Group nomenclature

REPRESENTATION THEORY
4mm
A1
A2
B1
B2
E
I
1
1
1
1
2
Q
1
1
1
1
−2
R, R 1
1
−1
−1
0
mx , my
1
−1
1
−1
0
md , md
1
−1
−1
1
0
Table 29.4 The character table deduced for the group 4mm. For an explanation of the entries in bold see the text.
with the characters of A1 requires that
1(1)(1) + 1(1)(1) + 2(1)(p) + 2(1)(q) + 2(1)(r) = 0.
The only possibility is that two of p, q, and r equal −1 and the other equals +1. This
can be achieved in three different ways, corresponding to the need to find three further
different one-dimensional irreps. Thus the first four lines of entries in character table 29.4
can be completed. The final line can be completed by requiring it to be orthogonal to the
other four. Property (v) has not been used here though it could have replaced part of the
argument given. 29.9 Group nomenclature
The nomenclature of published character tables, as we have said before, is erratic
and sometimes unfortunate; for example, often E is used to represent, not only
a two-dimensional irrep, but also the identity operation, where we have used I.
Thus the symbol E might appear in both the column and row headings of a
table, though with quite different meanings in the two cases. In this book we use
roman capitals to denote irreps.
One-dimensional irreps are regularly denoted by A and B, B being used if a
rotation about the principal axis of 2π/n has character −1. Here n is the highest
integer such that a rotation of 2π/n is a symmetry operation of the system, and
the principal axis is the one about which this occurs. For the group of operations
on a square, n = 4, the axis is the perpendicular to the square and the rotation
in question is R. The names for the group, 4mm and C4v , derive from the fact
that here n is equal to 4. Similarly, for the operations on an equilateral triangle,
n = 3 and the group names are 3m and C3v , but because the rotation by 2π/3 has
character +1 in all its one-dimensional irreps (see table 29.1), only A appears in
the irrep list.
Two-dimensional irreps are denoted by E, as we have already noted, and threedimensional irreps by T, although in many cases the symbols are modified by
primes and other alphabetic labels to denote variations in behaviour from one
irrep to another in respect of mirror reflections and parity inversions. In the study
of molecules, alternative names based on molecular angular momentum properties
are common. It is beyond the scope of this book to list all these variations, or to
1102