 Tensor square of the defining C2 crystal

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.