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After ordering a symplectic basis in two blocks, writeMatrices in the Symplectic Lie algebra have block formThe root-space decomposition isFor example, these one-dimensional spaces are spanned respectively byThus this is the Cn root systemThe upper-triangular choice givesIts simple roots, highest root, fundamental weights, and half-sum of positive roots are

Using the notation requested in the paper, the root lattice $P$ and weight lattice $Q$ areso $Q/P\cong \mathbb{Z}/2\mathbb{Z}$. This reverses the common notation in which the root lattice is called $Q$ and the weight lattice is called $P$.

Since a multiple-edge arrow in a Dynkin diagram points toward the shorter root, the finite and extended diagrams areand

Solved by gpt-5.6-sol high.

Choose a short simple root ${\alpha }_1$ and a long simple root ${\alpha }_2$, with $|{\alpha }_2{|}^2=3|{\alpha }_1{|}^2$ and angle $150^{\circ }$. The six positive roots of the G2 root system areand their negatives complete the two concentric hexagons of short and long roots. The fundamental weights areso ${\omega }_1$ is itself a short root and ${\omega }_2$ is the highest root.

The seven-dimensional representation $L_{{\omega }_1}$ has weight seteach with weight multiplicity one. Their positive heights are $1,2,3$, so the q-character of a highest-weight representation isThis is one ${\mathfrak{sl}}_2$ weight string, hence

The representation $L_{{\omega }_2}$ is the fourteen-dimensional Adjoint representation. Its nonzero weights are the twelve roots, and its zero-weight space is the two-dimensional Cartan subalgebra. The positive root heights are $1,1,2,3,4,5$, soSplitting this into ordinary ${\mathfrak{sl}}_2$ strings gives

Solved by gpt-5.6-sol high.

The fundamental weights satisfy $2({\omega }_i,{\alpha }_j)/({\alpha }_j,{\alpha }_j)={\delta }_{ij}$. Since $|{\alpha }_1{|}^2=1$ and $|{\alpha }_2{|}^2=2$, solving givesThe full B2 root system isThe weight lattice is generated by ${\omega }_1,omeg{a}_2$; geometrically it consists of the integer lattice together with the translate in which both coordinates are half-integers.

Solved by gpt-5.6-sol high.