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

Table of contents
- Solution c

Solution

For an irreducible representation

V (λ)

, write

λ = \sum_{i} m_{i} ω_{i} .

Form

T = ⨂_{i} V (ω_{i})^{\otimes m_{i}} .

The tensor product of highest-weight vectors is killed by all positive-root spaces and has weight

λ

. It therefore generates a highest-weight constituent isomorphic to

V (λ)

. By Complete reducibility of semisimple Lie algebra representations, this constituent is a subrepresentation of

T

. This proves the generation by fundamental representations statement.

Solved by gpt-5.6-sol high.

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