Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 102 / 2

Write an element of the diagonal Cartan subalgebra asand define the linear functionals ${\epsilon }_i(t)=t_i$. The root-space decomposition of the Symplectic Lie algebra ${\mathfrak{sp}}_6(\mathbb{C})$ then has the C3 root system

A convenient root basis isThe corresponding Cartan matrix isThus the Dynkin diagram is the three-node $C_3$ chain, with a double edge between ${\alpha }_2$ and ${\alpha }_3$ and its arrow pointing toward the shorter root ${\alpha }_2$.

The Weyl reflection formula $s_i(\beta )=\beta -⟨\beta ,{\alpha }_i^{\vee }⟩{\alpha }_i$, together with the Cartan matrix, givesThese are the images of every simple root under each simple reflection.

With the Euclidean inner product used above, the simple coroots areIndeed, if a positive root is $\beta ={\sum }_i{n}_i{\alpha }_i$, thenwhose coefficients are nonnegative. The negative roots give the negatives of these combinations, so the displayed coroots form a fundamental system of a root system for the dual root system.

Duality reverses root lengths. The dual of $C_3$ is therefore the B3 root system: its Dynkin diagram is again a three-node chain with a double final edge, but its arrow points toward the now-short root ${\alpha }_3^{\vee }$.