 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-102/4/e/solution

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

Let

V = V (ω_{1})

, so

V^{*} = V (ω_{2})

V \otimes V = Sym^{2} V \oplus Λ^{2} V ≅ V (2 ω_{1}) \oplus V (ω_{2}) .

The highest-weight tensor-product rule gives

V (2 ω_{1}) \otimes V (ω_{2}) ≅ V (2 ω_{1} + ω_{2}) \oplus V (ω_{1})

and

V (ω_{2}) \otimes V (ω_{2}) ≅ V (2 ω_{2}) \oplus V (ω_{1}) .

15 + 3 + 6 + 3 = 27

V \otimes V \otimes V^{*} ≅ V (2 ω_{1} + ω_{2}) \oplus V (2 ω_{2}) \oplus V (ω_{1})^{\oplus 2} .

Thus one may take

(λ_{1}, λ_{2}, λ_{3}, λ_{4}) = (2 ω_{1} + ω_{2}, 2 ω_{2}, ω_{1}, ω_{1}) .

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