Mathematical Principles of Theoretical Physics

5.2. QUARK MODEL 281

represents a particle system, then its conjugate

(5.2.50) N︸⊗ ··· ⊗︷︷ N︸

k 1

⊗N︸⊗ ··· ⊗︷︷ N︸

k 2

represents the antiparticle system. Due to the symmetry of particles and antiparticles, (4.2.49)
and (4.2.50) have the same irreducible representations. Inaddition each composite particle is
composed ofNparticles withN≤3. Hence it is enough to give the four cases 1)-4) above
for the physical purpose.

2.Computation of (5.2.43).In the right-hand side of the Young tableaux (4.2.45)-(4.2.48),
the number of square diagrams is theKas in (4.2.42), and each of which represents a dimen-
sionmj( 1 ≤j≤K)of an irreducible representation. The rule to computemjis as follows.
For example, for the following square diagram

(5.2.51)

the dimensionmis given by the formula

(5.2.52) m=
αN
βN

,

whereαNandβNcan be computed by the diagram as (5.2.51).

(a)Computation ofαN.Fill the blanks of (5.2.51) in numbers in the following fashion:

(5.2.53)

N N+1 N+2 N+2

N-1 N N+1

N-2

TheαNequals to the multiplication of all numbers in (5.2.53),

(5.2.54) αN=N(N+ 1 )(N+ 2 )(N+ 3 )(N− 1 )N(N+ 1 )(N− 2 ).

(b)Computation ofβN.Fill the blanks of (5.2.51) in the fashion:

(5.2.55)

6 4 3 1

4 2 1

1

where the data in a square equals to the number of all squares on its right-hand side and below
it adding one. For example, for the square marked 6, there are3 squares on its right-hand side,
and 2 square below it. Hence the numberkin this blank is

k= 3 + 2 + 1 = 6.

Then, the numberβNis the multiplication of all numbers in (4.2.55)

(5.2.56) βN= 6 × 4 × 3 × 1 × 4 × 2 × 1 × 1.

Thus, by (4.2.54) and (4.2.56) we can get the value of (4.2.52). In the following, we give
a few examples to show how to use the Young tableau to compute the irreducible representa-
tions ofSU(N).