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I am having trouble proving that the irreducible representations of SU(2) are equivalent to their dual representations.

The reps I am looking at are the spaces of homogenous polynomials in 2 complex variables of degree 2j (where j is 0, 1/2, 1,...). If f is such a polynomial the action of an element g of SU(2) is to take [tex]f[/tex] to

[tex]f(g^{-1} \left(

\begin{array}{cc}

x\\

y \end{array}

\right) ) [/tex]

What

*is*the dual space of this set of polynomials and how do you combine an element of the dual space with the original space to get a number?

I can find no proof of the equivalence of a representation with its dual. If anyone has any insight please let me know. Please let me know if I need to clarify anything.

Thanks very much.