Mathematical Principles of Theoretical Physics

282 CHAPTER 5. ELEMENTARY PARTICLES

Example 5.6.ForN= 3 ,consider the two cases given by

(5.2.57) 3 ⊗3 and 3⊗ 3 ⊗ 3 ,

which are the most important cases in particle physics. By (5.2.46) and (5.2.47), the Young
tableaux of (5.2.57) are as follows

(5.2.58) 3 ⊗ 3 = ⊗ = ⊕

(5.2.59) 3 ⊗ 3 ⊗ 3 = ⊗ ⊗ = ⊕ ⊕ ⊕

According to (5.2.52) and the methods to computeαNandβN(N= 3 ), we infer from (5.2.58)
and (5.2.59) that

αN= 3 × 2 × 1 , βN= 3 × 2 × 1 , m= 1 for

αN= 3 × 4 × 2 , βN= 3 × 1 × 1 , m= 8 for

αN= 3 × 4 × 5 , βN= 3 × 2 × 1 , m= 10 for

Consequently, we derive, from (5.2.58) and (5.2.59), the following irreducible representations
of (5.2.57):

(5.2.60)

3 ⊗ 3 = 1 ⊕ 8 ,

3 ⊗ 3 ⊗ 3 = 1 ⊕ 8 ⊕ 8 ⊕ 10.

Example 5.7.The other important cases in physics are the two irreduciblerepresentations
forN=4:

4 ⊗4 and 4⊗ 4 ⊗ 4.

Their Young tableaux are given by

⊗ = ⊕

⊗ ⊗ = ⊕ ⊕ ⊕