What extra observable(s) are needed to label the basis states of a representation of the rotation group in $N$-dimensional space?

Question

In $3$-dimensional space, any given irreducible representation of the rotation group has a basis whose states are uniquely labeled by the eigenvalues $m$ of a single observable $J_z$, which is one of the components of the angular momentum. The notation $|j,m\rangle$ is often used, where $j$ specifies the irreducible representation. The other two components $J_x$ and $J_y$ don't commute with $J_z$.

In $N$-dimensional space, with $N=2k$ or $N=2k+1$, angular momentum has $\binom{N}{2}$ linearly independent components. We can choose up to $k$ components that commute with each other. But when $N> 3$, specifying the eigenvalues of these $k$ commuting observables often does not uniquely specify a single state within a given irreducible representation.$^\dagger$ Therefore, to uniquely label the states in a basis, we need at least one additional observable that commutes with the $k$ commuting generators. A Casimir won't work, because Casimirs are invariant under rotations: they can't distinguish between states within an irreducible representation.

Question: What are these extra observable(s) that we need to uniquely label the basis states in an irreducible representation of $SO(N)$ when $N>3$?

Example: For $N=5$, let $J_{jk}$ denote the generator of rotations in the $j$-$k$ plane. Then $J_{12}$ and $J_{34}$ commute with each other, but we need at least one more observable that commutes with these. The combination $J_{15}^2+J_{25}^2+J_{35}^2+J_{45}^2$ is a candidate: it commutes with $J_{12}$ and $J_{34}$, and it's not invariant under rotations. But is this the only extra observable we need? What's the general pattern for arbitrary $N$?

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$^\dagger$ In the language of Lie-algebra representation theory, this is because an irreducible representations can have some weights with multiplicity $>1$. For examples of irreducible representations of $SO(5)$ having weights with multiplicity $>1$, see https://arxiv.org/abs/1511.02015.

Answer 1

Looks like you already specified your CSCOs, unless I am missing something. For even N, your k commuting su(2)~so(3) $j_z$s will completely characterize your N-vector by their k $j_z$s; while for odd N, you may extend your construction, which is manifestly adequate for complete characterization of any 5-vector: $$J_{12} \leadsto J_{12}^2; ~~ J_{34} \leadsto J_{34}^2; ~~J_{15}^2+J_{25}^2+J_{35}^2+J_{45}^2;$$ it should then work for any tensoring of such.

For generic N=2k+1, then, the manifestly commuting set $$ J_{12} ; J_{34}; ...; J_{2k-1,2k}; ~~J_{1, 2k+1}^2+J_{2, 2k+1}^2+ ...+ J_{2k,2k+1}^2 $$ appears adequate, no?

Answer 2

The hidden $SO(4)$ symmetry of the $1/r$ central potential is a very instructive example. I'll attempt to construct an answer to your question motivated by this example:

The conservation of the Runge-Lenz vector $\mathbf{M}$ around a $1/r$ potential leads to a hidden $SO(4)$ symmetry (see Section 4.1 of Sakurai's Modern Quantum Mechanics for an excellent discussion). Re-scaling to give it units of angular momentum, $\mathbf{M} \rightarrow \mathbf{N}$ you can define new operators from $\mathbf{N}$ and the $SO(3)$ physical angular momentum $\mathbf{L}$: $$\mathbf{I} = (\mathbf{L} + \mathbf{M})/2, $$ $$\mathbf{K} = (\mathbf{L} - \mathbf{M})/2, $$ which satisfy independent commutation relations $$ [I_i, I_j] = i \hbar \epsilon_{ijk} I_k,$$ $$ [K_i, K_j] = i \hbar \epsilon_{ijk} K_k,$$ $$ [I_i, K_j] = 0.$$ From this it might appear that there is a redundancy, and a state might be labelled with the eigenvalues of, say, $I_1$, $K_1$, $\mathbf{I}^2$, and $\mathbf{K}^2$, but it is easy to check (see Sakurai) that $\mathbf{I}^2 - \mathbf{K}^2 = \mathbf{L} \cdot \mathbf{N} = 0$, which enforces that the eigenvalues of $\mathbf{I}^2$ and $\mathbf{K}^2$ must be equal. Therefore, you have yourself a full set of 3 physical observables to label your quantum state $\lvert m_I, m_K, i\rangle$ or $\lvert m_I, m_K, k\rangle$ .

For a general group $SO(N)$, I posit without proof that it should be sufficient to replicate the above process by decomposing the space into independent (commuting) subspaces of the "angular momentum", and to identify relationships between the eigenvalues of these operators to eliminate redundancies.