Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 4 / e

Table of contents
- Solution e

Solution

The Fundamental representations of B2 have weight sets

d_{[1, 0]} : {(\frac{1}{2}, \frac{1}{2}), (\frac{1}{2}, - \frac{1}{2}), (- \frac{1}{2}, \frac{1}{2}), (- \frac{1}{2}, - \frac{1}{2})},

so

dim d_{[1, 0]} = 4

, and

d_{[0, 1]} : {(1, 0), (- 1, 0), (0, 1), (0, - 1), (0, 0)},

so

dim d_{[0, 1]} = 5

. The Adjoint representation has all eight roots as nonzero weights and zero with multiplicity two. Its highest root is

2 α_{1} + α_{2} = (1, 1) = 2 ω_{1},

so its highest-weight label is

[2, 0]

and its dimension is

8 + 2 = 10

.

Solved by gpt-5.6-sol high.

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