Mathematical Principles of Theoretical Physics

5.2. QUARK MODEL 271

∆− ∆

0

S= 0

S=− 1

S=− 2

Σ∗−

Σ∗^0

∆+ ∆++

Σ∗+

Ξ∗^0

Q=− 1 Q= 0 Q= 1

Y= 1

Y = 0

Y=− 1

− 1 −^120121

I 3

S=− 3

Ξ∗−

Ω−

Q= 2

Y=− 2

−^3232

Figure 5.4

If all block matricesHk(G) ( 1 ≤k≤n)in (5.2.8) cannot be split into smaller pieces anymore,

then the direct sum of all sub-representationsHk(G)as

H 1 (G)⊕ ··· ⊕Hn(G)

is called an irreducible representation ofG, which can be simply written in the following

form

(5.2.9) H(G) =m 1 ⊕ ··· ⊕mn,

wheremkis the order ofHk(G).

2.Fundamental representation SU(N).LetGbe a linear transformation group made up

of all linear norm-preserving mappings ofCN:

(5.2.10) g:CN→CN.

It is known that for each linear operatorg∈Gas defined in (5.2.10), there is a unique matrix
U∈SU(N)such that

(5.2.11) g(ψ) =Uψ ∀ψ∈CN.

Hence, relation (5.2.11) provides a correspondence

g7→U forg∈G andU∈SU(N),

which is a one to one and onto mapping

(5.2.12) H:G→SU(N),