Mathematical Methods for Physics and Engineering : A Comprehensive Guide

29.8 CONSTRUCTION OF A CHARACTER TABLE

mx

my

md′ md

Figure 29.3 The mirror planes associated with 4mm,thegroupoftwo-

dimensional symmetries of a square.

that at least one irrep has dimension 2 or greater. However, there can be no irrep with
dimension 3 or greater, since 3^2 >8, nor can there be more than one two-dimensional
irrep, since 2^2 +2^2 = 8 would rule out a contribution to the sum in (iii) of 1^2 from the
identity irrep, and this must be present. Thus the only possibility is one two-dimensional
irrep and, to make the sum in (iii) correct, four one-dimensional irreps.
Therefore using (i) we can now deduce that there are five classes. This same conclusion
can be reached by evaluatingX−^1 YXfor every pair of elements inG, as in the description
of conjugacy classes given in the previous chapter. However, it is tedious to do so and
certainly much longer than the above. The five classes areI,Q,{R,R′},{mx,my},{md,md′}.
It is straightforward to show that onlyIandQcommute with every element of the
group, so they are the only elements in classes of their own. Each other class must have
at least 2 members, but, as there are three classes to accommodate 8−2 = 6 elements,
there must be exactly 2 in each class. This does not pair up the remaining 6 elements, but
does say that the five classes have 1, 1, 2, 2, and 2 elements. Of course, if we had started
by dividing the group into classes, we would know the number of elements in each class
directly.
We cannot entirely ignore the group structure (though it sometimes happens that the
results are independent of the group structure – for example, all non-Abelian groups of
order 8 have the same character table!); thus we need to note in the present case that
m^2 i=Ifori=x, y, dord′and, as can be proved directly,Rmi=miR′for the same four
values of labeli. We also recall that for any pair of elementsXandY,D(XY)=D(X)D(Y).
We may conclude the following for the one-dimensional irreps.

(a) In view of result (vi),χ(mi)=D(mi)=±1.

(b) SinceR^4 =I, result (vi) requires thatχ(R) is one of 1,i,−1,−i.But,since

D(R)D(mi)=D(mi)D(R′), and theD(mi) are just numbers,D(R)=D(R′). Further

D(R)D(R)=D(R)D(R′)=D(RR′)=D(I)=1,

and soD(R)=±1=D(R′).

(c)D(Q)=D(RR)=D(R)D(R)=1.

If we add this to the fact that the characters of the identity irrep A 1 are all unity then we
can fill in those entries in character table 29.4 shown in bold.
Suppose now that the three missing entries in a one-dimensional irrep arep,qandr,
where each can only be±1. Then, allowing for the numbers in each class, orthogonality