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 as Fe$\text{}_{6}$, Cu$\text{}_{6}$, Fe$\text{}_{10}$, some planar macromolecules as Fe$\text{}_{12}$ or Fe$\text{}_{8}$) the symmetry group is isomorphic with the dihedral group D$\text{}_{N}$. 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 useful in many branches of physics, but here it is applied to finite spin systems, which serve as models for mesoscopic magnets.