I will start with the Hamiltonian of a isotropic two dimensional Harmonic Oscillator. The Hamiltonian is (N+1) fold degenerate for each value of N. Then I will use the Hermitian bilinear form aiaj+ which raises/lowers n1 or n2 keeping the energy constant. These operator will be s1 and s2.
I will compute the complete symmetry algebra by finding all possible commutators of s1, s2 and H. For two dimensions they will be Pauli Matrices.
We know that, any linear combination of the eigenstates of a 2D isotropic Harmonic oscillator is generated by a Unitary matrix with determinant 1. These unitary matrices commute with the Hamiltonian. These matrix form a group closed under commutation and are called SU(2) group.
I will follow the same procedure for an isotropic 3D Harmonic oscillator. The complete symmetry group will be the SU(3) group. I will find a complete set of commuting operator from the group of operators that leaves the Hamiltonian unchanged.
By looking at the potential of the harmonic oscillator, it can be seen that the Hamiltonian is conserved under rotation and parity operator. Also the Hamiltonian is symmetric under the action of U(3), the set of all possible 3X3 unitary matrices.
As before I will find the all possible commutators of the bilinear Hermitian combination of the shift operators. They will probably be related to the Gell-Mann matrices.
I will try to find all the operations that keeps the Hamiltonian unchanged. Then I will take the Maximal Set of Commuting Operators and diagonalize them. Then I will block diagonalize the Hamiltonian in this basis.
The block diagonalized Hamiltonian can be expanded as the direct sum of many smaller blocks.
The argument will be the same if we consider two or more non-interacting particles having same Hamiltonian. The combined Hilbert space will be the direct product of the two Hilbert space of individual particles.
We will find a complete set of commutating operators here also, and I will diagonalize the maximum possible set. Then using the eigenstates I will block diagonalize the Hamiltonian, and expand it as a direct sum of many blocks.
The above method is the Clebsch-Gordan decomposition.
The elements of transformation matrix that converts the direct product space to the direct sum space are the Clebsch-Gordan coefficient for SU(3).
As a example I will say that the combined Hilbert space of a Quark and an anti-Quark can be reduced into the direct sum of a singlet and an octet state.
I will also show the decomposition of product of SU(3) group representation using Young Tableaux.
