The question is given below:
And this is the mentioned exercise:
And this is 7.4:
Could anyone give me a hint about the solution of the question, I am stucked in it ?
2026-03-29 19:14:32.1774811672
Show that every irreducible representation of $SO_{3}$ is isomorphic to one of the representations $\Psi_{n}$.
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Vinberg book "Linear representations of finite groups "" has established the isomorphism between $SO_{3}$ and $SU_{2}/\{E, -E\}$ in pg. 76 Equation (4) and you can construct $\Psi_{n}$ from $\Phi_{n}$ with $n$ even from the last paragraph given in the question above and from Equation(8) mentioned in the question above. then use the second requirement that you have proved in question 7 to conclude what is required in question 9.