Groups irreducible matrix representations

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. 

When the direct product of two irreducible matrix representations of a group is reducible, it can be reduced to a direct sum of irreducible representations by cin equivalent transformation with a constant matrix, i.e. the same matrix for all the matrix representatives of the symmetry operators of the group (2). We shall assume the irreducible representations in unitary form then the constant matrix can be chosen as the real orthogonal matrix whose elements are the coupling coefficients occuring in Eq. (5). The orthogonality properties can be expressed as... 

For file three-dimensional rotation group Ej one obtains the highest symmetry of the irreducible matrix representations (7) by taking the A-functions in the order sine before cosine. For the two-dimensional dihedral group >co it is more natural (vide p. 230) to chose the opposite order of the A-functions. 

The sets of matrices e, aj, b( 62, a, b etc., therefore form representations of the group. The matrix representation e, a, b is said to be reducible and to have been reduced by the similarity transformation with the matrix p. If it is not possible to find a similarity transformation which will further reduce all the matrices of a given representation, the representation is said to be irreducible. 

The most important single theorem in group theory is that giving the orthogonality relation between the irreducible matrix representations of any group. As stated in Chapter 10, this theorem is... 

But this is just the expression that gives the elements of a matrix which is the product./ of two other matrices. Thus the matrices that describe the transformations of a set of k eigenfunctions corresponding to a /c-fold degenerate eigenvalue are a A-dimensional representation for the group. Moreover, this representation is irreducible. If it were reducible we could divide the k eigenfunctions. . . , or k linear combinations thereof, up... 

If each of the blocks in the matrices comprising the matrix system A cannot be reduced ftirther, the matrix system has been reduced completely and each of the matrix systems A1, A2, . .. in the direct sum is said to be irreducible. Matrix systems that are isomorphous to a group G are called matrix representations (Chapter 4). Irreducible representations (IRs) are of great importance in applications of group theory in physics and chemistry. A matrix representation in which the matrices are unitary matrices is called a unitary representation. Matrix representations are not necessarily unitary, but any representation of a finite group that consists of non-singular matrices is equivalent to a unitary representation, as will be demonstrated in Section A1.5. 

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

It ie dear that eqn (10-6.4) is much easier to solve than eqn (10-6.3), though the results, of course, are identical. In general, by using symmetry orbitals which are a basis for one of the irreducible representations of the point group, the matrix whose elements are... 

Since the operator YSn m nm is invariant upon the crystal space group, and the functions ( (k) and I g fk) transform upon irreducible representation of the space group, the matrix element (3.113) will not be zero only for excitonic states (p(/),k) and (p( ),k) which transform upon the same irreducible representation of the crystal group. 

Since each of the functions in VT forms a basis for an irreducible representation of the group, the matrix SD(i2)S must be diagonal or block diagonal for all R. The required matrix is therefore S. 

A criterion for reducibility or irreducibility of a matrix representation is required. It is desired to impose a simple test, which will indicate whether dr not a representation is reducible. It will be proved subsequently that a matrix that commutes with every matrix of an irreducible representation is a constant matrix, and conversely, if there exists a nonconstant matrix—that is, one that is not a simple multiple of the unit matrix—which commutes with all the matrices of a representation then the representation is reducible. This is the central theorem of group representations and we will use this result in other proofs later in the chapter. 

The tables have been constructed by considering possible symmetries (symmetry groups), creating suitable matrix representations, using similarity transformations to find the irreducible representations, summing up the diagonal elements we end up with the character tables in question. 

On account of the phase factor which appears in the matrix element of eq. (20), it is convenient to use the irreducible multiplier representation for non-symmorphic space groups instead of r, Rj), vdiich is defined by... 

For the matrix representation of the S5mimetry operations of symmetry groups, the most common representation, especially in crystallography, see the Volume IV of the present five-volume work, (Chiriae-Putz-Chiriae, 2005), the character of a symmetry operation corresponds to the sum of the numbers on the main diagonal of its irreducible matrix. 

In Section 4.8 we found how the matrix representation of a set of basis vector transformations for a point group can sometimes be made simpler. For example, the 2 x 2 matrix representation for x, y at O in H2O can be reduced to two T x 1 matrices (i.e. a simple number for each operation) for x and y. This process of simplifying a representation to a set of irreducible standard representations is central to the application of symmetry in chemistry and corresponds to finding the fundamental modes of vibration that underlie molecular motion. 

However, we can derive the irreducible representations before the patterns of atomic motion in the vibrational modes are actually identified. This is done by first imagining that each individual atom is independent of the others. For example. Figure 5.2a shows a basis of nine vectors for the atoms in the H2O molecule. This gives us the expected number of degrees of freedom, since 3N = 9 for H2O. If we were to set up a matrix representation for, say, the C2 operation in the C2V group for this basis, then we would have to use the 9x9 matrix shown in Figure 5.4a. 

Section 4.11 used the matrix representation to deal with a set of three basis vectors jc, y, z on the central atom of a square planar D41, complex. It was shown that this basis can be reduced to + A2U by inspection of the matrices for the operations in the Ah group. The characters for the reducible and irreducible representations are shown in Table 5.2. 

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 ... 

Since square matrices with any integral number of dimensions are possible, the number of possible (matrix) representations of any group is infinite. However, a small set of matrix representations of a given group have properties of particular significance these are called the irreducible representations of the group. 

From the relations already developed it is possible to obtain further interesting results. Any matrix representation of a group must be some one of the irreducible representations or some combination of them otherwise it would be an additional irreducible representation, but the number of irreducible representations is limited to the number of classes. Any reducible representation can be reduced to its irreducible representations by a similarity transformation which leaves the character unchanged. Thus we can write for the character of a matrix R of the reducible representation the expression... 

Having done this we solve the Scln-ddinger equation for the molecule by diagonalizing the Hamiltonian matrix in a complete set of known basis fiinctions. We choose the basis functions so that they transfonn according to the irreducible representations of the synnnetry group. 

The Hamiltonian matrix will be block diagonal in this basis set. There will be one block for each irreducible representation of the synnnetry group. 

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