Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 1 / 4

Solution

Φ \subset E ∖ {0}

r

, invariant under each Weyl reflection

s_{α} (x) = x - \frac{2 ( x , α )}{( α , α )} α,

with

2 (β, α) / (α, α) \in Z

for every pair of roots and

Φ \cap R α = {α, - α}

Δ = (α_{1}, \dots, α_{r})

is a basis such that the coefficients of every root are integers that are either all nonnegative or all nonpositive. These define the positive system of a root system. We use the existing Cartan matrix convention

A_{ij} = ⟨ α_{i}, α_{j}^{\lor} ⟩ = \frac{2 ( α _{i} , α _{j} )}{( α _{j} , α _{j} )} .

Using the alternative denominator

(α_{i}, α_{i})

The classification of rank-two root systems gives $A_{1} \times A_{1}$ , $A_{2}$ , $B_{2} = C_{2}$ , and $G_{2}$ . In orthonormal coordinates we may choose the following simple roots and Cartan matrices:

For $A_{1} \times A_{1}$ , take $α_{1} = (1, 0)$ , $α_{2} = (0, 1)$ and $A = (2002)$ . The positive roots are $α_{1}, α_{2}$ .
For $A_{2}$ , take $α_{1} = (1, 0)$ , $α_{2} = (- 1/2, 3 /2)$ and $A = (2 - 1 - 1 2)$ . The positive roots are $α_{1}, α_{2}, α_{1} + α_{2}$ .
For $B_{2}$ , take the long root $α_{1} = (1, - 1)$ and short root $α_{2} = (0, 1)$ , giving $A = (2 - 1 - 2 2)$ . The positive roots are $α_{1}, α_{2}, α_{1} + α_{2}, α_{1} + 2 α_{2}$ . Interchanging the long and short convention gives type $C_{2}$ , which is the same rank-two classification up to the usual identification.
For $G_{2}$ , take the short root $α_{1} = (1, 0)$ and long root $α_{2} = (- 3/2, 3 /2)$ , giving $A = (2 - 3 - 1 2)$ . The positive roots are $α_{1}, α_{2}, α_{1} + α_{2}, 2 α_{1} + α_{2}, 3 α_{1} + α_{2}, 3 α_{1} + 2 α_{2}$ .

Each complete root system consists of these positive roots and their negatives. To see why the list is exhaustive, the off-diagonal Cartan matrix entries of two distinct simple roots are nonpositive integers, and their product is