I refer to page 14 of Fulton and Harris' Representation Theory. I'm having some trouble understanding what they mean to say that this example gives another approach to the 'basic problem', which I assume to mean classifying the irreducible representations of $S_3$. Note that at this point in the book no theorems about characters have been proven (orthogonality relations for instance, or the fact that the number of irreps is bounded by the number of conjugacy classes). When we decompose $W$ into $U,U'$ and $V$ as in the book, we are already assuming that those are the only irreps available. So my question is what has this example proved?