# Projector operator in Representation theory

I am reading some introductory stuff on Representation theory applied to physics and I am a bit confused about some things. The book I use is Lie Algebra in Particle Physics by Georgi (you can find it here) and the part where I got confused is part 1.16 where he finds the normal modes for 3 particles connected by springs, forming a triangle. So I want to understand how all this works.

As far as I understood, you find the symmetry of the system and choose the proper dimensions for the representation (in this case 6D and S3). Then you get the characters of the matrices in this representation and calculate the projectors onto the space of irreducible representations. Now I am a bit lost. How do one gets the normal modes (or in general the eigenvectors associated to a certain irreducible representation, from the projector operators). For the projection operators associated to 1D representation I think I get it, as they are just the product between the column and row vector, that span the space of that representation. But for projector onto higher dimensional representations I am quite lost.

Moreover, I am not sure I understand what do I need to do when the same representation appears multiple times. Why do we have just a projection operator for both? The 6D vectors (in this case) forming the space on which each of them acts is different. So I would really appreciate if someone can give me a more clear explanation of all this. Thank you!

Now to get eigenvectors you need to choose operators to diagonalize. If the irrep is 1-d that's automatic since all operators act by multiplication. For the $2$-dimension irrep of $S_3$ you will need to select a subset of mutually commuting operators to get eigenvectors since $S_3$ is not abelian. The commuting operators are usually taken as the identity and $P_{12}$: they have distinct spectrum so getting the common eigenvectors is enough. (This part is extremely not clear from the discussion in Georgi; I would have to read the whole thing to get a handle on the notation.)