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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 19J / d / Solution

For

g

conjugate to

diag (z, z^{- 1})

, multiplication of the weight sums gives

χ_{n} (g)^{2} = \sum_{j = 0}^{n} χ_{2 n - 2 j} (g) .

Indeed, the coefficient of

z^{2 n - 2 m}

on the left counts pairs

(r, s)

with

0 \leq r, s \leq n

and

r + s = m

; the nested weight strings on the right give the same multiplicity. Characters determine finite-dimensional representations of the compact group

S U_{2}

, so

V_{n} \otimes V_{n} ≅ j = 0 ⨁ n V_{2 n - 2 j} .

This weight-counting argument proves the required case of the Clebsch-Gordan decomposition for SU2 rather than merely quoting it.

Solved by gpt-5.6-sol high.

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