Matrix representation of groups

An introduction to the mathematics of group theory for the non-mathematician. If you want to learn formal group theory but are uncomfortable with much of the mathematical literature, this book deserves your consideration. It does not treat matrix representations of groups or character tables in any significant detail, however. 

Many works[5, 6] on group theory describe matrix representations of groups. That is, we have a set of matrices, one for each element of a group, G, that satisfies... 

D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Oxford, New York and London, 1940. 

Reducibility of representations, irreducible representations Multidimensional matrix representations of groups are not unique and, if defined via bases of some carrier spaces, sensitively depend on the chosen basis. Any nonsingular linear transformation of the basis (pj j = 1, 2, 3,. .. to a new basis say yl/ = 1, 2, 3,. .. n], leads to the following well-known transformation formulae ... 

Symmetry tools are used to eombine these M objeets into M new objeets eaeh of whieh belongs to a speeifie symmetry of the point group. Beeause the hamiltonian (eleetronie in the m.o. ease and vibration/rotation in the latter ease) eommutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "bloek diagonal". That is, objeets of different symmetry will not interaet only interaetions among those of the same symmetry need be eonsidered. 

In an n-dimensional space L, the linear operators of the representation can be described by their matrix representatives. This procedure produces a homomorphic mapping of the group G on a group of n x n matrices D(G), i.e., a matrix representation of the group G. From equations (6) it follows that the matrices are non-singular, and that... 

It is reasonable to hope to assemble a complete set of representations to provide a full and non-redundant description of the symmetry species compatible with a point group The problem is that there are far too many representations of any group. On the one hand, matrices in representations derived from expressing symmetry operations in terms of coordinates - as in problem 5-18 - depend on the coordinate system. Thus there are an infinite number of matrix representations of C2v equivalent to example 7, derivable in different coordinate systems. These add no new information, but it is not necessarily easy to recognize that they are related. Even in the cases of representations not derived from geometric models via coordinate systems, an infinite number of other representations are derivable by similarity transformations. 

The idea of a group algebra is very powerful and allowed Frobenius to show constructively the entire structure of irreducible matrix representations of finite groups. The theory is outlined by Littlewood[37], who gives references to Frobenius s work. 

It is interesting that Weyl had a deep conviction that the harmony of nature could be expressed in mathematically beautiful laws and an outstanding characteristic of his work was his ability to unite previously unrelated subjects. He created a general theory of matrix representation of continuous groups and discovered that many of the regularities of quantum mechanics could be best understood by means of group theory. 

Now we come to a totally different method for producing matrix representations of a point group a method which involves the concept of a function space. The word space is used in this context in a mathematical sense and should not be confused with the more familiar three-dimensional physical space. A function space is a collection or family of mathematical functions which obeys certain rules. These rules are a generalization of those which apply to the family of position vectors in physical space and in order to help in understanding them, the corresponding vector rule will be put in square brackets after each function rule. 

An adjacency matrix representation of a graph has several nice properties. Many natural graph operations correspond to standard matrix operations (see (5) for some examples). The bits of M can be packed in groups into computer words, so that... 

Matrix representations are perhaps the most important objects for practical applications of group theory in quantum chemistry. We have seen how they can be defined in terms of a set of basis functions in Eq. 1.20. Evidently, by finding suitable sets of basis functions that transform among themselves under the operations of the group, we can find matrix representations of arbitrarily large dimension. Furthermore, we can apply an arbitrary similarity transformation X to our representations, since if D(G)D(tf) = D(F), XD(G)D(ff)X-1 = XD(F)X-1. We first restrict ourselves, therefore, to considering only representations that are not equivalent within a similarity transformation. Second, wc restrict ourselves to the consideration only of... 

Example 4.1-1 Find a matrix representation of the symmetry group C3v which consists of the symmetry operators associated with a regular triangular-based pyramid (see Section 2.2). 

Exercise 4.8-2 Verify closure in H. Is this sufficient reason to say that H is a group The character of the matrix representation of gj in the representation T induced from T, is... 

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

Rotations in a vector space of three orbitals are described by the group SO(3) of orthogonal 3x3 matrices with determinant +1. To embed the octahedral rotation group in this covering group one needs a matrix representation of O which also consists of orthogonal and unimodular 3x3 matrices. Such a matrix representation is sometimes called the fundamental vector representation of the point group. In the case of O the fundamental vector representation is Ti and not T2. Indeed the 7] matrices are unimodular, i.e. have determinant +1, while the determinants of the T2 matrices are equal to the characters of the one-dimensional representation A2. 

Symmetry considerations are instrumental in a qualitative discussion of spin-orbjt effects. Qualitatively, a phenomenological Hamiltonian of the form Aso Z matrix representation of the usual molecular point group. The same is true for the spatial and spin wave... 

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