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Name the four vertices $1,2,\bar{2},\bar{1}$ in their displayed order. This is the crystal of the defining symplectic representation in rank two, of highest weight ${\omega }_1$ for the C2 root system. Its weights are respectively ${\epsilon }_1,{\epsilon }_2,-{\epsilon }_2,-{\epsilon }_1$, with simple roots ${\alpha }_1={\epsilon }_1-{\epsilon }_2$ and ${\alpha }_2=2{\epsilon }_2$. Each lowering arrow subtracts the corresponding simple root. Thus ${\omega }_1={\epsilon }_1$ and ${\omega }_2={\epsilon }_1+{\epsilon }_2$.

Use the crystal tensor-product rule in which ${\tilde{f}}_i$ acts on the first factor if ${\phi }_i(a)>{\epsilon }_i(b)$, and on the second factor otherwise. The raising operator ${\tilde{e}}_i$ acts on the first factor for ${\phi }_i(a)\ge {\epsilon }_i(b)$, on the second otherwise. Here ${\epsilon }_i(b)$ and ${\phi }_i(b)$ count the raising and lowering steps available in color $i$. The data for the four vertices areThe strictness difference between raising and lowering ensures that they are inverse along each colored edge. Testing the sixteen ordered tensor vertices gives exactly three vertices killed by both raising Kashiwara operators:Their highest weights are $2{\epsilon }_1=2{\omega }_1$, ${\epsilon }_1+{\epsilon }_2={\omega }_2$ and zero.

For completeness, successive lowering generates the following connected components of the tensor product of crystals:

They have sizes ten, five and one, are disjoint, and exhaust all sixteen vertices. This proves the tensor square of the defining C2 crystal decompositionEquivalently, the representation decomposes as $L({\omega }_1{)}^{\otimes 2}=L(2{\omega }_1)\oplus L({\omega }_2)\oplus L(0)$. The dimensions agree with the Weyl dimension formula for C2, $\dim L(a{\omega }_1+b{\omega }_2)=(a+1)(b+1)(a+b+2)(a+2b+3)/6$. The ordered highest vertices would change under the opposite tensor convention, but the irreducible decomposition would be the same.

For $H=\mathrm{diag}(r,s,-r,-s)$, put $L_1(H)=r$, $L_2(H)=s$. Then $L_3=-L_1$, $L_4=-L_2$. The matrix unit identityreads off the root spaces directly. The complete list isThus the C2 root system isAll eight root spaces have dimension one. The zero-weight space is the two-dimensional Cartan subalgebra $\mathbb{C}(E_{11}-E_{33})\oplus \mathbb{C}(E_{22}-E_{44})$; zero itself is not a root.

Use the C2 root system realization in an orthonormal basis ${\epsilon }_1,{\epsilon }_2$ with simple roots ${\alpha }_1={\epsilon }_1-{\epsilon }_2$ and ${\alpha }_2=2{\epsilon }_2$. This numbers the short root first, as required. The fundamental weights and half-sum of positive roots areso $\lambda +\rho =(a+b+2){\epsilon }_1+(b+1){\epsilon }_2$. For the four positive roots, the factors of the Weyl dimension formula areMultiplying yields the Weyl dimension formula for C2:As checks, $V({\omega }_1)$ has dimension $4$, $V({\omega }_2)$ dimension $5$, and $V(2{\omega }_1)$ dimension $10$.

In the realization $\Phi (C_3)=\{\pm 2e_i,\pm e_i\pm e_j:i<j\}$, each coordinate two-plane cuts out a C2 root system. Each plane $\pm x_1\pm x_2\pm x_3=0$ cuts out an A2 root system; there are four such planes after identifying opposite normals. Thus the irreducible full rank-two root subsystems comprise three of type $C_2$ and four of type $A_2$. For an $A_2$ simple pair $p,q$, equal lengths and angle $120$ degrees give roots $\pm p,\pm q,\pm (p+q)$. For a mixed-length simple pair $s,l$ with $\Vert l{\Vert }^2=2\Vert s{\Vert }^2$ and $s\cdot l=-\Vert s{\Vert }^2$, the B2 root system has roots $\pm s,\pm l,\pm (s+l),\pm (2s+l)$. These lists also verify that the selected pairs are simple, rather than simply independent.

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.

For the C2 root system with its first simple root short, the irreducible highest-weight representation of highest weight $a{\omega }_1+b{\omega }_2$, where $a,b\in {\mathbb{Z}}_{\ge 0}$, has dimension