I am confused about something.
Group theory books written for physicists say that any reducible representation can be decomposed in terms of irreducible representations (so correct me if I am wrong, to me irreducible representations are like the unit vectors i j k in terms of which any 3D vector can be expanded, or they are like sines and cosines in terms of which any periodic function can be Fourier expanded.)
Now at the same time they say that any bigger representation of a group can be built out of irreducible ones.
What is unclear to me is the physical motivation for each direction. Of course those books contain physical applications but the big picture is never obvious (they get to applications after 200-300 pages of abstract details).
If someone could answer the following questions I would be really appreciated:
1-what is the physical motivation to write a representation in terms of irreducible ones?
2-what is the physical motivation to build bigger representations using irreducible ones?