On collineations and dualities of finite generalized polygons

Abstract

In this paper we generalize a result of Benson to all finite generalized polygons. In particular, given a collineation θ of a finite generalized polygon S, we obtain a relation between the parameters of S and, for various natural numbers i, the number of points x which are mapped to a point at distance i from x by θ. As a special case we consider generalized 2n-gons of order (1, t) and determine, in the generic case, the exact number of absolute points of a given duality of the underlying generalized n-gon of order t.

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References

  1. [1]

    C. T. Benson: On the structure of generalized quadrangles, J. Algebra15 (1970), 443–454.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    A. E. Brouwer, A. M. Cohen and A. Neumaier: Distance-Regular Graphs, Ergeb. Math. Grenzgeb. (3) 18, Springer-Verlag, Berlin, 1989.

    MATH  Google Scholar 

  3. [3]

    P. J. Cameron, J. A. Thas and S. E. Payne: Polarities of generalized hexagons and perfect codes, Geom. Dedicata5 (1976), 525–528.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    J. De Kaey and H. Van Maldeghem: A characterization of the split Cayley generalized hexagon H(q) using one subhexagon of order (1, q), Discr. Math.294 (2005), 109–118.

    MATH  Article  Google Scholar 

  5. [5]

    A. De Wispelaere and H. Van Maldeghem: Regular partitions of (weak) finite generalized polygons, Designs, Codes and Crypt.47(1–3) (2008), 53–73.

    Article  Google Scholar 

  6. [6]

    W. Feit and D. Higman: The nonexistence of certain generalized polygons, J. Algebra1 (1964), 114–131.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    J. B. Fraleigh: A First Course in Abstract Algebra, Addison-Wesley, 1994.

  8. [8]

    A. Offer: On the order of a generalized hexagon admitting an ovoid or spread, J. Combin. Theory Ser. A97 (2002), 184–186.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    S. E. Payne and J. A. Thas: Finite Generalized Quadrangles, Research Notes in Mathematics 110, Pitman Advanced Publishing Program, Boston-London-Melbourne, 1984.

    MATH  Google Scholar 

  10. [10]

    Cs. Schneider and H. Van Maldeghem: Primitive flag-transitive generalized hexagons and octagons, J. Combin. Theory Ser. A115(8) (2008), 1436–1455.

    MATH  Article  MathSciNet  Google Scholar 

  11. [11]

    J. A. Thas: A restriction on the parameters of a subhexagon, J. Combin. Theory Ser. A21 (1976), 115–117.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

    J. A. Thas: Generalized polygons, in: Handbook of Incidence Geometry, Buildings and Foundations (F. Buekenhout, ed.), Chapter 9, North-Holland, 383–431.

  13. [13]

    J. Tits: Sur la trialité et certains groupes qui s’en déduisent, Inst. Hautes Études Sci. Publ. Math.2 (1959), 13–60.

    MATH  Article  Google Scholar 

  14. [14]

    J. Tits: Classification of buildings of spherical type and Moufang polygons: a survey; in: Coll. Intern. Teorie Combin. Acc. Naz. Lincei, Proceedings; Roma 1973, Atti dei convegni Lincei17 (1976), 229–246.

  15. [15]

    H. Van Maldeghem: Generalized Polygons, Birkhäuser Verlag, Basel, Boston, Berlin, Monographs in Mathematics93, 1998.

    MATH  Google Scholar 

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Correspondence to Beukje Temmermans.

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Research supported by the Fund for Scientific Research — Flanders (FWO — Vlaanderen).

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Temmermans, B., Thas, J.A. & Van Maldeghem, H. On collineations and dualities of finite generalized polygons. Combinatorica 29, 569–594 (2009). https://doi.org/10.1007/s00493-009-2435-0

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Mathematics Subject Classification (2000)

  • 51E12