Recently I read some comment on Sakurai's book (which I have not read) that the writer of said comment didn't understand part of the text until they understood irreducible representations. I do not know to what they were referring, but it piqued my interest in representation theory. My question is: will an understanding of representations allow me to solve problems faster/more efficiently and/or reveal experimentally relevant aspects of quantum theory that are not accessible without an understanding of representations? As far as background, I have a good grasp on generators and Lie algebra stuff as pertains to quantum, and a solid understanding of the basics like Hilbert space and linear and what'not. My background is weaker with finite groups and product states. The cliff-notes version of irreducible representations seems clear enough, that if your matrix can be diagonalized into blocks then your system has structure you can exploit and factor into smaller chunks. It seems even a little self-evident, from a purely algebraic point of view. Does knowing about representation-y stuff let you decompose your system with less time and effort? Can someone supply an example from an applied problem? I would really appreciate it!