Mathematical Principles of Theoretical Physics

5.2. QUARK MODEL 279

The second step provides the rule and method to compute the dimensionalmjfrom thej-th
square diagrams j on the right-hand side of (5.2.42):

(5.2.43) Computation ofmj from j.

1.Rule to group the square diagram in (5.2.42).First of all, in the Young tableau we use
a square to stand for a fundamental representationNofSU(N), and use a column ofN− 1
squares to stand for the conjugate representationNin the right-hand side of (5.2.42):

N= , N=











N− 1.

For example, we can write 3⊗ 3 ⊗3 as

3 ⊗ 3 ⊗ 3 = ⊗ ⊗.

By this rule, the left-hand side of (5.2.42) can be expressed as

(5.2.44) N︸⊗ ··· ⊗︷︷ N︸

k 1

⊗N︸⊗ ··· ⊗︷︷ N︸

k 2

= ⊗ ··· ⊗

︸ ︷︷ ︸

k 1

⊗ ⊗ ··· ⊗

︸ ︷︷ ︸

k 2

Now, we give rules to group the square diagrams of the right-hand side of (5.2.42) for
differentk 1 andk 2.

(a)Case k 1 = 2 ,k 2 = 0 :By the rule (5.2.44), the left-hand side is

N⊗N= ⊗.

Then the right-hand side is defined as

(5.2.45) ⊗α= α⊕
α

= ⊕.

(b)Case k 1 = 1 and k 2 = 1 :We have

(5.2.46) α⊗











N− 1 =

α















N+

α











N− 1