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

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- Solution b

Solution

A convenient root basis is

α_{1} = ε_{1} - ε_{2}, α_{2} = ε_{2} - ε_{3}, α_{3} = 2 ε_{3} .

The corresponding Cartan matrix is

(⟨ α_{i}, α_{j}^{\lor} ⟩)_{ij} = 2 - 1 0 - 1 2 - 2 0 - 1 2 .

Thus the Dynkin diagram is the three-node

C_{3}

chain, with a double edge between

α_{2}

and

α_{3}

and its arrow pointing toward the shorter root

α_{2}

.

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