Irreducible performance

Irreducible performance $(\rho ,V)$ ${\ displaystyle (\ rho, V)}$ ${\ displaystyle (\ rho, V)}$ algebraic structure $A$ ${\ displaystyle A}$ $A$ Is a nonzero representation that does not have its own subview $(\rho |_{W},W),W\subset V$ ${\ displaystyle (\ rho | _ {W}, W), W \ subset V}$ ${\ displaystyle (\ rho | _ {W}, W), W \ subset V}$ closed by $\{\rho (a):a\in A\}$ ${\ displaystyle \ {\ rho (a): a \ in A \}}$ ${\ displaystyle \ {\ rho (a): a \ in A \}}$ .

Any finite-dimensional on a Hermitian vector space $V$ ${\ displaystyle V}$ $V$ ^[1] is the direct sum of irreducible representations. Since irreducible representations are always indecomposable (that is, cannot be further decomposed into the direct sum of representations), these terms are often confused. However, in the general case, there are many reducible but indecomposable representations, such as the two-dimensional representation of real numbers, acting by means of upper triangular unipentent matrices.

Content

History

The representation theory of groups was generalized by Richard Brower in the 1940s, giving a , in which matrix operations act on a vector space over a field $K$ ${\ displaystyle K}$ with an arbitrary characteristic , and not a vector space over a field of real numbers or over a field of complex numbers . A structure similar to the irreducible representation in the resulting theory is a simple module .

Overview

Let be $\rho$ ${\ displaystyle \ rho}$ will be a representation, i.e. a homomorphism $\rho :G\to GL(V)$ ${\ displaystyle \ rho: G \ to GL (V)}$ groups $G$ ${\ displaystyle G}$ where $V$ ${\ displaystyle V}$ is the vector space above the field $F$ ${\ displaystyle F}$ . If we choose a basis $B$ ${\ displaystyle B}$ for $V$ ${\ displaystyle V}$ , $\rho$ ${\ displaystyle \ rho}$ can be considered a function (homomorphism) from a group to a set of invertible matrices, and in this context, a representation is called a matrix representation . However, everything is greatly simplified if we consider the space $V$ ${\ displaystyle V}$ without basis.

Linear subspace $W\subset V$ ${\ displaystyle W \ subset V}$ called $G$ ${\ displaystyle G}$ -invariant if $\rho (g)w\in W$ ${\ displaystyle \ rho (g) w \ in W}$ for all $g\in G$ ${\ displaystyle g \ in G}$ and all $w\in W$ ${\ displaystyle w \ in W}$ . narrowing $\rho$ ${\ displaystyle \ rho}$ on $G$ ${\ displaystyle G}$ -invariant subspace $W\subset V$ ${\ displaystyle W \ subset V}$ known as subview . They say that performance $\rho :G\to GL(V)$ ${\ displaystyle \ rho: G \ to GL (V)}$ irreducible if it has only trivial subviews (all representations can form a subview with trivial $G$ ${\ displaystyle G}$ -invariant subviews, for example, with the whole vector space $V$ ${\ displaystyle V}$ and {0} ). If there exists a proper nontrivial invariant subspace $\rho$ ${\ displaystyle \ rho}$ They say the view is reducible .

Group designations and terminology

Elements of a group can be represented by matrices , although the term “represented” has a specific and precise meaning in this context. A group view is a mapping from the elements of a group to a complete linear group of matrices. Let a , b , c ... denote elements of the group G with a group product that is not reflected by any symbol, that is, ab is the group product of a and b , which is also an element of the group G. Let representations be denoted by the letter D. The representation of a is written as

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}}

{\ 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 the multiplication of the representation matrices :

D(ab)=D(a)D(b)

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

If e is a neutral element of the group (so that $ae=ea=a$ ${\ displaystyle ae = ea = a}$ ), then D ( e ) is the identity matrix , since we must have

D(ea)=D(ae)=D(a)D(e)=D(e)D(a)=D(a)

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

and the same for the other elements of the group. The last two statements correspond to the requirement that D be a group homomorphism .

Decomposable and indecomposable representations

The representation is decomposable if a similar matrix P can be found for the similarity transformation ^[2] :

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

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

,

which diagonalizes any matrix in the presentation into diagonal blocks - each of the blocks is a group representation independently of each other. It is said that the representations D ( a ) and D ′ ( a ) are equivalent ^[3] . A representation can be decomposed into a direct sum of k matrices :

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)

{\ 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 that D ( a ) is decomposable and usually labels on decomposition matrices are written in brackets, like D ^{( n )} ( a ) for n = 1, 2, ..., k , although some authors write numerical labels without brackets.

The dimension D ( a ) is equal to the sum of the dimensions of the blocks:

\dim[D(a)]=\dim[D^{(1)}(a)]+\dim[D^{(2)}(a)]+\cdots +\dim[D^{(k)}(a)]

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

If this is not possible, that is $k=1$ ${\ displaystyle k = 1}$ , then the representation is indecomposable ^[2] ^[4] .

Examples of irreducible representations

Trivial Presentation

All groups $G$ ${\ displaystyle G}$ have a one-dimensional irreducible trivial representation. More generally, any one-dimensional representation is irreducible due to the absence of its own nontrivial subspaces.

Irreducible Integrated Representations

Irreducible complex representations of a finite group G can be described using results from character theory . In particular, all such representations are decomposable into a direct sum of irreducible representations and the number of irreducible representations of the group $G$ ${\ displaystyle G}$ equal to the number of conjugacy classes $G$ ${\ displaystyle G}$ ^[5] .

Irreducible complex representations $\mathbb {Z} /n\mathbb {Z}$ ${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$ exactly defined by the mappings $1\mapsto \gamma$ ${\ displaystyle 1 \ mapsto \ gamma}$ where $\gamma$ ${\ displaystyle \ gamma}$ is an $n$ ${\ displaystyle n}$ th root of unity .

Let be $V$ ${\ displaystyle V}$ will be $n$ ${\ displaystyle n}$ -dimensional integrated representation $S_{n}$ ${\ displaystyle S_ {n}}$ with basis $\{v_{i}\}_{i=1}^{n}$ ${\ displaystyle \ {v_ {i} \} _ {i = 1} ^ {n}}$ . Then $V$ ${\ displaystyle V}$ decomposes as a direct sum of irreducible representations

V_{triv}=\mathbb {C} \left(\sum _{i=1}^{n}v_{i}\right)

{\ displaystyle V_ {triv} = \ mathbb {C} \ left (\ sum _ {i = 1} ^ {n} v_ {i} \ right)}

and the orthogonal subspace is given by the formula:

V_{std}=\left\{\sum _{i=1}^{n}a_{i}v_{i}:a_{i}\in \mathbb {C} ,\sum _{i=1}^{n}a_{i}=0\right\}

{\ displaystyle V_ {std} = \ left \ {\ sum _ {i = 1} ^ {n} a_ {i} v_ {i}: a_ {i} \ in \ mathbb {C}, \ sum _ {i = 1} ^ {n} a_ {i} = 0 \ right \}}

The first irreducible representation is one-dimensional and isomorphic to the trivial representation

S_{n}

{\ displaystyle S_ {n}}

. Second is

n-1

{\ displaystyle n-1}

dimensional and known as standard representation

S_{n}

{\ displaystyle S_ {n}}

^[5] .

Let be $G$ ${\ displaystyle G}$ - Group. group $G$ ${\ displaystyle G}$ is a free complex vector space with basis $\{e_{g}\}_{g\in G}$ ${\ displaystyle \ {e_ {g} \} _ {g \ in G}}$ with group action $g\cdot e_{g'}=e_{gg'}$ ${\ displaystyle g \ cdot e_ {g '} = e_ {gg'}}$ denoted by $\mathbb {C} G.$ ${\ displaystyle \ mathbb {C} G.}$ All irreducible representations $G$ ${\ displaystyle G}$ appear in decomposition $\mathbb {C} G$ ${\ displaystyle \ mathbb {C} G}$ as a direct sum of irreducible representations.

Applications in Theoretical Physics and Chemistry

In quantum mechanics and quantum chemistry, each set of degenerate eigenstates of a Hamiltonian operator constitutes a vector space V to represent the symmetry group of the Hamiltonian, a “multiplet” that is best studied through reduction to irreducible parts. The notation of irreducible representations therefore allows us to assign labels to states and to predict how they upon perturbation or go into another state in V. Thus, in quantum mechanics, the irreducible representations of the symmetry group of the system partially or completely determine the labels for the energy levels of the system, which allows us to determine the selection rules ^[6] .

Lee Groups

Lorentz Group

The irreducible representations D ( K ) and D ( J ) , where J is a generator of rotations and K is a generator of boosts , can be used to construct the of the Lorentz group , since they are related to quantum mechanics . This allows us to use them to derive ^[7] .

Notes

↑ Definition A finite-dimensional vector space over a field C equipped with a positive definite Hermitian form is called a Hermitian space ( Nikitin 2010 ), ( Timofeeva 2017 )
↑ ¹ ² Wigner, 1959 , p. 73.
↑ Tung, 1985 , p. 32.
↑ Tung, 1985 , p. 33.
↑ ¹ ² Serre, 1977 .
↑ A Dictionary of Chemistry, Answers.com (unopened) . Oxford Dictionary of Chemistry.
↑ Jaroszewicz, Kurzepa, 1992 , p. 226-267.

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Articles

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Links

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McGraw-Hill dictionary of scientific and technical terms (neopr.) .

[1] Definition A finite-dimensional vector space over a field C equipped with a positive definite Hermitian form is called a Hermitian space ( Nikitin 2010 ), ( Timofeeva 2017 )

[_864b5a71e2de7995-2] ¹ ² Wigner, 1959 , p. 73.

[_c89bcc5546be333d-3] Tung, 1985 , p. 32.

[_c89bcc5546be333c-4] Tung, 1985 , p. 33.

[_282c36d813307094-5] ¹ ² Serre, 1977 .

[6] A Dictionary of Chemistry, Answers.com (unopened) . Oxford Dictionary of Chemistry.

[_9dee0341cb36a5df-7] Jaroszewicz, Kurzepa, 1992 , p. 226-267.

Irreducible performance

Content

History

Overview

Group designations and terminology

Decomposable and indecomposable representations

Examples of irreducible representations

Trivial Presentation

Irreducible Integrated Representations

Applications in Theoretical Physics and Chemistry

Lee Groups

Lorentz Group

See also

Associative Algebra

Lee Groups

Notes

Literature

Books

Articles

Further Reading

Links

More articles: