Mathematical Principles of Theoretical Physics

272 CHAPTER 5. ELEMENTARY PARTICLES

K∗^0 K∗+

S= 1

S= 0

S=− 1

ρ− ρ^0 ρ+

ω

I 3

K

∗− K ̄

∗ 0

Q=− 1 Q= 0 Q= 1

Y = 1

Y= 0

Y =− 1

-1 −^1

2 0

1

2 1

I 3

Figure 5.5

and satisfies the multiplication relation (5.2.6).
Usually, the representation given by (5.2.12):

SU(N) =H(G)

is called anN-dimensional fundamental representation of linear norm-preserving transforma-
tion groupG, which for simplicity is denoted bySU(N).

3.Conjugate representation SU(N). The conjugate groupSU(N)ofSU(N)is called the
conjugate representation, expressed as

(5.2.13) SU(N) ={U|U∈SU(N)},

whereUis the complex conjugate ofU.
IfSU(N)andSU(N)are regarded as linear norm-preserving transformation groups of
N-dimensional complex spaceCN, then they represent such linear operators as follows. Let

{e 1 ,···,eN} ⊂CN

constitute a complex orthogonal basis ofCN, i.e.

CN=

{

N

∑

k= 1

ckek | ck∈C

}

.

Then the conjugate spaceC
N
ofCNcan be written as

C

N

=

{

N

∑

k= 1

βkek | βk∈C

}

.