Let $\mathfrak{sp}(2d, \mathbb C)$ be a symplectic Lie algebra over $\mathbb C$. Let $V$ be the standard representation of $\mathfrak{sp}(2d, \mathbb C).$
I'm looking for a reference in which exterior powers of standard representation
$\Lambda^k V$ are decomposed into a sum of irreducible representations of $\mathfrak{sp}(2d, \mathbb C)$.
Best Answer
Let $\varpi_1$, $\ldots$, $\varpi_d$ be the fundamental highest weights (with the usual labeling, $\varpi_d$ corresponding to a simple long root).
One can show that the contraction map $\varphi: \wedge^k(V) \rightarrow \wedge^{k-2}(V)$ is surjective and that $\operatorname{Ker} \varphi$ is irreducible of highest weight $\varpi_k$. It follows that the composition factors of $\wedge^k(V)$ have highest weights $\varpi_{k-2i}$ for $0 \leq i \leq k/2$. (Here $\varpi_0 = 0$).
For details, see for example ยง17.2 in the representation theory book by Fulton and Harris.