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

Table of contents
- Solution a

Solution

Choose positive roots

Φ^{+}

and let

ρ = \frac{1}{2} \sum_{α \in Φ^{+}} α

be the Weyl vector. For a dominant integral weight

λ

, the Weyl dimension formula is

dim V (λ) = α \in Φ^{+} \prod \frac{⟨ λ + ρ , α ^{\lor} ⟩}{⟨ ρ , α ^{\lor} ⟩} .

Here

α^{\lor}

is the coroot and

⟨, ⟩

is the natural weight-coroot pairing.

Solved by gpt-5.6-sol high.

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