Hello! I am reading some representation theory (the book is Lie Algebra in Particle Physics, by Georgi, part 1.17) and the author solves a problem of 3 bodies connected by springs forming a triangle, aiming to find the normal modes. He builds a 6 dimensional vector formed of the 3 particles and their x and y coordinates and a 6 dimensional representation of S3, as the symmetry of the system. Firstly, can someone explain to me why does the system has a 6D S3 symmetry. I understand you can permute the 3 balls, but I am not sure I understand why changing the x direction motion of one particle with the y direction of another one, is still a symmetry of the system (while keeping everything else the same). Then, he finds the projector operators onto the subspaces on which the irreducible representations act on. The 1D representations appear only once, so finding the normal modes associated to them are easy to find. However the 2D representation appears 2 times and he tries to find the normal modes associated to that. He calculates the projector - P2 - associated to this (P2a.png) and he takes the x and y translations as already known modes and subtract them from P2 and then in order to find the other 2 modes he does the stuff in P2b.png. Can someone explain that to me, please? Why does he pick that vector and why does he rotates it by $2\pi/3$. I assume it is something trivial, but I can't seem to figure it out. Thank you!