Mathematical Methods for Physics and Engineering : A Comprehensive Guide

REPRESENTATION THEORY

where theλiare the eigenvalues ofD(X). Therefore, from (29.22), we have that













λm 1 0 ··· 0

0 λm 2

..

.

..

.

..

. 0
0 ··· 0 λmn













=













10 ··· 0

01

..

.

..

.

..

. 0
0 ··· 01













.

Hence all the eigenvaluesλiaremth roots of unity, and soχ(X), the trace of

D(X), is the sum ofnof these. In view of the implications of Lagrange’s theorem

(section 28.6 and subsection 28.7.2), the only values ofmallowed are the divisors

of the ordergof the group.

29.8 Construction of a character table

In order to decompose representations into irreps on a routine basis using

characters, it is necessary to have available a character table for the group in

question. Such a table gives, for each irrepμof the group, the characterχ(μ)(X)

of the class to which group elementXbelongs. To construct such a table the

following properties of a group, established earlier in this chapter, may be used:

(i) the number of classes equals the number of irreps;

(ii) the ‘vector’ formed by the characters from a given irrep is orthogonal to

the ‘vector’ formed by the characters from a different irrep;

(iii)

∑

μn

2

μ=g,wherenμis the dimension of theμth irrep andgis the order

of the group;

(iv) the identity irrep (one-dimensional with all characters equal to 1) is present

for every group;

(v)

∑

X

∣

∣χ(μ)(X)

∣

∣^2 =g.

(vi)χ(μ)(X)isthesumofnμmth roots of unity, wheremis the order ofX.

Construct the character table for the group 4 mm(orC 4 v) using the properties of classes,

irreps and characters so far established.

The group 4mmis the group of two-dimensional symmetries of a square, namely rotations
of 0,π/2,πand 3π/2 and reflections in the mirror planes parallel to the coordinate axes
and along the main diagonals. These are illustrated in figure 29.3. For this group there are
eight elements:

• the identity,I;


• rotations byπ/2and3π/2,RandR′;


• arotationbyπ,Q;


• four mirror reflectionsmx,my,mdandmd′.

Requirements (i) to (iv) at the start of this section put tight constraints on the possible
character sets, as the following argument shows.
The group is non-Abelian (clearlyRmx=mxR), and so there are fewer than eight
classes, and hence fewer than eight irreps. But requirement (iii), withg= 8, then implies