278 CHAPTER 5. ELEMENTARY PARTICLES
then under the new basisE ̃={Ψ ̃ 1 ,···,Ψ ̃Nk}, the linear mapping
H(U):X→X (X as in (5.2.22))becomes the block diagonal form
H(U) =
H 1 (U) 0
..
.
0 HK(U)
∀U∈SU(N).
This means that the new basisE ̃ofXis divided intoKsub-bases
G 1 ={E ̃ 1 ,···,E ̃m 1 },···,GK={E ̃JK+ 1 ,···,E ̃Jk+mK},withJK+mK=NK, such that each subspaceXJofX, spanned by thej-th sub-basisGjas
Xj=spanGj ( 1 ≤j≤K)is an invariant subspace of the mappingH(U)for allU∈SU(N). In particular, the block
matrixHj(U)is the restriction ofH(U)onXj( 1 ≤j≤K):
(5.2.40) H(U)|Xj=Hj(U):Xj→Xj.
Hence, when we take any linear transformation onCNas
U:CN→CN, U∈SU(N),then the subspacesXjwill undergo themselves a linear transformation in the fashion as given
by (5.2.40).
Thus, from the mathematical viewpoint, the Physical Explanation5.5is sound, and pro-
vides the mathematical foundation for the quark model.
5.2.4 Computations for irreducible representations
We have understood the physical implications of the irreducible representations ofSU(N).
The remaining crucial problem is how to compute the irreducible representations. Namely,
for anN-dimensional representation as (5.2.21), we need to determine its irreducible decom-
position (5.2.24). In other words, we need to determinem 1 ,···,mKandKin
(5.2.41) N︸⊗ ··· ⊗︷︷ N︸
k 1⊗N︸⊗ ··· ⊗︷︷ N︸
k 2=m 1 ⊕ ··· ⊕mK.A very effective method to compute (5.2.41) is the Young tableaux, which uses square
diagrams to deduce these numbersmj( 1 ≤j≤K). The method is divided in two steps.
The first step is to obtain the rule to group together square diagrams, and to obtain the
irreducible representation (5.2.41) in the form
(5.2.42) N︸⊗ ··· ⊗︷︷ N︸
k 1⊗N︸⊗ ··· ⊗︷︷ N︸
k 2