I am working through the exercises in Fulton and Harris's Representation Theory, and am stuck on two on page 189.
Let $\text{Sym}^2V$ denote the second symmetric power of the standard 3-dimensional representation $V$ of $\mathfrak{sl}_3\mathbb{C}$ and $\Gamma_{a,b}$ be an irreducible, finite-dimensional representation of $\mathfrak{sl}_3\mathbb{C}$ with highest weight $aL_1-bL_3$.
Show that the representation $\wedge^2(\text{Sym}^2V)$ is isomorphic to $\Gamma_{2,1}$.
Determine the irreducible factors of the representation $\wedge^3(\text{Sym}^2V)$.
It seems that both questions can be done by drawing the weight diagrams, but alas I do not think I have developed the intuition for what irreducible representations I should try, or how to proceed further. Would anybody have any pointers? Thank you in advance.