Representations irreducible inequivalent

The reader may quickly verify that the scalar product of any two different vectors from among the set of six in the C3V representation table is zero. Thus these six vectors are orthogonal. Once again, this result always holds between representation vectors in irreducible inequivalent representations for any group. Combining this orthogonality property with the normality property mentioned earlier, we have... 

Many of the properties of IRs that are used in applications of group theory in chemistry and physics follow from one fundamental theorem called the orthogonality theorem (OT). If F, F are two irreducible unitary representations of G which are inequivalent if i -/ j and identical if i = j, then... 

One speaks of Eqs. (9-144) and (9-145) as a representation of the operators a and o satisfying the commutation rules (9-128), (9-124), and (9-125). The states 1, - , ) = 0,1,2,- are the basis vectors spanning the Hilbert space in which the operators a and oj operate. The representation (9-144) and (9-145) is characterized by the fact that a no-particle state 0> exists which is annihilated by a, furthermore this representation is irreducible since in this representation a(a ) operating upon an n-particle state, results in an n — 1 ( + 1) particle state so that there are no invariant subspaces. Besides the above representation there exist other inequivalent irreducible representations of the commutation rules for which neither a no-particle state nor a number operator exists.8... 

Case (c).- a0< produces a set of functions , which is independent of the set la, which forms a basis for the irreducible representation Ay(u) of H, which is inequivalent to A (u), but which has the same dimension. Df corresponds to two inequivalent irreducible representations of H, A (u), and A (u), such that in this case the anti-unitary operators cause A (u) and A (u) to become degenerate. 

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... 

We stated above that there is an inequivalent irreducible representation of S associated with each partition of n, and we use the symbol fx to represent the number of standard tableaux corresponding to the partition, X. Using induction on n. Young proved the theorem... 

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

The number of inequivalent irreducible representations is equal to the number of classes in the group of symmetry operators. 

Efficient use of symmetry can greatly speed up localized-orbital density-functional-exchange-and-correlation calculations. The local potential of density functional theory makes this process simpler than it is in Hartree-Fock-based methods. The greatest efficiency can be achieved by using non-Abelian point-group symmetry. Such groups have multidimensional irreducible representations. Only one member of each such representation need be used in the calculation. However efficient localized-orbital evaluation of the chosen matrix element requires the sum of the magnitude squared of the components of all the members on one of the symmetry inequivalent atoms, based on Eq. 13. 

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... 

The Z>5 group has four classes of operation and has order 10. How many inequivalent irreducible representations are there and what are their dimensions ... 

The first equation of (69) implies that an n-dimensional reducible representation D(R) can be expressed in terms of representations of lower dimensions. Here D(R) is expressed as a direct sum of an ni-dimensional representation Di(R) and an /i2-dimensional representation D R)y where ni + ri2 = n. This suggests the possibility of reducing a reducible representation down to a point where it can be expressed in terms of inequivalent irreducible representations. Furthermore, representations can be constructed from these inequivalent irreducible representations. The second equation of (69) implies that the character of a reducible representation can be written as the sum of the characters of the irreducible representations. From the theory of group representations, one finds that the number of nonequivalent irreducible representations is equal to the number of classes with respect to the relation of conjugation. Also the sum of the squares of the dimensions of the irreducible representations is equal to the order of the group. Then, with the aid of the above theorems one can express this reducibility for a reducible representation D(R) as... 

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