 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-102/4/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Let

R^{+}

W

ℓ (w)

ρ

α^{\lor}

a coroot. For a dominant integral highest weight

λ

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

Taking the value at the identity gives the Weyl dimension formula

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

For the q-character convention relevant to the Principal sl2 subalgebra, set

h_{pr} = 2 ρ^{\lor}

, so

α_{i} (h_{pr}) = 2

ch_{q} L_{λ} = \sum_{μ} (dim L_{λ} [μ]) q^{μ (h_{pr})} .

The q-character formula is the principal specialization of the Weyl character formula:

ch_{q} L_{λ} = \frac{\sum _{w \in W} ( - 1 ) ^{ℓ (w)} q ^{⟨ w (λ + ρ), 2 ρ^{\lor} ⟩}}{\sum _{w \in W} ( - 1 ) ^{ℓ (w)} q ^{⟨ wρ, 2 ρ^{\lor} ⟩}} .

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