Generalized projection operator of a representation

Given a (unitary) representation of a compact group $𝑈 : 𝐺 \to G L (𝑉)$ , the generalized projection operators¹ $𝑃 𝜇 𝑗 𝑘$ are given by #m/def/rep

𝑃 𝜇 𝑗 𝑘 = 𝑑 𝜇 \int 𝐺 [Γ 𝜇 (𝑔) - 1] 𝑗 𝑘 𝑈 (𝑔) 𝑑 𝜇 (𝑔) = 𝑑 𝜇 \int 𝐺 ―――― Γ 𝜇 𝑘 𝑗 (𝑔) 𝑈 (𝑔) 𝑑 𝜇 (𝑔) 𝑈 (𝑔) = \sum 𝜇; 𝑗, 𝑘 Γ 𝜇 𝑘 𝑗 (𝑔) 𝑃 𝜇 𝑗 𝑘

where the second line is allowed for finite groups since Every finite complex representation of a compact group is equivalent to a unitary representation, and $𝑑 𝜇$ is the normalized Haar measure.

#to/complete

While the definition above is for all compact groups, I haven't fully formulated this yet.

Explanation

Considering Irreducible orthonormal basis $𝑒 𝜇 𝛼 𝑗$ for each $𝑉 𝜇 𝛼$ , then the generalized projection operator $𝑃 𝜇 ℓ 𝑘$ sends $𝑒 𝜇 𝛼 ℓ$ to $𝑒 𝜇 𝛼 𝑘$ and all other basis vectors to $⃗ 𝟎$ , that is

𝑃 𝜇 ℓ 𝑘 𝑒 𝜈 𝛼 𝑗 = 𝛿 𝜇 𝜈 𝛿 ℓ 𝑗 𝑒 𝜇 𝛼 𝑘

As a notational mnemonic one can imagine $𝑃 𝜇 ℓ \to 𝑘$ . We may then define projection operators,

𝑃 𝜇 𝑗 = 𝑃 𝜇 𝑗 𝑗 𝑃 𝜇 = 𝑑 𝜇 \sum 𝑗 = 1 𝑃 𝜇 𝑗

the former onto the subspace spanned by $𝑒 𝜇 𝛼 𝑗$ , the latter being onto the subspace $⨁ 𝛼 𝑉 𝜇 𝛼$ transforming under irrep $Γ 𝜇$ .

If $𝑃 𝜇 𝑗 𝑘 𝜓 \neq 0$ for any $𝑗, 𝑘$ , then ${𝑃 𝜇 𝑗 𝑘} 𝑑 𝜇 𝑘 = 1$ with fixed $𝑗$ transform in $Γ 𝜇$ .

Properties

For given $𝜓 \in 𝑉$ and fixed $𝜇, 𝑗$ , either $𝑃 𝜇 𝑗 𝑘 𝜓$ vanish for all $1 \leq 𝑘 \leq 𝑑 𝜇$ or they transform under $𝑈$ in the irrep $Γ 𝜇$ carried by an invariant subspace $𝑉 𝜇 𝛼$ for some $𝛼$ $𝑈 (𝑔) 𝑃 𝜇 𝑗 𝑘 = \sum ℓ 𝑃 𝜇 𝑗 ℓ Γ 𝜇 ℓ 𝑘$
$𝑃 𝜈 𝑗 𝑖 𝑃 𝜇 ℓ 𝑘 = 𝛿 𝜇 𝜈 𝛿 𝑗 𝑘 𝑃 𝜇 ℓ 𝑖$
$\sum 𝜇 𝑃 𝜇 = 𝐈$ , assuming $𝑈$ is completely reducible.

#state/tidy | #lang/en | #SemBr

2023, Groups and representations, pp. 50–51. ↩

Generalized projection operator of a representation

Explanation

Properties

Footnotes