Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 3 / d / i

Table of contents
- Solution i

Solution

In Dynkin label coordinates, subtracting

α_{i}

subtracts row

i

of the stated Cartan matrix. Starting from

(1, 0, 0)

gives the weight chain

(1, 0, 0) - α_{1} (- 1, 1, 0) - α_{2} (0, - 1, 1) - α_{3} (0, 0, - 1) .

These are the four weights of the defining representation of

A_{3} ≅ sl_{4}

.

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