## 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)

Zbl 0743.00058