Designs in Grassmannian Spaces and Lattices

AbstractWe introduce the notion of a t-design on the Grassmann manifold $$\mathcal{G}_{m,n} $$ of the m-subspaces of the Euclidean space $$\mathbb{R}$$ n. It generalizes the notion of antipodal spherical design which was introduced by P. Delsarte, J.-M. Goethals and J.-J. Seidel. We characterize the finite subgroups of the orthogonal group which have the property that all its orbits are t-designs. Generalizing a result due to B. Venkov, we prove that, if the minimal m-sections of a lattice L form a 4-design, then L is a local maximum for the Rankin function γn,m.