[[Group ring]]
# Idempotent of the complex group ring

An [[Idempotent]] $e_{\mu} \in \mathbb{C}[G]$ of the complex group ring is an element satisfying $(e_{\mu})^2 = e_{\mu}$.
Iff $(e_{\mu})^2 = z_{\mu} e_{\mu}$ for some $z_{\mu} \in \mathbb{C}$ then $e_{\mu}$ is called **essentially idempotent**. #m/def/rep
Idempotents of the group ring generate [[Ideal of the complex group ring|left ideals]] by right convolution,
and each left ideal is generated by some idempotent. #m/thm/rep 

> [!check]- Proof
> Let $e_{\mu} \in \mathbb{C}[G]$ be an idempotent. 
> Then $q * r * e_{\mu} * e_{\mu} = q * r * e_{\mu}$ for any $q, r \in \mathbb{C}[G]$,
> so by associativity $\Rho_{\mathbb{C}[G]}(e_{\mu}) \Lambda_{\mathbb{C}[G]}(q) \Rho_{\mathbb{C}[G]}(e_{\mu}) =  \Lambda_{\mathbb{C}[G]}(q) \Rho_{\mathbb{C}[G]}(e_{\mu})$.
> Thus $\Rho_{\mathbb{C}[G]}(e_{\mu}) \mathbb{C}[G]$ is a left-ideäl with projection operator $P^\mu = \Rho_{\mathbb{C}[G]}(e_{\mu})$.
> 
> Let $L$ be a [[Ideal of the complex group ring|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 $q_{2} \in L^\perp$.
> In particular, the identity $e = e_{1} + e_{2}$, so
> $$
> \begin{align*}
> q*e = \underbrace{ q*e_{1} }_{ \in L } + \underbrace{ q*e_{2} }_{ \in L^\perp }
> \end{align*}
> $$
> so $\Rho_{\mathbb{C}[G]}(e_{1})$ projects onto $L$.
> Clearly $e_{1}$ is idempotent since $e_{1}*e_{1} = \Rho_{\mathbb{C}[G]}(e_{1})e_{1} = e_{1}$.
> <span class="QED"/>

Thus $\Rho_{\mathbb{C}[G]}(e_{\mu})$ is a [[Generalized projection operator of a representation|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

> [!check]- 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 $e_{1}*e_{2} = 0 = e_{2}e_{1}$.
> <span class="QED"/>

## 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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