Spinor representation of the double group of icosahedron and entangled states of fullerene C$_{60}$ wave functions

Symmetry of the icosahedron group is a basic principle producing the analytical description of the fullerene by means of the theory of molecular orbitals. Further development of the fullerene model requires consideration of the half-integer statistics of the electron states. Double group of the icosahedron describes both statistics cases, but we use only fermions case, in other words, the spinor representation of the icosahedron group. In this representation, it is possible to remove restriction defining the parity of the wave function coordinate part and to build a table of characters for arbitrary meaning of the total momentum. Symmetry of the icosahedron produces algebraic structure of the residue ring on the modulo 5, when using momentum projection on chosen axis. If the momentum projection equals both $\pm$ 5/2 the states come into one representation. If the momentum projection by absolute value is higher 5/2, the wave function occupies state with several entangled spinor.