The set of states with the same total angular momentum and the angular momentum operators which act on them are often represented by vectors and matrices. For example the different states for will be represented by a 3 component vector and the angular momentum operators represented by 3X3 matrices. There are both practical and theoretical reasons why this set of states is separated from the states with different total angular momentum quantum numbers. The states are often (nearly) degenerate and therefore should be treated as a group for practical reasons. Also, a rotation of the coordinate axes will not change the total angular momentum quantum number so the rotation operator works within this group of states.
We write our 3 component vectors as follows.
The rotation operators (symmetry operators) are given by
Jim Branson 2013-04-22