7 SL₃

We now discuss the finite-dimensional complex-linear representation theory of $𝔰{𝔩}_{3,ℂ}$. This then tells us everything about the holomorphic representations of ${\mathrm{SL}}_3(ℂ)$ and, via the machinery of real forms, tells us about representations of $\mathrm{SU}(3)$ and ${\mathrm{SL}}_3(ℝ)$.

Much of what we say, particularly at first, will generalise in a fairly obvious way to $𝔰{𝔩}_{n,ℂ}$.

In this section, all representations of $𝔰{𝔩}_{3,ℂ}$ will be assumed to be complex-linear.