Let $\pi$ be an irreducible representation of $\mathfrak{sl}(3,\mathbb{C})$ and let $\pi^*$ be the dual representation of $\pi$ defined by $\pi^*(X) = - \pi(X)^T$, where $T$ stands for transpose. Show that if $\pi$ has highest weight $(m_1,m_2)$ then $\pi^*$ has highest weight $(m_2,m_1)$.
As a hint I am asked to establish this first for $(1,0)$ and $(0,1)$.
From reading the text I am a bit puzzled about to show this. By definition 5.8 on page 132: Let $\alpha_1= (2,-1) , \alpha_2 = (-1,2)$ , let $\mu_1 ,\mu_2$ be two weights; $\mu_1$ is higher than $\mu_2$ if $\mu_1-\mu_2 = a\alpha_1+b\alpha_2$ for $a,b \ge 0$; a weight is said to be a highest weight if it's higher than all weights $\mu$ of $\pi$ where $\pi$ is a representation of $\mathfrak{sl}(3,\mathbb{C})$.
Iv'e seen before that the weights of $\pi^*$ are the negatives of weights of $\pi$, how to use this here?
Thanks in advance.