Tensor-square decomposition of the defining symplectic representation ($V\otimes V=L(2{\omega }_1)\oplus L({\omega }_2)\oplus L(0)$)

For $n\ge 2$, the defining Lie algebra representation $V$ of ${\mathfrak{sp}}_{2n}$ has the displayed tensor product decomposition. The crystal of the defining symplectic representation and the crystal tensor-product rule give exactly these three highest vertices, each occurring once. The symmetric square is $L(2{\omega }_1)$ of dimension $n(2n+1)$. The exterior square splits into its invariant inverse-form line and the primitive exterior square $L({\omega }_2)$ of dimension $n(2n-1)-1$. Their dimensions sum to $4n^2$. At $n=1$, the exterior square is only the invariant line, giving $L(2{\omega }_1)\oplus L(0)$.