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

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- Solution b

Solution

The classification of finite-dimensional representations of SU2 says that every irreducible is one of the

V_{n}

, and compactness gives complete reducibility. The standard alternating form

ω ((u_{1}, u_{2}), (v_{1}, v_{2})) = u_{1} v_{2} - u_{2} v_{1}

is

S U_{2}

-invariant and identifies

V_{1}

with

V_{1}^{*}

. Taking symmetric powers gives

V_{n} ≅ V_{n}^{*}

for every

n

. Therefore, if

V ≅ ⨁_{n} m_{n} V_{n},

then

V^{*} ≅ ⨁_{n} m_{n} V_{n}^{*} ≅ V .

This proves self-duality of finite-dimensional SU2 representations.

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