Mathematical Methods for Physics and Engineering - Matematica.NET

29.6 CHARACTERS3m I A, B C, D, EA 1 1 1 1 z; z 2 ; x 2 + y 2A 2 1 1 −1 R zE 2 −1 0 (x, y); (xz, yz); (R x ,R y ); (x 2 − y 2 , 2xy)Table 29.1 The character table for the irreps of group 3m (C 3v or S 3 ). Theright-hand column lists some common functions that transform according tothe irrep against which each is shown (see text).though, in general, these will vary from one representation to another. However,it might also happen that two or more conjugacy classes have the same charactersin a representation – indeed, in the trivial irrep A 1 , see (29.12), every elementinevitably has the character 1.For the irrep A 2 of the group 3m, the classes {I}, {A, B} and {C,D,E} havecharacters 1, 1 and −1, respectively, whilst they have characters 2, −1 and0respectively in irrep E.We are thus able to draw up a character table for the group 3m as shownin table 29.1. This table holds in compact form most of the important informationon the behaviour of functions under the two-dimensional rotational andreflection symmetries of an equilateral triangle, i.e. under the elements of group3m. The entry under I for any irrep gives the dimension of the irrep, since itis equal to the trace of the unit matrix whose dimension is equal to that ofthe irrep. In other words, for the λth irrep χ (λ) (I) =n λ ,wheren λ is its dimension.In the extreme right-hand column we list some common functions of Cartesiancoordinates that transform, under the group 3m, according to the irrep on whoseline they are listed. Thus, as we have seen, z, z 2 ,andx 2 + y 2 are all unchangedby the group operations (though x and y individually are affected) and so arelisted against the one-dimensional irrep A 1 . Each of the pairs (x, y), (xz, yz), and(x 2 − y 2 , 2xy), however, is mixed as a pair by some of the operations, and so thesepairs are listed against the two-dimensional irrep E: each pair forms a basis setfor this irrep.The quantities R x , R y and R z refer to rotations about the indicated axes;they transform in the same way as the corresponding components of angularmomentum J, and their behaviour can be established by examining how thecomponents of J = r × p transform under the operations of the group. To dothis explicitly is beyond the scope of this book. However, it can be noted thatR z , being listed opposite the one-dimensional A 2 , is unchanged by I and by therotations A and B but changes sign under the mirror reflections C, D, andE, aswould be expected.1093