Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 19H / a

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- Solution a

Solution

Every element of

S U (2)

is unitarily conjugate to an element of the maximal torus

T = {diag (z, z^{- 1}) : z \in S^{1}} .

Since characters are class functions,

χ_{V}

is determined on

T

. The restriction decomposes into finitely many torus weights, so

χ_{V} (z) = \sum_{n \in Z} a_{n} z^{n}, a_{n} \in Z_{\geq 0}, \sum_{n} a_{n} = dim V .

A Weyl-group element conjugates

diag (z, z^{- 1})

to

diag (z^{- 1}, z)

, so

χ_{V} (z) = χ_{V} (z^{- 1})

and

a_{n} = a_{- n}

.

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