Formal Duality in Finite Abelian Groups

Shuxing Li, Alexander Pott, Robert Schüler

Published 2018-04-17Version 1

Inspired by an experimental study of energy-minimizing periodic configurations in Euclidean space, Cohn, Kumar and Sch\"urmann proposed the concept of formal duality between a pair of periodic configurations, which indicates an unexpected symmetry possessed by the energy-minimizing periodic configurations. Later on, Cohn, Kumar, Reiher and Sch\"urmann translated the formal duality between a pair of periodic configurations into the formal duality of a pair of subsets in a finite abelian group. This insight suggests to study the combinatorial counterpart of formal duality, which is a configuration named formal dual pair. In this paper, we initiate a systematic investigation on formal dual pairs in finite abelian groups, which involves basic concepts, constructions, characterizations and nonexistence results. In contrast to the belief that primitive formal dual pairs are very rare in cyclic groups, we construct three families of primitive formal dual pairs in noncyclic groups. These constructions enlighten us to propose the concept of even set, which reveals more structural information of formal dual pairs and leads to a characterization of rank three primitive formal dual pairs. Finally, we derive some nonexistence results of primitive formal dual pairs, which are in favor of the main conjecture that except two small examples, no primitive formal dual pair exists in cyclic group.