Formal duality and generalizations of the Poisson summation formula
Henry Cohn, Abhinav Kumar, Christian Reiher, Achill Sch\"urmann
TL;DR
This paper explores formal duality in Euclidean space configurations, reformulating it via the Poisson summation formula as a combinatorial property in finite abelian groups, and presents new examples and classification progress.
Contribution
It introduces a new combinatorial perspective on formal duality using the Poisson summation formula and provides novel examples and classification insights.
Findings
Reformulation of formal duality as a combinatorial phenomenon
New examples related to Gauss sums
Progress towards classifying formally dual configurations
Abstract
We study the notion of formal duality introduced by Cohn, Kumar, and Sch\"urmann in their computational study of energy-minimizing particle configurations in Euclidean space. In particular, using the Poisson summation formula we reformulate formal duality as a combinatorial phenomenon in finite abelian groups. We give new examples related to Gauss sums and make some progress towards classifying formally dual configurations.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Graph Theory Research · Limits and Structures in Graph Theory