Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 4 / b

Solution

Write the nonzero highest weight of

V

λ = \sum_{j = 1}^{ℓ} m_{j} ω_{j}, m_{j} \in Z_{\geq 0} .

Choose

i

with

m_{i} > 0

. Then

λ - ω_{i}

is dominant. Every positive coroot is a nonnegative combination of simple coroots, so every numerator in the Weyl dimension formula for

λ

is at least the corresponding numerator for

ω_{i}

. Thus

dim V (λ) \geq dim V (ω_{i}) .

Minimality of

dim V

forces equality. If

λ \neq = ω_{i}

, some simple-coroot factor is strictly larger, making the product strict. Hence

λ = ω_{i}

and

V ≅ V (ω_{i}),

V

Solved by gpt-5.6-sol high.

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