We study the notion of formal duality introduced by Cohn, Kumar, andSch\"urmann in their computational study of energy-minimizing particleconfigurations in Euclidean space. In particular, using the Poisson summationformula we reformulate formal duality as a combinatorial phenomenon in finiteabelian groups. We give new examples related to Gauss sums and make someprogress towards classifying formally dual configurations.
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