Irreducible matrix representations

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 

The theory of group representation proves a number of results. 

There is a set of inequivalent irreducible representations. The number of these is equal to the number of equivalence classes among the group elements. If the a irreducible representation is an / x / , matrix, then 

The elements of the irreducible representation matrices satisfy a somewhat complicated law of composition  

If we specify in the previous item that rj = /, the orthogonality theorem results  

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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 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... 

Now we may find the matrix representation, U, of the operators. The dimensions of the matrices will be the same as the dimensions of the irreducible representation used. The matrix representation of the identity operator, U E), will of course be the identity matrix. If it is noted that any permutation may be written as a product of transpositions (permutations of order 2), and any transposition may be written as a product of elementary transpositions p p + 1) [74], then it is only nessesary to find matrix representations of the elementary transpositions. The diagonal elements of the elementary transposition p p + 1) are given by... 

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. 

In this case, the transformation of 11 to T has brought 1 (R) into block-diagonal form and the matrix representation T was therefore reducible. But if T is irreducible, then dk = dh V k, and D is a constant matrix, that is the constant d, times the unit matrix. But if UHU- 1 is a multiple of the unit matrix, then so is H. And if H i and H2 are multiples of the unit matrix, then so also is M = VifH, — iH2), which proves Schur s lemma. 

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

If an eigenvalue X - as defined by the characteristic equation for T in any matrix representation - is degenerate, the situation is more complicated, and the eigenvalue problems (2.3) have to be replaced by the associated stability problems see ref. B, Sec. 4. In matrix theory, the search for the irreducible stable subspaces of T is reflected in the block-diagonalization of the matrix... 

We can say that this set of characters also forms a representation. It is an alternate shorthand version of the matrix representation. Whether in matrix or character format, this representation is called a reducible representation, a combination of more fundamental irreducible representations as described in the next section. Reducible representations are frequently designated with a capital gamma (F). 

The set of 3 X 3 matrices obtained for H2O is called a reducible matrix representation because it is the sum of irreducible representations (the block-diagonalized 1 X 1 matrices), which cannot be reduced to smaller component parts. The set of characters of these matrices also forms the reducible representation F, for the same reason. 

The relationships among symmetry operations, matrix representations, reducible and irreducible representations, and character tables are conveniently illustrated in a flowchart, as shown for C2v symmetry in Table 4.8. 

It is not so straightforward to reduce the matrix representation in Eq. 4.37a into its irreducible components and in the general case a reducible representation will contain off-diagonal matrix elements. We therefore require a general method of analyzing reducible representations. 

Note that they would not all be diagonal since otherwise the representation would be reducible. The partner of a given function would not necessarily be the same for these two equivalent representations. Notice also that the Eqs. 5.64 would be affected by this change in the representation matrices. We are emphasizing the point that the matrix elements of a representation are not unique, they are specified only up to a similarity transformation. The character of each matrix representation is, however, unique. It is neither mandatory, nor is it alw ays expedient to use coordinate transformation matrices as the matrices of an irreducible representation. 

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. 

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