Abstract
We consider lattices generated by finite Abelian groups. We prove that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame. Further, our result combined with a recent theorem of R. Bacher produces (via the classical Voronoi criterion) a new infinite family of extreme lattices. Additionally, we investigate the structure of the automorphism groups of these lattices, strengthening our previous results in this direction.
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Acknowledgements
We wish to thank Professor Gabriele Nebe for her helpful comments on the subject of this paper. We are also grateful to the referee for a thorough reading and several useful remarks. Fukshansky acknowledges support by NSA Grant H98230-1510051, Garcia acknowledges support by NSF Grant DMS-1265973.
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Böttcher, A., Eisenbarth, S., Fukshansky, L. et al. Spherical 2-Designs and Lattices from Abelian Groups. Discrete Comput Geom 61, 123–135 (2019). https://doi.org/10.1007/s00454-017-9909-4
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DOI: https://doi.org/10.1007/s00454-017-9909-4