Irreducible Representations and Invariant Integration

Irreducible representations are the building blocks of all other representations. Just as each molecule is made up of particular atoms, each representation is made up of particular irreducible representations. Unlike a molecule, whose properties are determined not only by which atoms it is made of, but also by their configuration, a representation is merely the sum of its irreducible parts. Mathematically, irreducible representations are useful because one can often reduce an idea or a calculation involving representations to an easier one involving only irreducible representations. Physically, irreducible representations correspond to fundamental physical entities. 

In Section 6.1 we define irreducible representations. Then we state, prove and illustrate Schur s lemma. Schur s lemma is the statement of the all-or-nothing personality of irreducible representations. In the Section 6.2 we discuss the physical importance of irreducible representations. In Section 6.3 we introduce invariant integration and apply it to show that characters of irreducible representations form an orthonormal set. In the optional Section 6.4 we use the technology we have developed to show that finite-dimensional unitary representations are no more than the sum of their irreducible parts. The remainder of the chapter is devoted to classifying the irreducible representations of 5(7(2) and 50(3). 

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Exercise 6.12 Suppose that G is a Lie group with a volume-one invariant integral. Suppose that (G, V, p) is a representation with character y. Show that p is irreducible if and only if /(g) dg = 1. 

In contrast to HF, where full use of symmetry in evaluating two-electron integrals is limited to one-dimensional representations, it is possible to use the full symmetry of V in Xa-like methods. Assume V is invariant under the generators / , of a group G. Then we need only concern ourselves with wavefunctions /./i > that transform as some irreducible representation 2 and basis partner g of G ... 

Depending on the problem, i i, and may be atomic orbitals used to construct molecular orbitals, or they may represent two different electronic states of the same atom or molecule, etc. The energy, then, expresses the extent of interaction between the two wave functions i[<. and As was shown in Chapter 4, an integral will have a nonzero value only if the integrand is invariant to the symmetry operations of the point group, i.e., belongs to the totally symmetric irreducible representation. 

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