Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 102 / 4 / i

Table of contents
- Solution i

Solution

For a dominant integral weight

λ

, the Weyl character formula is

ch L_{λ} = \frac{\sum _{w \in W} ( - 1 ) ^{ℓ (w)} e ^{w (λ + ρ)}}{\sum _{w \in W} ( - 1 ) ^{ℓ (w)} e ^{wρ}} .

Here

W

is the Weyl group,

ℓ (w)

its Coxeter length,

ρ

the Weyl vector, and

ch

the formal character of a weight module. Taking the limit at the identity gives the Weyl dimension formula

dim L_{λ} = \prod_{α \in R^{+}} \frac{⟨ λ + ρ , α ^{\lor} ⟩}{⟨ ρ , α ^{\lor} ⟩} .

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