$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Sym{Sym}$Let $V = \mathbb{R}^{2n}$ be the standard representation of the symplectic group $\Sp(2n,\mathbb{R})$, and let $\{a_1,b_1,\dotsc,a_n,b_n\}$ be the standard symplectic basis for $V$. Let $$W \subset \left(\bigwedge\nolimits^2 V\right)^{\otimes 2}$$ be the $\Sp(2n,\mathbb{R})$-subrepresentation spanned by the elements $$\{\text{$(a_i \wedge b_i) \otimes (a_j \wedge b_j)$ $|$ $1 \leq i,j \leq n$}\}.$$ Is there a nice description of $W$?
EDIT: Here is one small observation that breaks this up into two problems. The tensor square is the direct sum of the alternating and symmetric tensors, so $W$ is the direct sum of its projections to these two factors. In other words, what we need to do is determine the $\Sp(2n,\mathbb{R})$-subrepresentation spanned by
$$\{(a_i \wedge b_i) \wedge (a_j \wedge b_j) \mid 1 \leq i < j \leq n\}$$ in $\bigwedge\nolimits^2(\bigwedge\nolimits^2 V))$ and by $$\{(a_i \wedge b_i) \cdot (a_j \wedge b_j) \mid 1 \leq i,j \leq n\}$$ in $\Sym^2(\bigwedge\nolimits^2 V))$. We've now reached the limits of my knowledge of representation theory!
More generally, many variants of this question have been arising in my research recently and I'm sure there is some way to answer them with a computer (at least for specific values of $n$). How can I do this?
By the way, I am not a representation theorist, so I apologize if this question is too easy.
