I'm working out the irreducible representation of $SO(3)$. Let's call $R_{\theta}$ as a general rotation and $Y_{m}^{l}\left(\eta,\varphi\right)$ the spherical harmonics. I now like to have the matrix of the representation $\rho$ of a general rotation $R_{\theta}$ of $SO(3)$. Taking the spherical harmonics as a basis I have

$$\left(\rho^{l}\left(R_{\theta}\right)Y_{m}^{l}\right)\left(\eta,\varphi\right)= {\sum}\rho_{m,n}^{l}\left(R_{\theta}\right)Y_{l}^{n}\left(\eta,\varphi\right),$$ where $m,n=-l,...,l$.

The book now says that the coefficient are given by $$\rho_{m,n}^{l}\left(R\left(\theta_{x},\theta_{y},\theta_{z}\right)\right)=e^{-\imath\theta_{x}n}P_{n,m}^{l}(\cos\theta_{y})e^{-\imath\theta_{z}m},$$ What I'd like to understand is why this asymmetry between rotations along the x and the z axis in respect to rotations along the y axes. Is he under impliying that these axes are aligned in some kind of way in respect of the parameter $\left(\eta,\varphi\right)$ to which respect I defined the spherical harmonics?

Edit: few pages before he defined the parametrization of the sphere like this $$\\ x=\cos\varphi\cos\eta, \\y=\cos\varphi\sin\eta, \\z=\sin\varphi.$$ But still I don't get why the y axis is different from the x and the z and not instead the z axes differrent from the x and the y axis.


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