Crystal of the defining symplectic representation ($B({\omega }_1)$)

For the symplectic Lie algebra ${\mathfrak{sp}}_{2n}$, the defining crystal basis is the chain $1,2,\dots ,n,\bar{n},\dots ,\bar{1}$, with weights ${\epsilon }_1,\dots ,{\epsilon }_n,-{\epsilon }_n,\dots ,-{\epsilon }_1$. Its successive lowering-edge colors are $1,2,\dots ,n-1,n,n-1,\dots ,1$. 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 $n\ge 2$, its only highest vertices are $1\otimes 1$, $1\otimes 2$, and $1\otimes \bar{1}$, 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.

Under the crystal tensor-product rule that uses the first factor for lowering when ${\phi }_i(a)>{\epsilon }_i(b)$, the four-letter C2 root system crystal has highest tensor vertices $1\otimes 1$, $1\otimes 2$ and $1\otimes \bar{1}$. Their highest weights are $2{\omega }_1$, ${\omega }_2$ and zero. Lowering generates connected components of sizes ten, five and one, respectively; their union exhausts all sixteen vertices. The corresponding representation decomposition is the symmetric square plus the primitive exterior square and the invariant symplectic line.