Abstract
Combining theoretical arguments with calculations in the computer algebra package GAP, we are able to show that there are 27 isomorphism classes of hyperplanes in the near hexagon for the group U 4(3). We give an explicit construction of a representative of each class and we list several combinatorial properties of such a representative.
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References
Aschbacher M.: Flag structures on Tits geometries. Geom. Dedicata 14, 21–32 (1983)
Bardoe M.K.: On the universal embedding of the near-hexagon for U 4(3). Geom. Dedicata 56, 7–17 (1995)
Brouwer A.E., Cohen A.M., Hall J.I., Wilbrink H.A.: Near polygons and Fischer spaces. Geom. Dedicata 49, 349–368 (1994)
Brouwer A.E., Cuypers H., Lambeck E.W.: The hyperplanes of the M 24 near polygon. Graphs Combin. 18, 415–420 (2002)
Brouwer A.E., Wilbrink H.A.: The structure of near polygons with quads. Geom. Dedicata 14, 145–176 (1983)
Cameron P.J.: Dual polar spaces. Geom. Dedicata 12, 75–85 (1982)
Cardinali I., De Bruyn B., Pasini A.: Minimal full polarized embeddings of dual polar spaces. J. Algebraic Combin. 25, 7–23 (2007)
Cooperstein B.N.: On the generation of dual polar spaces of unitary type over finite fields. Eur. J. Combin. 18, 849–856 (1997)
De Bruyn B.: On near hexagons and spreads of generalized quadrangles. J. Algebraic Combin. 11, 211–226 (2000)
De Bruyn B.: The generating rank of the U 4(3) near hexagon. Discrete Math. 307, 2900–2905 (2007)
De Bruyn, B.: The universal embedding of the near polygon \({\mathbb{G}_n}\). Electron. J. Comb. 14. Research paper R39, 12p. (2007)
De Bruyn B.: On the Grassmann-embeddings of the Hermitian dual polar spaces. Linear Multilinear Algebra 56, 665–677 (2008)
De Bruyn B.: Two new classes of hyperplanes of the dual polar space DH(2n−1, 4) not arising from the Grassmann-embedding. Linear Algebra Appl. 429, 2030–2045 (2008)
De Bruyn B.: The hyperplanes of DW(5, 2h) which arise from embedding. Discrete Math. 309, 304–321 (2009)
De Bruyn, B.: Dual embeddings of dense near polygons. Ars Combinatoria (to appear)
De Bruyn B., Pralle H.: The exceptional hyperplanes of DH(5, 4). Eur. J. Combin. 28, 1412–1418 (2007)
De Bruyn B., Pralle H.: The hyperplanes of DH(5, q 2). Forum Math. 20, 239–264 (2008)
De Bruyn B., Pralle H.: The hyperplanes of the near hexagon on the 2-factors of the complete graph K 8. Discrete Math. 308, 5656–5671 (2008)
De Bruyn, B., Shpectorov, S.: The hyperplanes of the near hexagon related to the extended ternary Golay code. (in preparation)
De Bruyn B., Vandecasteele P.: Valuations of near polygons. Glasg. Math. J. 47, 347–361 (2005)
De Bruyn B., Vandecasteele P.: The distance-2-sets of the slim dense near hexagons. Ann. Comb. 10, 193–210 (2006)
The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.4.12 (http://www.gap-system.org) (2008)
Kantor W.M.: Some geometries that are almost buildings. Eur. J. Combin. 2, 239–247 (1981)
Li P.: On the universal embedding of the U 2n (2) dual polar space. J. Combin. Theory Ser. A 98, 235–252 (2002)
Pasini A., Shpectorov S.: Uniform hyperplanes of finite dual polar spaces of rank 3. J. Combin. Theory Ser. A 94, 276–288 (2001)
Payne S.E., Thas J.A.: Finite Generalized Quadrangles. Research Notes in Mathematics, vol. 110. Pitman, Boston (1984)
Pralle H.: The hyperplanes of DW(5, 2). Experiment. Math. 14, 373–384 (2005)
Ronan M.A.: Embeddings and hyperplanes of discrete geometries. Eur. J. Combin. 8, 179–185 (1987)
Ronan, M.A., Smith, S.D.: 2-Local geometries for some sporadic groups. The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979). Proc. Sympos. Pure Math. 37, 283–289 (1980)
Shult E.E.: On Veldkamp lines. Bull. Belg. Math. Soc. Simon Stevin 4, 299–316 (1997)
Yoshiara S.: Embeddings of flag-transitive classical locally polar geometries of rank 3. Geom. Dedicata 43, 121–165 (1992)
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De Bruyn, B., Shpectorov, S. The Hyperplanes of the U 4(3) Near Hexagon. Graphs and Combinatorics 26, 647–671 (2010). https://doi.org/10.1007/s00373-010-0926-9
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DOI: https://doi.org/10.1007/s00373-010-0926-9