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(Georgi chapter 14) Why does the n=2$n=2$ states transform under the 5+1 of the angular momentum?

In Georgi's Lie algebra in particle physics, chapter 14, the 3D harmonic oscillator is studied. The systems exhibits SU(3)$SU(3)$ symmetry in the energy levels, we can construct the 8 generators of the SU(3)$SU(3)$ using the creation/annihilation operators. We also notice that a subgroup of the 8 generators generate angular momentum (so the 3 generators of SO(3)$SO(3)$ symmetry).

But then Georgi take an example: if we consider the 6 states that have n=2 $n=2$ (that lie in the (2,0)=6 rep of SU(3)$SU(3)$), they transform like 5+1 under angular momentum. I'm not sure that I understand what this means. Does it mean that we can find some linear combinations of the 6 states that is invariant under the the action of the 3 generators of SO(3) $SO(3)$ (giving the singlet state), and that we can find 5 states that the just transform in each other under the action of the SO(3) generators $SO(3)$ generators? This may be a dumb question, but I'm a bit stuck on this.

Thank you for your help !

(Georgi chapter 14) Why does the n=2 states transform under the 5+1 of the angular momentum?

In Georgi's Lie algebra in particle physics, chapter 14, the 3D harmonic oscillator is studied. The systems exhibits SU(3) symmetry in the energy levels, we can construct the 8 generators of the SU(3) using the creation/annihilation operators. We also notice that a subgroup of the 8 generators generate angular momentum (so the 3 generators of SO(3) symmetry).

But then Georgi take an example: if we consider the 6 states that have n=2 (that lie in the (2,0)=6 rep of SU(3)), they transform like 5+1 under angular momentum. I'm not sure that I understand what this means. Does it mean that we can find some linear combinations of the 6 states that is invariant under the the action of the 3 generators of SO(3) (giving the singlet state), and that we can find 5 states that the just transform in each other under the action of the SO(3) generators ? This may be a dumb question, but I'm a bit stuck on this.

Thank you for your help !

(Georgi chapter 14) Why does the $n=2$ states transform under the 5+1 of the angular momentum?

In Georgi's Lie algebra in particle physics, chapter 14, the 3D harmonic oscillator is studied. The systems exhibits $SU(3)$ symmetry in the energy levels, we can construct the 8 generators of the $SU(3)$ using the creation/annihilation operators. We also notice that a subgroup of the 8 generators generate angular momentum (so the 3 generators of $SO(3)$ symmetry).

But then Georgi take an example: if we consider the 6 states that have $n=2$ (that lie in the (2,0)=6 rep of $SU(3)$), they transform like 5+1 under angular momentum. I'm not sure that I understand what this means. Does it mean that we can find some linear combinations of the 6 states that is invariant under the the action of the 3 generators of $SO(3)$ (giving the singlet state), and that we can find 5 states that the just transform in each other under the action of the $SO(3)$ generators? This may be a dumb question, but I'm a bit stuck on this.