The artin braid group actions on the set of spin structures on a surface

We study the action of the Artin braid group B2g+2 on the set of spin structures on a hyperelliptic curve of genus g, which reduces to that of the symmetric group S2g+2. It has been already described in terms of the classical theory of Riemann surfaces. In this paper, we compute the S2g+2-orbits of the spin structures of genus g and the isotropy group Gᵢ of each orbit in a purely combinatorial way.