Group representation theory MOC

Generalized projection operator of a representation

Given a (unitary) representation of a compact group , the generalized projection operators¹ are given by rep

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 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 for each , then the generalized projection operator sends to and all other basis vectors to , that is

As a notational mnemonic one can imagine . We may then define projection operators,

the former onto the subspace spanned by , the latter being onto the subspace transforming under irrep .

If for any , then with fixed transform in .

Properties

• For given and fixed , either vanish for all or they transform under in the irrep carried by an invariant subspace for some
• , assuming is completely reducible.

---

tidy | en | sembr

Footnotes

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