Idempotent of the complex group ring

An Idempotent of the complex group ring is an element satisfying . Iff for some then is called essentially idempotent. #m/def/rep Idempotents of the group ring generate left ideals by right convolution, and each left ideal is generated by some idempotent. #m/thm/rep

Proof

Let be an idempotent. Then for any , so by associativity . Thus is a left-ideäl with projection operator .

Let be a left ideal and . The orthogonal complement of an invariant subspace under a unitary operator is invariant, so we may decompose with and . In particular, the identity , so

so projects onto . Clearly is idempotent since .

Thus is a projection operator onto some left ideal. Those idempotents that generate minimal left ideäls are called primitive idempotents. #m/def/rep Non-primitive idempotents can be written as the sum of two non-zero idempotents such that . #m/thm/rep

Proof

Let be the non-minimal ideäl generated by . Then for ideäls generated by respectively. Clearly , and .

Properties

Idempotent primitivity criterion
Equivalence of irreps on left ideals criterion
Irreducible character as function of an idempotent
A set of primitive idempotents generating every irrep adds to identity.

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