Irreducible representation of a group

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... 

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. 

We have found three distinct irreducible 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 irreducible representations of a group is equal to the number of classes. Since there are three classes of operation, we have found all the irreducible representations of the C3v point group. There are no more. 

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 (T, V) is an irreducible representation of a group G on a finite-dimensional complex vector space V. If A is a linear map such that A T(g) = T(g) A for all g in G, then there is a complex number c such that A = cl, where I is the identity operator (identity matrix in this case). 

The process of finding all the matrices of lowest order (irreducible representations) of a group is usually very tedious. In many cases it is necessary to know only their trace, (40), 10.VIIIN, which in group theory is called the character, denoted by %. The trace of a representation f(<) is denoted by —" (A ), X(i)(A2),etc.,Au A2, A3i. .. being members of a group. The character of every element in a single class is identical. The class is represented by a subscript, Xi(t), etc. If the column gives the class and the row the representation, then ... 

The irreducible representations of a group are generally reducible as representations of a sub-group. In Racah s lemma this statement is considered by introducing the concept of a group which is reduced with respect to a sub-group. By this is meant that the irreducible representa-... 

Generalization of the characteristic value problem. The characteristic value problem can be formulated as the quest for the irreducible linear manifolds which are invariant under an operator. The principal result of the spectral theory of normal operators can be formulated, from this point of view, as the statement that all irreducible linear manifolds of normal operators are one-dimensional. Similarly, one can ask for irreducible closed linear manifolds which are invariant under a set of operators. Since a closed linear manifold which is invariant under a set of operators is also invariant under the group or algebra generated by these operators, one is naturally led in this way to a linear manifold which belongs to an irreducible representation of a group or an algebra. 

Theorem 6 Let L2 and T2 be two irreducible representations of a group G, and consider vectors formed by taking elements [i j] and kl from the respective representation matrices for every element of the group. Then these vectors are orthogonal to each other, and their squared norm is equal to the order of the group, divided by the dimension of the irrep ... 

One immediate result of the relation is that it enables us to tell when we have completed the task of finding all the inequivalent irreducible representations of a group. If we consider the C3V group, for example, we note that it is of order six, since there are six symmetry operations. This means that each representation vector will have six elements, i.e., is a vector in six-dimensional space. The maximum number of orthogonal vectors we can have in six-dimensional space is six. Therefore, the number of representation vectors cannot exceed the order of the group. Furthermore, since the number of vectors provided by an -dimensional representation is (e.g., E is two-dimensional and gives four vectors), we can state that the sum of the squares of the... 

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