Application of algebraic combinatorics to finite spin systems with dihedral symmetry

Properties of a given symmetry group G are very important in investigation of a physical system invariant under its action. In the case of finite spin systems (magnetic rings, some planar macromolecules) the symmetry group is isomorphic with the dihedral group D_N. In this paper group-theoretical `parameters' of such groups are determined, especially decompositions of transitive representations into irreducible ones and double cosets. These results are necessary to construct matrix elements of any operator commuting with G in an efficient way. The approach proposed can be usefull in many branches of physics, but here it is applied to finite spin systems, which serve as models for mesoscopic magnets.