Irreducible representations group theoretical properties

One of the most important aspects of spin functions is their group theoretical properties. In particular, the set of spin functions, for a given N and S, forms a basis for an irreducible representation of the group of permutations of N objects. This group is of order Nl and is often referred to as the symmetric group . It is denoted by the symbol Sn- Thus if P" denotes a permutation of the N spin coordinates [Pg.2675]

The value of the branching-diagram method is that it leads to spin eigenfunctions that provide standard irreducible representations (irreps) of the group when the permutations are applied to spin variables, as in (4.2.1) and for many pmposes it is possible to exploit the group-theoretical properties of the basis functions without ever explicitly constructing them. This approach is developed in Sections 4.3-4.6 and in later chapters. 

W heii every matrix in a representation can be brought into the form of E

similarity transformation then the re])reseTitation is said to be reducible. When this is not possible the representation is irreducible. Clearly irreducible representations can never be equivalent unless they are identical. The remainder of the chapter will be devoted to the construction and properties of irreducible representations that behave like orthogonal vectors in the space of the group elements. The theorems that are outlined below form the basis of representation theory and the means of applying group theoretic ideas to quantum chemical problems. 

The representations provided in the basis of degenerate eigenfunctions are usually irreducible and can be chosen to be unitary matrices (in fact usually orthogonal matrices if the functions are real functions). In practice, what one is usually faced with is a collection of functions which have arbitrary (but known) transformation properties and what one actually wants to do is to adapt these functions so that they actually transform like the true eigenfunctions of the problem. This can be done by means of the group theoretical projection operator. 

---