Any mathematician, stranded on a desert island with only these ideas and the definition of a particular Lie algebra $\mathfrak{g}$ such as $\mathfrak{sl}_n \mathbb{C}$, $\mathfrak{so}_n \mathbb{C}$, or $\mathfrak{sp}_n \mathbb{C}$, would in short order have a complete description of all the objects defined above in the case of $\mathfrak{g}$. We should say as well, however, that at the conclusion of this procedure we are left without one vital piece of information about the representations of $\mathfrak{g}$ ... this is, of course, a description of the multiplicities of the basic representations $\Gamma_a$. As we said, we will, in fact, describe and prove such a formula (the Weyl character formula); but it is of a much les straightforward character (our hypothetical shipwrecked mathematician would have to have what could only be described as a pretty good day to come up the idea) and will be left until later.