If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of $\mathfrak{sp}_{2n}\mathbb{C}$?
For example, does $\wedge^kV$ decompose into a direct sum of $\wedge^jW$'s, and if so, why?
As an aside: I would think that it should if there is an induced homomorphism between representation rings $R(\mathfrak{sl}_{2n}\mathbb{C})\to R(\mathfrak{sp}_{2n}\mathbb{C})$ by the natural inclusion $\mathfrak{sp}_{2n}\mathbb{C}\subset \mathfrak{sl}_{2n}\mathbb{C}$, but unfortunately I don't see how $\wedge^kV$ decomposes so that I can write down the map between polynomial generators.
Even a reference would be very helpful here! Thanks in advamce.