Dual representations of fundamental representations of a Lie algebra.

Question

Let $g$ be a Lie algebra. Let $V(\omega_i)$, $i=1,\ldots,n$, be the fundamental representations. Are the dual representations $V(\omega_i)^*$ highest weight representations?

The dual representation $(\pi^*, V^*)$ of $(\pi, V)$ is defined as $\pi^*(X) = - \pi(X)^T$, $X \in sl_n$.

If yes, how to compute the $\lambda$ in $V(\omega_i)^* = V(\lambda)$? Thank you very much.

Answer 1

Let $V\cong \mathbb{C}^n\cong V(\omega_1)$ be the natural representation of $\frak{sl}_n$. $V(\omega_k)\cong \wedge_{i=1}^{k}V$. Taking dual will make all the weight space into its negative weight space. Hence negative of the lowest weight of $\wedge_{i=1}^{k}V$ is the highest weight of $V(\omega_k)^*$. Hence, $\omega_{n-k}$ is the higest weight of $V(\omega_k)^*$.