 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/19h/b/solution

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

For the

(n + 1)

-dimensional irreducible,

χ_{V_{n}} (z) = z^{n} + z^{n - 2} + \dots + z^{- n} .

The Clebsch--Gordan rule gives

V_{4} \otimes V_{4} ≅ V_{8} \oplus V_{6} \oplus V_{4} \oplus V_{2} \oplus V_{0} .

The even total-spin summands are symmetric and the odd ones alternating, hence

S^{2} V_{4} ≅ V_{8} \oplus V_{4} \oplus V_{0}, Λ^{2} V_{4} ≅ V_{6} \oplus V_{2} .

 New to topics? Read the docs here!