблон:Math and is also an element of Шаблон:Math, and let representations be indicated by Шаблон:Math. The representation of a is written as

<math>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}</math>

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

<math>D(ab) = D(a)D(b) </math>

If Шаблон:Math is the identity element of the group (so that Шаблон:Math, etc.), then Шаблон:Math is an identity matrix, or identically a block matrix of identity matrices, since we must have

<math>D(ea) = D(ae) = D(a)D(e) = D(e)D(a) = D(a)</math>

and similarly for all other group elements. The last two statements correspond to the requirement that Шаблон:Math 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 <math>D(a)</math> can be put in upper triangular block form by the same invertible matrix <math>P</math>. In other words, if there is a similarity transformation:

<math> D'(a) \equiv P^{-1} D(a) P,</math>

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, for example, of dimension 2, then we have: <math display="block">D'(a) = P^{-1} D(a) P = \begin{pmatrix} D^{(11)}(a) & D^{(12)}(a) \\ 0 & D^{(22)}(a) \end{pmatrix}, </math>

where <math>D^{(11)}(a)</math> is a nontrivial subrepresentation. If we are able to find a matrix <math>P </math> that makes <math>D^{(12)}(a) = 0</math> as well, then <math>D(a)</math> 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 <math>P^{-1}</math> above to the standard basis.

Decomposable and indecomposable representations

A representation is decomposable if all the matrices <math>D(a)</math> can be put in block-diagonal form by the same invertible matrix <math>P</math>. In other words, if there is a similarity transformation:[1]

<math> D'(a) \equiv P^{-1} D(a) P,</math>

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 Шаблон:Math and Шаблон:Math are said to be equivalent representations.[2] The (k-dimensional, say) representation can be decomposed into a [[direct sum of matrices|direct sum of Шаблон:Math matrices]]:

<math>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),</math>

so Шаблон:Math is decomposable, and it is customary to label the decomposed matrices by a superscript in brackets, as in Шаблон:Math for Шаблон:Math, although some authors just write the numerical label without parentheses.

The dimension of Шаблон:Math is the sum of the dimensions of the blocks:

<math>\dim[D(a)] = \dim[D^{(1)}(a)] + \dim[D^{(2)}(a)] + \cdots + \dim[D^{(k)}(a)].</math>

If this is not possible, i.e. Шаблон:Math, then the representation is indecomposable.[1][3]

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 <math>P^{-1}</math> above to the standard basis.

Connection between irreducible representation and indecomposable 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.

  • When group <math>G</math> is finite, and it has a representation over field <math>\Complex</math>, then an indecomposable representation is an irreducible representation.[4]
  • When group <math>G</math> is finite, and it has a representation over field <math>K</math>, if we have <math>char(K)\nmid |G|</math>, then an indecomposable representation is an irreducible representation.

Examples of irreducible representations

Trivial representation

All groups <math>G</math> have a one-dimensional, irreducible trivial representation by mapping all group elements to the identity transformation.

One-dimensional representation

Any one-dimensional representation is irreducible 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 complex representations decompose as a direct sum of irreps, and the number of irreps of <math>G</math> is equal to the number of conjugacy classes of <math>G</math>.[5]

  • The irreducible complex representations of <math>\Z / n\Z</math> are exactly given by the maps <math>1 \mapsto \gamma</math>, where <math>\gamma</math> is an <math>n</math>th root of unity.
  • Let <math>V</math> be an <math>n</math>-dimensional complex representation of <math>S_n</math> with basis <math>\{v_i\}^n_{i=1}</math>. Then <math>V</math> decomposes as a direct sum of the irreps <math display="block">V_\text{triv} = \Complex \left ( \sum^n_{i=1} v_i \right )</math> and the orthogonal subspace given by <math display="block">V_\text{std} = \left \{ \sum^n_{i=1} a_i v_i : a_i \in \Complex, \sum^n_{i=1} a_i = 0 \right \}.</math> The former irrep is one-dimensional and isomorphic to the trivial representation of <math>S_n</math>. The latter is <math>n-1</math> dimensional and is known as the standard representation of <math>S_n</math>.[5]
  • Let <math>G</math> be a group. The regular representation of <math>G</math> is the free complex vector space on the basis <math>\{e_g\}_{g \in G}</math> with the group action <math>g \cdot e_{g'} = e_{gg'}</math>, denoted <math>\Complex G.</math> All irreducible representations of <math>G</math> appear in the decomposition of <math>\Complex G</math> as a direct sum of irreps.

Example of an irreducible representation over Шаблон:Math

  • Let <math>G</math> be a <math>p</math> group and <math>V = \mathbb{F}_p^{n}</math> be a finite dimensional irreducible representation of G over <math>\mathbb{F}_p</math>. By Orbit-stabilizer theorem, the orbit of every <math>V</math> element acted by the <math>p</math> group <math>G</math> has size being power of <math>p</math>. Since the sizes of all these orbits sum up to the size of <math>G</math>, and <math>0 \in V</math> is in a size 1 orbit only containing itself, there must be other orbits of size 1 for the sum to match. That is, there exists some <math>v\in V</math> such that <math>gv = v</math> for all <math>g \in G</math>. This forces every irreducible representation of a <math>p</math> group over <math> \mathbb{F}_p</math> to be one dimensional.

Applications in theoretical physics and chemistry

Шаблон:See also

In quantum physics and quantum chemistry, each set of degenerate eigenstates of the Hamiltonian operator comprises a vector space Шаблон:Mvar 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 Шаблон:Mvar. 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]Шаблон:Better source needed

Lie groups

Шаблон:Main

Lorentz group

Шаблон:Main

The irreps of Шаблон:Math and Шаблон:Math, where Шаблон:Math is the generator of rotations and Шаблон:Math 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.[7]

See also

Associative algebras

Lie groups

References

Шаблон:Reflist

Books

Articles

Further reading

External links

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

  1. ↑ 1,0 1,1 Шаблон:Cite book
  2. ↑ Шаблон:Cite book
  3. ↑ Шаблон:Cite book
  4. ↑ Шаблон:Cite book
  5. ↑ 5,0 5,1 Шаблон:Cite book
  6. ↑ Шаблон:Cite web
  7. ↑ Шаблон:Cite journal
Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Irreducible_representation&oldid=15447240