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Extremal lattices for an automorphism group. (Réseaux extrêmes pour un groupe d’automorphismes.) (French) Zbl 0753.11026

Journées arithmétiques, Exp. Congr., Luminy/Fr. 1989, Astérisque 198-200, 41-66 (1991).
[For the entire collection see Zbl 0743.00058.]
In this paper the authors study \(G\)-lattices. These are lattices that admit a faithful action by a finite group of isometries \(G\). First, following Korkine-Zolotareff and Barnes, the fundamental theory of \(G\)-lattices is developed. Then the “\(G\)-Hermite constants” \(\gamma_{n,G}\) are determined for some small dimensions \(n\) and groups \(G\). These constants are generalizations of the usual Hermite constants: \(\gamma_{n,G}=\sup_ \Lambda\gamma_ n(\Lambda)\), where \(\Lambda\) runs over the \(G\)-lattices of rank \(n\) and \[ \gamma_ n(\Lambda)={\inf\{\| {\mathbf x}\|^ 2:{\mathbf x}\in\Lambda,{\mathbf x}\neq 0\}\over \text{Disc}(\Lambda)^{2/n}}. \] In some cases \(\gamma_{n,G}<\gamma_ n\): \[ \begin{array}{|c|c|c|c|} \hline n & \gamma_ n & G & \gamma_{n,G} \\ \hline 2& 2/\sqrt3 & \mathbb{Z}/4\mathbb{Z} &1 \\ 4& \sqrt2 & \mathbb{Z}/5\mathbb{Z} & 2/\sqrt[4]5 \\ 6& 2/\sqrt[6]3 & \mathbb{Z}/7\mathbb{Z} & 4/\sqrt7 \\ 8&2& \mathbb{Z}/16\mathbb{Z} & 8/\sqrt[4]{482} \\ \hline \end{array} \] In each case the authors exhibit a \(G\)-lattice \(\Lambda\) for which \(\gamma_ n(\Lambda)=\gamma_{n,G}\).
Reviewer: R.Schoof (Povo)

MSC:

11H06 Lattices and convex bodies (number-theoretic aspects)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)

Citations:

Zbl 0743.00058
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