Kybernetika 56 no. 5, 934-947, 2020

A note on representing dowling geometries by partitions

František Matúš and Aner Ben-EfraimDOI: 10.14736/kyb-2020-5-0934

Abstract:

We prove that a rank $\geq 3$ Dowling geometry of a group $H$ is partition representable if and only if $H$ is a Frobenius complement. This implies that Dowling group geometries are secret-sharing if and only if they are multilinearly representable.

Keywords:

matroid representations, partition representations, Dowling geometries, Frobenius groups

Classification:

05B35

paper.pdf

References:

  1. A. Beimel, A. Ben-Efraim, C. Padró and I. Tyomkin: Multi-linear secret-sharing schemes. Theory Cryptogr. Conf. 14 (2014), 394-418.   DOI:10.1007/978-3-642-54242-8\_17
  2. A. Ben-Efraim: Secret-sharing matroids need not be Algebraic. Discrete Math. 339 (2015), 8, 2136-2145.   DOI:10.1016/j.disc.2016.02.012
  3. E. F. Brickell and D. M. Davenport: On the classification of ideal secret sharing schemes.    CrossRef
  4. R. Brown: Frobenius groups and classical maximal orders. Memoirs Amer. Math. Soc. 151 (2001), 717.   DOI:10.1090/memo/0717
  5. T. A. Dowling: A class of geometric lattices based on finite groups. J. Combinat. Theory, Ser. B 14 (1973), 61-86.   DOI:10.1016/s0095-8956(73)80007-3
  6. D. M. Evans and E. Hrushovski: Projective planes in algebraically closed fields. Proc. London Math. Soc. 62 (1989), 3, 1-24.   DOI:10.1112/plms/s3-62.1.1
  7. W. Feit: Characters of Finite Groups. W. A. Benjamin Company, Inc., New York 1967.   CrossRef
  8. F. Matúš: Matroid representations by partitions. Discrete Math. 203 (1999), 169-194.   DOI:10.1016/s0012-365x(99)00004-7
  9. N. Jacobson: Basic Algebra II. (Second Edition) W. H. Freeman and Co., New York 1989.   CrossRef
  10. J. G. Oxley: Matroid Theory. (Second Edition) Oxford University Press Inc., New York 2011.   DOI:10.1093/acprof:oso/9780198566946.001.0001
  11. D. S. Passman: Permutation Groups. Dover Publications, Inc. Mineola, New York 2012.   CrossRef
  12. R. A. Pendavingh and S. H. M. van Zwam: Skew partial fields, multilinear representations of matroids, and a matrix tree theorem. Adv. Appl. Math. 50 (2013), 1, 201-227.   DOI:10.1016/j.aam.2011.08.003
  13. P. D. Seymour: On secret-sharing matroids. J. Combinat. Theory, Ser. B 56 (1992), 69-73.   DOI:10.1016/0095-8956(92)90007-k
  14. J. Simonis and A. Ashikhmin: Almost affine codes. Designs Codes Cryptogr. 14 (1998), 2, 179-197.   DOI:10.1023/a:1008244215660
  15. M. Suzuki: Group Theory I. Springer-Verlag, Berlin 1982.   CrossRef
  16. D. Vertigan: Dowling Geometries representable over rings. Ann. Combinat. 19 (2015), 225-233.   DOI:10.1007/s00026-015-0250-4