Subduction and Induction

This also implies that the diagonal operator Pkk is idempotent, i.e., applying it twice gives exactly the same result as applying it once  

Finally, summing over all diagonal projectors gives rise to the unit element  

we have made use of the fact that the sum over all characters multiplied by the dimension of the irrep is the character of the regular representation, and this vanishes for all R except for the unit element, where it is equal to G (see Eq. (4.41)). In Dirac terminology this reads 

This relation is also known as the closure relation. It is frequently applied in the context of the crystal field theory of the lanthanides. 

In some cases a subgroup can be reached via two different symmetry breakings. An example is the subduction Oh i Dzh Here, two pathways for symmetry breaking 

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The opposite process to subduction is induction. Here, we start from an irrep in a subgroup H. By coset expansion, this subgroup is put on an orbit inside a higher symmetry group. This leads to an extension of the function space and generates... 

C. Definitions of subduction, induction, and regular induction are important, as is the main theorem on induction embodied in Eq. (34), but not its proof. 

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