= Tensor-square decomposition of the defining symplectic representation
{title2=$V\otimes V=L(2\omega_1)\oplus L(\omega_2)\oplus L(0)$}
For $n\ge2$, 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 $L(2\omega_1)\oplus L(0)$.
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