Group representation theory MOC

Generalized projection operator of a representation

Given a (unitary) representation of a compact group π‘ˆ :𝐺 β†’GL(𝑉), the generalized projection operators1 π‘ƒπœ‡π‘—π‘˜ are given by rep

π‘ƒπœ‡π‘—π‘˜=π‘‘πœ‡βˆ«πΊ[Ξ“πœ‡(𝑔)βˆ’1]π‘—π‘˜π‘ˆ(𝑔)π‘‘πœ‡(𝑔)=π‘‘πœ‡βˆ«πΊβ€•β€•β€•β€•Ξ“πœ‡π‘˜π‘—(𝑔)π‘ˆ(𝑔)π‘‘πœ‡(𝑔)π‘ˆ(𝑔)=βˆ‘πœ‡;𝑗,π‘˜Ξ“πœ‡π‘˜π‘—(𝑔)π‘ƒπœ‡π‘—π‘˜

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 π‘‘πœ‡ is the normalized Haar measure.

complete

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

Explanation

Considering Irreducible orthonormal basis π‘’πœ‡π›Όπ‘— for each π‘‰πœ‡π›Ό, then the generalized projection operator π‘ƒπœ‡β„“π‘˜ sends π‘’πœ‡π›Όβ„“ to π‘’πœ‡π›Όπ‘˜ and all other basis vectors to βƒ—πŸŽ, that is

π‘ƒπœ‡β„“π‘˜π‘’πœˆπ›Όπ‘—=π›Ώπœ‡πœˆπ›Ώβ„“π‘—π‘’πœ‡π›Όπ‘˜

As a notational mnemonic one can imagine π‘ƒπœ‡β„“β†’π‘˜. We may then define projection operators,

π‘ƒπœ‡π‘—=π‘ƒπœ‡π‘—π‘—π‘ƒπœ‡=π‘‘πœ‡βˆ‘π‘—=1π‘ƒπœ‡π‘—

the former onto the subspace spanned by π‘’πœ‡π›Όπ‘—, the latter being onto the subspace β¨π›Όπ‘‰πœ‡π›Ό transforming under irrep Ξ“πœ‡.

If π‘ƒπœ‡π‘—π‘˜πœ“ β‰ 0 for any 𝑗,π‘˜, then {π‘ƒπœ‡π‘—π‘˜}π‘‘πœ‡π‘˜=1 with fixed 𝑗 transform in Ξ“πœ‡.

Properties

  • For given πœ“ βˆˆπ‘‰ and fixed πœ‡,𝑗, either π‘ƒπœ‡π‘—π‘˜πœ“ vanish for all 1 β‰€π‘˜ β‰€π‘‘πœ‡ or they transform under π‘ˆ in the irrep Ξ“πœ‡ carried by an invariant subspace π‘‰πœ‡π›Ό for some 𝛼

    π‘ˆ(𝑔)π‘ƒπœ‡π‘—π‘˜=βˆ‘β„“π‘ƒπœ‡π‘—β„“Ξ“πœ‡β„“π‘˜

  • π‘ƒπœˆπ‘—π‘–π‘ƒπœ‡β„“π‘˜ =π›Ώπœ‡πœˆπ›Ώπ‘—π‘˜π‘ƒπœ‡β„“π‘–

  • βˆ‘πœ‡π‘ƒπœ‡ =𝐈, assuming π‘ˆ is completely reducible.

tidy | en | SemBr

Footnotes

  1. 2023, Groups and representations, pp. 50–51. ↩

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