Application of algebraic combinatorics to finite spin systems with dihedral symmetry

TL;DR

This paper applies algebraic combinatorics to analyze dihedral symmetry groups in finite spin systems, enabling efficient computation of operators and advancing understanding of mesoscopic magnetic models.

Contribution

It introduces a group-theoretical framework for decomposing dihedral group representations, facilitating calculations in finite spin systems with dihedral symmetry.

Findings

Decomposition of transitive representations into irreducible components.

Determination of double cosets for dihedral groups.

Framework for constructing matrix elements of symmetric operators.

Abstract

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.