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Journal of Algebraic Combinatorics

, Volume 16, Issue 1, pp 5–19 | Cite as

Designs in Grassmannian Spaces and Lattices

  • Christine Bachoc
  • Renaud Coulangeon
  • Gabriele Nebe
Article

Abstract

We 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 γ n,m .

lattice Grassmann manifold orthogonal group zonal function 

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Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Christine Bachoc
    • 1
  • Renaud Coulangeon
    • 1
  • Gabriele Nebe
    • 2
  1. 1.Laboratoire A2XUniversité Bordeaux ITalenceFrance
  2. 2.Abteilung Reine MathematikUniversität UlmUlmGermany

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