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A NOTE ON NULL DESIGNS OF DUAL POLAR SPACES

CHO, SOO-JIN

  • Published : 2005.01.01

Abstract

Null designs on the poset of dual polar spaces are considered. A poset of dual polar spaces is the set of isotropic subspaces of a finite vector space equipped with a nondegenerate bilinear form, ordered by inclusion. We show that the minimum number of isotropic subspaces to construct a nonzero null t-design is ${\prod}^{t}_{i=0}(1+q^{i})$ for the types $B_N,\;D_N$, whereas for the case of type $C_N$, more isotropic subspaces are needed.

Keywords

null designs;minimal null designs;dual polar spaces

References

  1. E. Artin, Geometric algebra, Interscience, New York, Interscience Tracks in Pure and Applied Mathematics 3 (1957)
  2. E. Bannai and T. Ito, Algebraic combinatorics I, Association Schemes, The Ben- jamin/Cummings Publishing Company, Inc., 1984
  3. R. Carter, Simple groups of Lie type, Wiley-Interscience, London, 1972
  4. S. Cho, Minimal null designs and a density theorem of posets, European J. Combin. 19 (1998), 433-440 https://doi.org/10.1006/eujc.1997.0201
  5. S. Cho, Minimal null designs of subspace lattice over Finite Fields, Linear Algebra Appl. 282 (1998), 199-220 https://doi.org/10.1016/S0024-3795(98)10062-9
  6. S. Cho, On the support size of null designs of Finite ranked posets, Combinatorica 19 (1999), 589-595 https://doi.org/10.1007/s004939970009
  7. P. Frankl and J. Pach, On the number of sets in a null t-design, European J. Combin. 4 (1983), 21-23 https://doi.org/10.1093/eurheartj/4.suppl_G.21
  8. S. Li, R. Graham and W. Li,On the structure of t-designs, SIAM J. Alg. Disc. Math. 1 (1980), 8-14 https://doi.org/10.1137/0601002
  9. G. James, Representations of general linear groups, LMS Lecture Note Series 94, Cambridge University Press, 1984
  10. R. Liebler and K. Zimmermann, Combinatorial Sn-modules as codes, J. Algebraic Combin. 4 (1995), 47-68 https://doi.org/10.1023/A:1022485624417
  11. R. Stanley, Enumerative combinatorics vol 1, Wadsworth & Brooks/Cole, 1986
  12. D. Stanton, Some q-Krawtchouk polynomials on Chevalley groups, Amer. J. Math. 102 (1986), 625-662 https://doi.org/10.2307/2374091
  13. D. Stanton, t-designs in classical association schemes, Graphs Combin. 2 (1980), 283-286 https://doi.org/10.1007/BF01788103