Nonequivalent periodic subsets of the lattice.

Abstract

The use of Pólya's theorem in crystallography and other applications has greatly simplified many counting and coloring problems. Given a group of equivalences acting on a set, Pólya's theorem equates the number of unique subsets with the orbits of the group action. For a lattice and a given group of periodic equivalences, the number of nonequivalent subsets of the lattice can be solved using Pólya's counting on the group of relevant symmetries acting on the lattice. When equivalence is defined via a sublattice, the use of Pólya's theorem is equivalent to knowing the cycle index of the action of the group elements on a related finite group structure. A simple algebraic method is presented to determine the cycle index for a group element acting on a lattice subject to certain periodicity arguments.