I am currently in my second undergraduate quantum course and just finished studying the addition of angular momenta. I am also in my third abstract algebra course and am now covering product groups and group actions. In my QM book (griffiths) there was a reference made to group theory. it said " what we are talking about is the decomposition of the direct product of two irreducible representations of the rotation group into a direct sum of irreducible representations" I am familiar with direct products and sums and representations and read a bit about the rotation group on wikipedia, but am still not really understanding what groups we are talking about. for example, for a spin 3 particle and a spin 3/2 particle, we would have 3X3/2=3/2+6/2+9/2 the left hand side representing the spins of the particles and the right side representing the total spin of the system. Now from my understanding, direct products and sums are done on groups, not numbers. So I am guessing that each number here represents a subgroup of the group of rotations SO3. Would this be somewhat accurate? If yes then what subgroups are they, and if no then what groups are they? or am i completely mistaken as to what this notation means? Thanks!