Representations of a group

In applications of group theory we often obtain a reducible representation, and we then need to reduce it to its irreducible components. The way that a given representation of a group is reduced to its irreducible components depends only on the characters of the matrices in the representation and on the characters of the matrices in the irreducible representations of the group. Suppose that the reducible representation is F and that the group involved... 

We have found three distinet irredueible representations for the C3V symmetry group two different one-dimensional and one two dimensional representations. Are there any more An important theorem of group theory shows that the number of irredueible representations of a group is equal to the number of elasses. Sinee there are three elasses of operation, we have found all the irredueible representations of the C3V point group. There are no more. 

There are two theorems of fundamental importance, known as Schur s lemmas, which are useful in the study of the irreducible representations of a group. 

This equation (16) is known as the great orthogonality theorem for the irreducible representations of a group and occupies a central position in the theory of group representations. 

Each irreducible representation of a group consists of a set of square matrices of order lt. The set of matrix elements with the same index, grouped together, one from each matrix in the set, constitutes a vector in -dimensional space. The great orthogonality theorem (16) states that all these vectors are mutually orthogonal and that each of them is normalized so that the square of its length is equal to g/li. This interpretation becomes more obvious when (16) is unpacked into separate expressions ... 

Here we find a new concept, the direct product between irreducible representations of a symmetry group. This direct product is related to the product of their corresponding space functions. For our purposes, we will only mention that the direct product between two, Pj and A, (or more) irreducible representations of a group is a new... 

The shape of the Young frame corresponding to the desired irreducible representation of the symmetric group is obtained from the physics of the system. For example, for the totally antisymmetric representation of a group S we use a frame that is completely horizontal with n boxes. For example, for four particles we have... 

To verify that the product formula for characters holds even for functions that transform according to representations of higher dimensions, suppose that the functions /i, /2 / and gi, g2 . gn form bases for n— and m—dimensional representations of a group. Thus under any group operation A, each fi is transformed into a linear combination of all the fk,k = 1,... n and similarly each gj is transformed into a linear combination of all the gi,l = 1,... n. 

Projection operators are a technique for constructing linear combinations of basis functions that transform according to irreducible representations of a group. Projection operators can be used to form molecular orbitals from a basis set of atomic orbitals, or to form normal modes of vibration from a basis of displacement vectors. With projection operators we can revisit a number of topics considered previously but which can now be treated in a uniform way. 

Exercise 4.40 (Used in Proposition 6.12) Suppose p and p are isomorphic representations of a group G. Show that their characters are equal. 

Suppose that T is an n-dimensional representation of a group of transformation operators Om acting on the functions of an n-dimensional function space and that we have basis functions flt /s,. ., fn with the property that the first m (m < n) are transformed among themselves for all Om (e.g. in 6-3, the p-orbitals Pi and p5 were transformed among themselves by all O, and so m = 2 for this case) ... 

The connection between group theory and quantum mechanics will be established in Section 9.7, where the importance of the representations of a group will become evident. Instead of dealing with the matrices of a representation, for many purposes the information provided by their traces... 

The number of nonequivalent irreducible representations of a group is equal to the number of classes in the group. 

Theorem (2) shows that there are only a finite number of nonequivalent irreducible representations of a group of finite order. Any reducible representation must either be the direct sum of two or more irreducible representations or be convertible to such a direct sum by the performance of a similarity transformation on its matrices. In the former case, it is easy to see by inspection what irreducible representations make up the reducible representation in the latter case, this is not obvious, since the matrices... 

Suppose that we have a set of matrices, (f,. r/, sA, (<, . . . , which form a representation of a group. If we make the same similarity transformation on each matrix, we obtain a new set of matrices, namely,... 

Any set of algebraic functions or vectors may serve as the basis for a representation of a group. In order to use them for a basis, we consider them to be the components of a vector and then determine the matrices which show how that vector is transformed by each symmetry operation. The resulting matrices, naturally, constitute a representation of the group. We have previously used the coordinates jc, y, and z as a basis for representations of groups C2r (page 78) and T (page 74). In the present case it will be easily seen that the matrices for one operation in each of the three classes are as follows ... 

That is, each element of 1 is the element of, fY divided by A. Since division by zero is not defined, only matrices for which the corresponding determinants are nonzero can have inverses. A matrix. < /such that A = 0 is said to be singular (no inverse), whereas matrices of which the corresponding determinants are nonzero are said to be nonsingular. Only nonsingular matrices can occur in representations of a group. It is also clear that since only square matrices can have corresponding determinants, only square matrices can have inverses. 

We begin by reiterating the definition of a PR and listing some conventions regarding PFs. A projective unitary representation of a group G = g, of dimension g is a set of matrices that satisfy the relations... 

The orthogonality theorem The inequivalent irreducible unitary matrix representations of a group G satisfy the orthogonality relations... 

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