Designs in Grassmannian Spaces and Lattices
Christine Bachoc
, Renaud Coulangeon2
and Gabriele Nebe3
2Laboratoire A2X, Université Bordeaux I, 351, cours de la Libération, 33405 Talence, France
DOI: 10.1023/A:1020826329555
Abstract
We introduce the notion of a t-design on the Grassmann manifold G m, n \mathcal{G}_{m,n} of the m-subspaces of the Euclidean space \mathbb R \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 
n,m .
n,m .Pages: 5–19
Keywords: lattice; Grassmann manifold; orthogonal group; zonal function
Full Text: PDF
References
1. C. Bachoc and B. Venkov, “Modular forms, lattices and spherical designs,” in “Réseaux euclidiens, “designs” sphériques et groupes,” in L'Enseignement Mathématique J. Martinet ( Éd.), Monographie no. 37, Gen`eve (2001), to appear.
2. J.H. Conway, R.H. Hardin, and N.J.A. Sloane, “Packing lines, planes, etc., packings in Grassmannian spaces,” Experimental Mathematics 5 (1996), 139-159.
3. R. Coulangeon,“Réseaux k-extr\hat emes,” Proc. London Math. Soc. 73 (3) (1996), 555-574.
4. P. Delsarte, J.M. Goethals, and J.J. Seidel, “Spherical codes and designs,” Geom. Dedicata 6 (1977), 363-388.
5. W. Fulton and J. Harris, Representation Theory, a First Course, GTM, Vol. 129, Springer, 1991.
6. G.H. Golub and C.F. Van Loan, Matrix Computations, 2nd edn., John Hopkins University Press, 1989.
7. R. Goodman and N.R. Wallach, Representations and Invariants of the Classical Groups, Encyclopedia of Mathematics and its Applications, Vol. 68, Cambridge University Press, 1998.
8. A.T. James and A.G. Constantine, “Generalized Jacobi polynomials as spherical functions of the Grassmann manifold,” Proc. London Math. Soc. 29(3) (1974), 174-192.
9. W. Lempken, B. Schr\ddot oder, and P.H. Tiep, “Symmetric squares, spherical designs and lattice minima,” J. Algebra 240 (2001), 185-208.
10. R.A. Rankin, “On positive definite quadratic forms,” J. London Math. Soc. 28 (1953), 309-314.
11. B. Venkov, “Even unimodular extremal lattices,” Proc. Steklov Inst. Math. 165 (1984), 47-52.
12. B. Venkov, “Réseaux et designs sphériques,” in “Réseaux euclidiens, “designs” sphériques et groupes,” J. Martinet ( Éd.), L'Enseignement Mathématique, Monographie no 37, Gen`eve (2001), to appear.
2. J.H. Conway, R.H. Hardin, and N.J.A. Sloane, “Packing lines, planes, etc., packings in Grassmannian spaces,” Experimental Mathematics 5 (1996), 139-159.
3. R. Coulangeon,“Réseaux k-extr\hat emes,” Proc. London Math. Soc. 73 (3) (1996), 555-574.
4. P. Delsarte, J.M. Goethals, and J.J. Seidel, “Spherical codes and designs,” Geom. Dedicata 6 (1977), 363-388.
5. W. Fulton and J. Harris, Representation Theory, a First Course, GTM, Vol. 129, Springer, 1991.
6. G.H. Golub and C.F. Van Loan, Matrix Computations, 2nd edn., John Hopkins University Press, 1989.
7. R. Goodman and N.R. Wallach, Representations and Invariants of the Classical Groups, Encyclopedia of Mathematics and its Applications, Vol. 68, Cambridge University Press, 1998.
8. A.T. James and A.G. Constantine, “Generalized Jacobi polynomials as spherical functions of the Grassmann manifold,” Proc. London Math. Soc. 29(3) (1974), 174-192.
9. W. Lempken, B. Schr\ddot oder, and P.H. Tiep, “Symmetric squares, spherical designs and lattice minima,” J. Algebra 240 (2001), 185-208.
10. R.A. Rankin, “On positive definite quadratic forms,” J. London Math. Soc. 28 (1953), 309-314.
11. B. Venkov, “Even unimodular extremal lattices,” Proc. Steklov Inst. Math. 165 (1984), 47-52.
12. B. Venkov, “Réseaux et designs sphériques,” in “Réseaux euclidiens, “designs” sphériques et groupes,” J. Martinet ( Éd.), L'Enseignement Mathématique, Monographie no 37, Gen`eve (2001), to appear.