In the chapter IV.1 "Reducible or Irreducible?" of Zee's Group Theory book (p. 188-), the author breaks a 2nd rank tensor $T^{ij}$ into invariant subspaces with respect to the action of $\mathrm{SO(3)}$ group. The tensor $T^{ij}$ breaks into a five-dimensional (symmetric traceless), three-dimensional (antisymmetric) and one-dimensional invariant subspaces. The author then claims that each of the subspaces corresponds to an irreducible representation and proceeds to use this as a fact in the further discussions. I do not see a reason to be making claims about irreducibility of the five- and three-dimensional representations because it is not clear that they do not have their own non-trivial invariant subspaces. I wonder if someone could help me here. Why are the three- and five-dimensional representations are irreducible?
Asked
Active
Viewed 148 times
