Application of algebraic combinatorics to finite spin systems

A finite spin system invariant under a symmetry group G is a very illustrat ive example of a, finite group action on mappings f : X --> Y (X is a set o f spin carriers, Y contains spin projections for a given spin number s). Or bits and stabilizers are used as additional indices of the symmetry adapted basis. Their mathematical nature does not decrease a dimension of a given eigenproblem, but they label states in a systematic way. It allows construc tion of general formulas for vectors of symmetry adapted basis and matrix e lements of operators commuting with the action of G in the space of states. The special role is played by double cosets, since they label nonequivalen t (from the symmetry point of view) matrix elements <x/H/y > for an operato r H between Ising configurations /x >, /y >. Considerations presented in th is paper should be followed by a detailed discussion of different symmetry groups (e.g.) cyclic or dihedral ones) and optimal implementation of algori thms. The paradigmatic example, i.e. a finite spin system, can be useful in investigations of magnetic macromolecules like Fe-6 or Mn(12)acetate.