M.-C. Chen et al. / Nuclear Physics B 883 (2014) 267–305 273
First we start with the consistency condition equation (2.16) for a class-inverting u.Byre-
placing g by u(g) (and u(g) by u
2
(g))inEq.(2.16) and bringing the U
r
i
’s to the other side, we
obtain
ρ
r
i
u(g)
=U
T
r
i
ρ
r
i
u
2
(g)
∗
U
∗
r
i
=U
T
r
i
ρ
r
i
(g)
∗
U
∗
r
i
∀g ∈G and ∀i. (2.21)
This shows that with U
r
i
also the transpose U
T
r
i
fulfills equation (2.16). Since ρ
r
i
is an irreducible
representation, Schur’s Lemma implies that
U
T
r
i
=e
iα
U
r
i
∀i, (2.22)
which is only possible if each U
r
i
is either symmetric or anti-symmetric, i.e. α is either 0 or π .
Thus, V
r
i
=U
r
i
U
∗
r
i
=±1. Hence, V consists of blocks identical to ±1.
Now assume that all V
r
i
are ±1. Then, by inserting Eq. (2.16) into itself,
ρ
r
i
u
2
(g)
=
U
r
i
U
∗
r
i
ρ
r
i
(g)
U
r
i
U
∗
r
i
†
=ρ
r
i
(g) ∀g ∈G and ∀i. (2.23)
Since this equation is true for all irreducible representations, it follows that u
2
(g) = g for all g
in G and the order of u is thus either one or two (i.e. u is involuntary). This completes the proof
that V is different from a diagonal matrix with only ±1 on the diagonal if and only if u is of
order n>2.
We therefore conclude that, if an involutory u is
imposed as a symmetry, G may be amended
by an additional Z
2
symmetry. This is possible if and only if V
r
i
= +1 for some r
i
. We will
discuss this case in more detail in Section 2.7. In what follows, we refer to such an enlargement
as “trivial” extension of G to G × Z
2
. Note that the assignment of Z
2
charges to the fields of
a model is not arbitrary but is given by the signs of the V
r
i
for their respective representations
under G. We will also discuss this Z
2
factor in an example in Section 3.3.
The second logical possibility, case (ii), is that v is
an inner automorphism.
3
In this case,
the order of u is larger than two but one can still show that the flavor group only gets enlarged
by some Abelian factor. However, CP transformations connected to automorphisms that square
to an inner automorphism do not seem to yield any CP transformations which are physically
different from those that are connected to involutory automorphisms. The reason is that if two
automorphisms u and u
are related by an inner automorphism,
u(g) = bu
(g)b
−1
∀g ∈G and some b ∈ G, (2.24)
the resulting CP transformations only differ by a transformation with the group element ρ(b).
Since the latter transformation certainly is a symmetry of the Lagrangean, the two CP transfor-
mations are indistinguishable. In fact, it turns out that we were not able to find an example where
there is a class-inverting automorphism of higher order that is not related to an involutory class-
inverting automorphism in the prescribed way. We were able to prove that such an automorphism
cannot exist for some cases, see Appendix C, and have explicitly checked this for all non-Abelian
groups of order less than 150 (with the exception of some groups of order 128) with the group
theory program GAP [15].
The last logical possibility, case (iii), is that u
2
=v is an outer automorphism. Then the addi-
tional generator h with ρ
r
i
(h) =V
r
i
does not commute with all group elements of G, and, hence,
3
The property that u should square to the identity or an inner automorphism has also been stressed in [14].However,
the discussion there misses the point that this does not imply U
r
i
U
∗
r
i
= ρ
r
i
(g) for some g in G but that the group still
might be extended by a Z
2
factor.