# Show that every irreducible representation of $SO_{3}$ is isomorphic to one of the representations $\Psi_{n}$.

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 ?

• You should formulate your question in a more concise way, and showing more efforts than posting a lengthy scan. – YCor Apr 11 at 9:51
• @YCor okay I will do this within 6 hrs ..... I am sorry – Intuition Apr 11 at 10:31
• @YCor Sorry for being late ..... I have edited my question ...... shall I edit it more or this is fine ? Actually I have a difficulty in understanding how we get $\Psi {n}$ from $\Phi_{n}$. – Intuition Apr 13 at 21:17

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.