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5) Its dimension is 1. The group C3v has another one-dimensional representation: D(A2)(E) = 1 , D(A2)( C 3 ) = 1 , D(A2)(U d = -1 , D(A2)(C 3 1 ) = 1, D(A2)(U2) = -1 , D(A2)(U 3 ) = -1 . 6) 46 4. Representations of a Group I As will be seen in due course, C 3v has only the three independent representations given above. The nomenclature for the representations AI' A2 and E will be explained in Sect. 4. 1 Basis for a Representation Let 1/1 1, 1/12' ... ,1/1d be independent elements in a vector space, and suppose that the linear operators 2 Gi operating on them form a group i'§.

27); that is, it must include group elements in complete classes. 28). Then the residual set f(J' satisfies the same equation Gf(J' G -1 = f(J' for any G, if it is a nonempty set at all. Since this equality is assumed to hold for any Gj (i = 1, 2, ... , g), f(J' includes all the transformed sets Gjf(J' G j- 1 • Such a f(J' must contain all mutually conjugate elements. In other words, it must consist of classes. D) Having established the basic properties of classes, we next consider class multiplication.

If the inner product is defined in this space, by application of an appropriate linear transformation to this basis (Gram-Schmidt orthogonalization), we can construct an orthonormal basis {e 1,e2, ... 4 Metric Vector Spaces 39 using their components with respect to this basis. 36) To sum up, the metric vector space is provided with the inner product. 34). 3 Unitary Operators and Unitary Matrices Suppose u and v belong to a metric vector space "Y. 38) then U is said to be unitary. 37) are written as Therefore, we have (u', v') = L ui*vi = L (L ute UiI ) utV, .