Group representation theory MOC

Abelian representation

An abelian representation is a representation whose range is abelian, rep i.e. for all

Main theorem

A representation is abelian iff it is the direct sum of 1-dimensional irrep. rep

Proof

If then it is immediately abelian since there exists a reälization in which matrices are simultaneously diagonal, and hence commute. Conversely, Let be the decomposition of into irreducible invariant subspaces. Since is abelian in each of these subspaces, the irrep carried thereby they must be 1-dimensional. Hence is the direct sum of 1-dimensional irreps.

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