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Optimal Line Packings from Nonabelian Groups

  • Joseph W. IversonEmail author
  • John Jasper
  • Dustin G. Mixon
Article
  • 6 Downloads

Abstract

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.

Keywords

Equiangular tight frames Association schemes Difference sets Group frames Heisenberg group 

Mathematics Subject Classification

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

Notes

Acknowledgements

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Joseph W. Iverson
    • 1
    • 2
    Email author
  • John Jasper
    • 3
  • Dustin G. Mixon
    • 4
  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsIowa State UniversityAmesUSA
  3. 3.Department of Mathematics and StatisticsSouth Dakota State UniversityBrookingsUSA
  4. 4.Department of MathematicsThe Ohio State UniversityColumbusUSA

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