= Crystal of the defining symplectic representation
{title2=$B(\omega_1)$}
For the <symplectic Lie algebra> $\mathfrak{sp}_{2n}$, the defining <crystal basis> is the chain $1,2,\ldots,n,\bar n,\ldots,\bar1$, with <weights> $\varepsilon_1,\ldots,\varepsilon_n,-\varepsilon_n,\ldots,-\varepsilon_1$. Its successive lowering-edge colors are $1,2,\ldots,n-1,n,n-1,\ldots,1$. Every step subtracts its color's <simple root>. Applying the <crystal tensor-product rule> to two copies gives a complete grid description of the tensor-square crystal. For $n\ge2$, its only highest vertices are $1\otimes1$, $1\otimes2$, and $1\otimes\bar1$, of <highest weights> $2\omega_1$, $\omega_2$, and zero. For $n=1$, the middle vertex is absent and only the other two highest vertices remain.
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