PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

alternating group which is not a direct product of cyclic groups, and is isomorphic to the
tetrahedral group T d . The group A 4 has 12 elements, which can be written in terms of
the generators of the group S and T. The generators [35,36] of A 4 satisfy the relation
S 2 = (ST) 3 = (T) 3 = 1 (2.30)
There are three one-dimensional irreducible representations of the group A 4 denoted as
1 S = 1 T = 1 (2.31)
1 ′ S = 1 T = ω 2 (2.32)
1 ′′ S = 1 T = ω (2.33)
It is easy to check that there is no two-dimensional irreducible representation of this
group. The three-dimensional unitary representations of T and S are
⎛ ⎞ ⎛ ⎞
1 0 0
T = ⎝ 0 ω 2 0 ⎠, S = 1 −1 2 2
⎝ 2 −1 2 ⎠ . (2.34)
3
0 0 ω
2 2 −1
where T has been chosen to be diagonal. The multiplication rules for the singlet and
triplet representations are as follows
For two triplets
one can write
1×1 = 1, 1 ′ ×1 ′′ = 1 3×3 = 3+3 A +1+1 ′ +1 ′′ . (2.35)
a = (a 1 ,a 2 ,a 3 ), b = (b 1 ,b 2 ,b 3 ) (2.36)
1 ≡ (ab) = (a 1 b 1 +a 2 b 3 +a 3 b 2 ) (2.37)
1 ′ ≡ (ab) ′ = (a 3 b 3 +a 1 b 2 +a 2 b 1 ) (2.38)
1 ′′ ≡ (ab) ′′ = (a 2 b 2 +a 1 b 3 +a 3 b 1 ) . (2.39)
Note that while 1 remains invariant under the exchange of the second and third elements
of a and b, 1 ′ is symmetric under the exchange of the first and second elements while 1 ′′
is symmetric under the exchange of the first and third elements.
3 ≡ (ab) S = 1 )
(2a 1 b 1 −a 2 b 3 −a 3 b 2 ,2a 3 b 3 −a 1 b 2 −a 2 b 1 ,2a 2 b 2 −a 1 b 3 −a 3 b 1 (2.40)
3
3 A ≡ (ab) A = 1 )
(a 2 b 3 −a 3 b 2 ,a 1 b 2 −a 2 b 1 ,a 1 b 3 −a 3 b 1 . (2.41)
3
The representation 3 has been considered in the literature [35] to generate the neutrino
massmatrix. Wecanseethatthefirstelement herehas2-3exchange symmetry,the second
element has 1-2 exchange symmetry, while the third element has 1-3 exchange symmetry.
26