Mathematical Methods for Physics and Engineering : A Comprehensive Guide

29.6 CHARACTERS

3 m IA,BC,D,E

A 1 11 1 z;z^2 ;x^2 +y^2

A 2 11 − 1 Rz

E 2 − 10 (x, y); (xz, yz); (Rx,Ry); (x^2 −y^2 , 2 xy)

Table 29.1 The character table for the irreps of group 3m(C 3 vorS 3 ). The

right-hand column lists some common functions that transform according to

the 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 characters

in a representation – indeed, in the trivial irrep A 1 , see (29.12), every element

inevitably has the character 1.

For the irrep A 2 of the group 3m, the classes{I},{A, B}and{C, D, E}have

characters 1, 1 and−1, respectively, whilst they have characters 2,−1and0

respectively in irrep E.

We are thus able to draw up acharacter tablefor the group 3mas shown

in table 29.1. This table holds in compact form most of the important infor-

mation on the behaviour of functions under the two-dimensional rotational and

reflection symmetries of an equilateral triangle, i.e. under the elements of group

3 m. The entry underIfor any irrep gives the dimension of the irrep, since it

is equal to the trace of the unit matrix whose dimension is equal to that of

the irrep. In other words, for theλth irrepχ(λ)(I)=nλ,wherenλis its dimen-

sion.

In the extreme right-hand column we list some common functions of Cartesian

coordinates that transform, under the group 3m, according to the irrep on whose

line they are listed. Thus, as we have seen,z,z^2 ,andx^2 +y^2 are all unchanged

by the group operations (thoughxandyindividually are affected) and so are

listed against the one-dimensional irrep A 1. Each of the pairs (x, y), (xz, yz), and

(x^2 −y^2 , 2 xy), however, is mixed as a pair by some of the operations, and so these

pairs are listed against the two-dimensional irrep E: each pair forms a basis set

for this irrep.

The quantitiesRx,RyandRzrefer to rotations about the indicated axes;

they transform in the same way as the corresponding components of angular

momentumJ, and their behaviour can be established by examining how the

components ofJ=r×ptransform under the operations of the group. To do

this explicitly is beyond the scope of this book. However, it can be noted that

Rz, being listed opposite the one-dimensional A 2 , is unchanged byIand by the

rotationsAandBbut changes sign under the mirror reflectionsC,D,andE,as

would be expected.