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arXiv · 1510.08804

Spherical 2-designs and lattices from Abelian groups

Albrecht Boettcher·Simon Eisenbarth·Lenny Fukshansky·Stephan Ramon Garcia·Hiren Maharaj
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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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Albrecht Boettcher, Simon Eisenbarth, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj. 2015-10-29. Spherical 2-designs and lattices from Abelian groups. https://arxiv.org/abs/1510.08804

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