# Why an element of SU(2) acts as a rotation for Majorana representation of states?

I know that for a given spin-j quantum state, say $$\vert\psi\rangle = (\psi_0 , \psi_1 , \cdots , \psi_{2j})$$, we can construct a polynomial as follows

$$w(z) = \sum_{k = 0}^{2j} (-1)^k \psi_k \sqrt{\binom{2j}{k} } z^{2j-k}$$

and by means of the inverse stereographic projection of the roots of $$w(z)$$, the Majorana's representation of $$\vert\psi\rangle$$ is obtained on sphere. I also know that the Majorana's representation for the eigen-state of $$\mathbf{J.n}$$ operator with eigen-value $$m$$ is the configuration of points on sphere in which there are $$j+m$$ points in the $$\mathbf{n}$$ direction and other $$j-m$$ points in the antipodal point (equivalently $$-\mathbf{n}$$ direction).

My question is why the Majorana's representation of $$e^{i\mathbf{n.J} \theta} \vert \psi \rangle$$ is just that of $$\vert\psi\rangle$$ rotated around $$\mathbf{n}$$ by an angle of $$\theta$$? (This is transparent when $$\vert \psi \rangle$$ is an eigen-vector of angular momentum in some direction but I do not see why this is true in general.)

Thank you

• This type of question might be more appropriate over in the physics stack exchange. To my knowledge the Majorana representation has nothing to do with the bloch sphere. Aug 4, 2020 at 13:53
• Have you seen Borel-Weil-Bott? Aug 4, 2020 at 19:51