280 CHAPTER 5. ELEMENTARY PARTICLES
(c)Case k 1 = 3 and k 2 = 0 :We define the right-hand side of (5.2.42) as shown(5.2.47) N⊗N⊗N= ⊗ ⊗
= ( ⊗ )⊗
=
(
⊕
)
⊗ by (5.2.45)= ⊗ ⊕ ⊗
= ⊗α⊕ ⊗ use (5.2.46) for 2nd term= α⊕
α⊕
β
⊕ β.(d)Case k 1 = 2 and k 2 = 1 :We haveN⊗N⊗N= ⊗ ⊗
(5.2.48) N− 1
=
⊗
N− 1
⊗ use (5.2.46)=
N+
N− 1
⊗α=α⊗
N⊕β⊗
N− 1
=
α
N⊕
β
N⊕ β
N− 1 ⊕
β
N− 1.
We know that the representations ofSU(N)for particles and antiparticles are conjugate
relation, i.e. if
(5.2.49) N︸⊗ ··· ⊗︷︷ N︸
k 1⊗N︸⊗ ··· ⊗︷︷ N︸
k 2