Chapter 4 Representations of $\mathfrak{sl}_{3,\mathbb{C}}$

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

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

In this section, all representations of $\mathfrak{sl}_{3,\mathbb{C}}$ will be assumed to be complex-linear.