Group ring

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. rep Idempotents of the group ring generate left ideals by right convolution, and each left ideal is generated by some idempotent. 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. rep Non-primitive idempotents can be written as the sum of two non-zero idempotents such that . 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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