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What does this mean:

"The standard Cartan weight operators for SO(4) are $L_{12}$ and $L_{34}$. An SO(4) irrep is labeled by the highest weight defined by these operators, which is of the form $\left[pq\right]$, with $p\geq|q|$, both integer or both odd half integer."

I'm confused about what a cartan weight operator is, and what the "highest weight" of an operator would be.

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I posted some explanation of weights and highest weights here. Unfortunately the answer is fairly involved. –  Bill Cook Jan 24 '12 at 20:27
    
Also, if you want an answer which specifically addresses your example in SO(4), you'll need to explain the notation "$L_{12}$" and "$L_{34}$" and "$[pq]$". –  Bill Cook Jan 24 '12 at 20:28
    
@Bill Cook: the definitions of the operators are as follows: $L_{rs}\equiv-i(x_r\partial_s-x_s\partial_r)$ angular momentum operators, and as far as I can tell the "$[pq]$" is a 2-tuple that labels the irreps of SO(4). –  okj Jan 24 '12 at 21:11

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