Construction of a character table

REPRESENTATION THEORY
where the λi are the eigenvalues
 m
λ1 0
···

m
 0
λ2

 .
..
.
 .
.
0
··· 0
of D(X). Therefore, from (29.22), we have that

 
1 0
··· 0
0
.. 
..  

. 
. 
.
= 0 1

  .
.
.
.
. 0 
0   .
0 ··· 0
1
λm
n
Hence all the eigenvalues λi are mth roots of unity, and so χ(X), the trace of
D(X), is the sum of n of these. In view of the implications of Lagrange’s theorem
(section 28.6 and subsection 28.7.2), the only values of m allowed are the divisors
of the order g of 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 irrep µ of the group, the character χ(µ) (X)
of the class to which group element X belongs. 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;
2
(iii)
µ nµ = g, where nµ is the dimension of the µth irrep and g is the order
of the group;
(iv) the identity irrep (one-dimensional with all characters equal to 1) is present
for every
group;
2
(µ)
(v)
X χ (X) = g.
(µ)
(vi) χ (X) is the sum of nµ mth roots of unity, where m is the order of X.
Construct the character table for the group 4mm (or C4v ) using the properties of classes,
irreps and characters so far established.
The group 4mm is the group of two-dimensional symmetries of a square, namely rotations
of 0, π/2, π and 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 by π/2 and 3π/2, R and R ;
a rotation by π, Q ;
four mirror reflections mx , my , md and md .
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 (clearly Rmx = mx R), and so there are fewer than eight
classes, and hence fewer than eight irreps. But requirement (iii), with g = 8, then implies
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29.8 CONSTRUCTION OF A CHARACTER TABLE
mx
md
md
my
Figure 29.3 The mirror planes associated with 4mm, the group of twodimensional 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 32 > 8, nor can there be more than one two-dimensional
irrep, since 22 + 22 = 8 would rule out a contribution to the sum in (iii) of 12 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 evaluating X −1 Y X for every pair of elements in G, 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 are I, Q, {R, R }, {mx , my }, {md , md }.
It is straightforward to show that only I and Q commute 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
m2i = I for i = x, y, d or d and, as can be proved directly, Rmi = mi R for the same four
values of label i. We also recall that for any pair of elements X and Y , 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) Since R 4 = I, result (vi) requires that χ(R) is one of 1, i, −1, −i. But, since
D(R)D(mi ) = D(mi )D(R ), and the D(mi ) are just numbers, D(R) = D(R ). Further
D(R)D(R) = D(R)D(R ) = D(RR ) = D(I) = 1,
and so D(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 A1 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 are p, q and r,
where each can only be ±1. Then, allowing for the numbers in each class, orthogonality
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