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A set of objects with the definition of the operation of multiplication constitute a group. Multiplication should be associative and the group should be closed under multiplication. A multiplicative identity as well as multiplicative inverses for all the group elements should exit. The number of elements in a group is called the order of the group and a finite group is the one with a finite order. For every element of the group,a∈G, we have aha _{=ewhereeis the identity element and h}

a is said to

be the order of the elementa.

Let f be a map from a group G to another G0. This map is homomorphic only if it preserves the multiplicative structure, i.e. f(a)f(b) =f(ab). The special case where the map is one-to-one is known as the isomorphic map. The element g−1_{ag for g ∈ G is called the element conjugate to a. The set of all the elements} conjugate to a form a conjugacy class of G, i.e. g−1ag,∀g∈G. All elements in a

conjugacy class have the same order. Every group can be divided into a specific number of conjugacy classes.

If the subset H of a group G is itself a group, then H is said to be the subgroup ofG. If a subgroup N of Gsatisfiesg−1N g=N for any elementg of G, thenN is called a Normal subgroup ofG. LetHandN be a subgroup and a normal subgroup ofGrespectively. The groupHN is defined as

{hini|hi ∈H, ni ∈N} (3.11)

It can be shown that HN =N H and this group is a subgroup of G. The normal subgroup is important in the definition of semidirect product of two groups. In the special case whereG=HN =N HwithH∩N ={e}, the semidirect productNoH

is isomorphic to G. Leta1, a2 ∈ N and b1, b2 ∈ H. The multiplication rule for the semidirect productN oH is given by

(a1, a2)(b1, b2) = (a1, fa2(b1), a2b2) (3.12) wherefa2(b1) denotes the homomorphic map

fa2(b1) =a2b1a −1

2 (3.13)

from H to N. The definition of direct product is simpler. Let a1, a2 ∈ G1 and

b1, b2 ∈ G2, where G1, G2 are two groups. The multiplication rule for the direct productG1×G2 is given by

(a1, a2)(b1, b2) = (a1b1, a2b2). (3.14)

A representation ofG is a homomorphic map from the abstract elements g of G onto matrices D(g). The representation is said to be faithful if the map is injective, i.e. if all the representation matrices are distinct. The vector space on which the representation matrices act is said to the representation space and the dimension of the vector space is the dimension of the representation. Ifvj constitute

a subspace of the representation space andD(g)ijvj∀galso lies in the same subspace,

then it is called an invariant subspace. A representation with an invariant subspace is reducible, and one without is irreducible. Every reducible representation can be block diagonalised into irreducible representations using similarity transformations. In other words, every reducible representation is a direct sum of the constituting

irreducible representations: D(g) = r X α=1 ⊕Dα(g) (3.15)

whereD(g)is reducible andDα(g)are the various irreducible representations. Given

two irreducible representations, we may obtain the tensor product of the correspond- ing irreducible vector spaces. The tensor product representation acting on this tensor product space in general is reducible:

Dm(g)⊗Dn(g) = r

X

α=1

⊕Dα(g) (3.16)

CharacterχD(g) of a representation D(g) is the trace of the representation

matrix D(g). Similarity transformations do not change the character. By the same logic, all the elements in a conjugacy class have the same character. The number of irreducible representations for a group will be equal to the number of its conjugacy classes. In a character table, characters of all the conjugacy classes for all the irreducible representations are tabulated.

Suppose we are given the tensor product of any two irreducible representa- tions and we need the decomposition of the direct product into the direct sum of irreducible representations. As a first step, we can use the character table and the orthogonality relationship for characters to enumerate the irreducible representations contained in the direct sum. LetDaandDb be two irreducible representations. The

orthogonality relation [53] for characters is

1 N X g∈G χDa(g) ∗ χDb(g) =δab (3.17) whereN is the order of the group Gand χs are the characters. The direct product a⊗bcan be represented using the Kronecker product of matricesDaandDb. So the

characters of the direct product representation will be the product of the characters of the representationsDa and Db.

LetDa⊗b be the representation of the direct product of two irreducible rep-

resentations, Da and Db. Let χDa⊗b(g) be the characters of this direct product representation. SinceDa⊗b is a reducible representation, it will contain each of the

irreducible representations (Dc) some integer number of times, mc. We can use the

orthogonality relation, Eq. (3.17), to computemc:

mc= 1 N X g∈G χDc(g) ∗_χ Da⊗b(g) = 1 N X g∈G χDc(g) ∗_χ Da(g)χDb(g). (3.18)