Idempotent of the complex group ring

An Idempotent $𝑒 𝜇 \in ℂ [𝐺]$ of the complex group ring is an element satisfying $(𝑒 𝜇) 2 = 𝑒 𝜇$ . Iff $(𝑒 𝜇) 2 = 𝑧 𝜇 𝑒 𝜇$ for some $𝑧 𝜇 \in ℂ$ then $𝑒 𝜇$ 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 $𝑒 𝜇 \in ℂ [𝐺]$ be an idempotent. Then $𝑞 * 𝑟 * 𝑒 𝜇 * 𝑒 𝜇 = 𝑞 * 𝑟 * 𝑒 𝜇$ for any $𝑞, 𝑟 \in ℂ [𝐺]$ , so by associativity $P ℂ [𝐺] (𝑒 𝜇) Λ ℂ [𝐺] (𝑞) P ℂ [𝐺] (𝑒 𝜇) = Λ ℂ [𝐺] (𝑞) P ℂ [𝐺] (𝑒 𝜇)$ . Thus $P ℂ [𝐺] (𝑒 𝜇) ℂ [𝐺]$ is a left-ideäl with projection operator $𝑃 𝜇 = P ℂ [𝐺] (𝑒 𝜇)$ .

Let $𝐿$ be a left ideal and $𝑞 \in 𝐿$ . The orthogonal complement of an invariant subspace under a unitary operator is invariant, so we may decompose $𝑞 = 𝑞 1 + 𝑞 2$ with $𝑞 1 \in 𝐿$ and $𝑞 2 \in 𝐿 ⟂$ . In particular, the identity $𝑒 = 𝑒 1 + 𝑒 2$ , so

𝑞 * 𝑒 = 𝑞 * 𝑒 1 ⏟ \in 𝐿 + 𝑞 * 𝑒 2 ⏟ \in 𝐿 ⟂

so $P ℂ [𝐺] (𝑒 1)$ projects onto $𝐿$ . Clearly $𝑒 1$ is idempotent since $𝑒 1 * 𝑒 1 = P ℂ [𝐺] (𝑒 1) 𝑒 1 = 𝑒 1$ .

Thus $P ℂ [𝐺] (𝑒 𝜇)$ 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 $𝑒 1 + 𝑒 2$ such that $𝑒 1 * 𝑒 2 = 0 = 𝑒 2 * 𝑒 1$ . #m/thm/rep

Proof

Let $𝐿$ be the non-minimal ideäl generated by $𝑒$ . Then $𝐿 = 𝐿 1 \oplus 𝐿 2$ for ideäls $𝐿 1, 𝐿 2$ generated by $𝑒 1, 𝑒 2$ respectively. Clearly $𝑒 = 𝑒 1 + 𝑒 2$ , and $𝑒 1 * 𝑒 2 = 0 = 𝑒 2 𝑒 1$ .

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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