Optimal Line Packings from Nonabelian Groups


We use group schemes to construct optimal packings of lines through the origin. In this setting, optimal line packings are naturally characterized using representation theory, which in turn leads to a necessary integrality condition for the existence of equiangular central group frames. We conclude with an infinite family of optimal line packings using the group schemes associated with certain Suzuki 2-groups, specifically, extensions of Heisenberg groups. Notably, this is the first known infinite family of equiangular tight frames generated by representations of nonabelian groups.

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The authors thank the anonymous referees for thoughtful recommendations that significantly altered and greatly improved the manuscript. Part of this work was conducted during the SOFT 2016: Summer of Frame Theory workshop at the Air Force Institute of Technology. The authors thank Nathaniel Hammen for helpful discussions during this workshop. This work was partially supported by NSF DMS 1321779, ARO W911NF-16-1-0008, AFOSR F4FGA05076J002, an AFOSR Young Investigator Research Program award, and an AFRL Summer Faculty Fellowship Program award. The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.

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Iverson, J.W., Jasper, J. & Mixon, D.G. Optimal Line Packings from Nonabelian Groups. Discrete Comput Geom 63, 731–763 (2020). https://doi.org/10.1007/s00454-019-00084-z

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  • Equiangular tight frames
  • Association schemes
  • Difference sets
  • Group frames
  • Heisenberg group

Mathematics Subject Classification

  • Primary: 05B10
  • 42C15
  • 94C30
  • Secondary: 20C15
  • 52C99