A general definition of spherical harmonic super basis of a -multiplet problem can be expressed as 
where the parentheses denote a 3-j symbol; K is the rank which ranges ; Q is the
projection index of rank K which ranges from −K to +K.  A cubic harmonic super basis where all the tensor operators are hermitian can be defined as 
Then, any quantum operator  defined in the -multiplet Hilbert space can be expanded as
where the expansion coefficients can be obtained by taking the trace inner product, e.g. .
Apparently, one can make linear combination of these operators to form a new super basis that have different symmetries.