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Diagrams

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Diagram Geometry

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Abstract

A diagram is a structure defined on a set of types I. This structure generally is close to a labelled graph and provides information on the isomorphism class of residues of rank two of geometries over I. This way diagrams lead naturally to classification questions like all residually connected geometries pertaining to a given diagram.

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References

  1. M. Aschbacher, Finite geometries of type C 3 with flag-transitive groups. Geom. Dedic. 16, 195–200 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. C.T. Benson, N.E. Losey, On a graph of Hoffman and Singleton. J. Comb. Theory, Ser. B 11, 67–79 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Blass, F. Harary, Z. Miller, Which trees are link graphs? J. Comb. Theory, Ser. B 29, 277–292 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Blokhuis, A.E. Brouwer, D. Buset, A.M. Cohen, The locally icosahedral graphs, in Finite Geometries, Winnipeg, Man., 1984. Lecture Notes in Pure and Appl. Math., vol. 103 (Dekker, New York, 1985), pp. 19–22

    Google Scholar 

  5. A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-Regular Graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 18 (Springer, Berlin, 1989)

    Book  MATH  Google Scholar 

  6. F. Bruhat, J. Tits, Groupes réductifs sur un corps local. Publ. Math. Inst. Hautes Études Sci. 5–251 (1972)

    Google Scholar 

  7. F. Buekenhout, Diagrams for geometries and groups. J. Comb. Theory, Ser. A 27, 121–151 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. F. Buekenhout, The basic diagram of a geometry, in Geometries and Groups, Berlin, 1981. Lecture Notes in Math., vol. 893 (Springer, Berlin, 1981), pp. 1–29

    Chapter  Google Scholar 

  9. F. Buekenhout, (g, d ∗, d)-gons, in Finite Geometries, Pullman, Wash., 1981. Lecture Notes in Pure and Appl. Math., vol. 82 (Dekker, New York, 1981), pp. 93–111

    Google Scholar 

  10. F. Buekenhout, X. Hubaut, Locally polar spaces and related rank 3 groups. J. Algebra 45, 391–434 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  11. A.M. Cohen, Finite complex reflection groups. Ann. Sci. Éc. Norm. Super. (4) 9, 379–436 (1976)

    MATH  Google Scholar 

  12. A.M. Cohen, Exceptional presentations of three generalized hexagons of order 2. J. Comb. Theory, Ser. A 35, 79–88 (1983)

    Article  MATH  Google Scholar 

  13. A.M. Cohen, B.N. Cooperstein, On the local recognition of finite metasymplectic spaces. J. Algebra 125, 348–366 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. A.M. Cohen, E.J. Postma, Covers of point-hyperplane graphs. J. Algebr. Comb. 22, 317–329 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. A.M. Cohen, J. Tits, On generalized hexagons and a near octagon whose lines have three points. Eur. J. Comb. 6, 13–27 (1985)

    MathSciNet  MATH  Google Scholar 

  16. A.M. Cohen, H. Cuypers, H. Sterk (eds.), Some Tapas of Computer Algebra. Algorithms and Computation in Mathematics, vol. 4 (Springer, Berlin, 1999)

    MATH  Google Scholar 

  17. A.M. Cohen, H. Cuypers, R. Gramlich, Local recognition of non-incident point-hyperplane graphs. Combinatorica 25, 271–296 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. K. Coolsaet, A construction of the simple group of Rudvalis from the group U3(5):2. J. Group Theory 1, 143–163 (1998)

    MathSciNet  MATH  Google Scholar 

  19. H. Cuypers, A. Pasini, Locally polar geometries with affine planes. Eur. J. Comb. 13, 39–57 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Doyen, X. Hubaut, Finite regular locally projective spaces. Math. Z. 119, 83–88 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Gramlich, J.I. Hall, A. Straub, The local recognition of reflection graphs of spherical Coxeter groups. J. Algebr. Comb. 32, 1–14 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Hall, Projective planes. Trans. Am. Math. Soc. 54, 229–277 (1943)

    Article  MATH  Google Scholar 

  23. J.I. Hall, Locally Petersen graphs. J. Graph Theory 4, 173–187 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. J.I. Hall, E.E. Shult, Locally cotriangular graphs. Geom. Dedic. 18, 113–159 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  25. S. Heiss, Two sporadic geometries related to the Hoffman-Singleton graph. J. Comb. Theory, Ser. A 53, 68–80 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  26. D.G. Higman, C.C. Sims, A simple group of order 44,352,000. Math. Z. 105, 110–113 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  27. A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3. IBM J. Res. Dev. 4, 497–504 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  28. D.R. Hughes, F.C. Piper, Projective Planes. Graduate Texts in Mathematics, vol. 6 (Springer, New York, 1973)

    MATH  Google Scholar 

  29. A.A. Ivanov, Petersen and tilde geometries, in Geometry of Sporadic Groups. I. Encyclopedia of Mathematics and Its Applications, vol. 76 (Cambridge University Press, Cambridge, 1999)

    Chapter  Google Scholar 

  30. A.A. Ivanov, S.V. Shpectorov, Representations and amalgams, in Geometry of Sporadic Groups. II. Encyclopedia of Mathematics and Its Applications, vol. 91 (Cambridge University Press, Cambridge, 2002)

    Chapter  Google Scholar 

  31. L.O. James, A combinatorial proof that the Moore (7, 2) graph is unique. Util. Math. 5, 79–84 (1974)

    MATH  Google Scholar 

  32. T. Januszkiewicz, J. Świątkowski, Simplicial nonpositive curvature. Publ. Math. Inst. Hautes Études Sci. 1–85 (2006)

    Google Scholar 

  33. W.M. Kantor, Some geometries that are almost buildings. Eur. J. Comb. 2, 239–247 (1981)

    MathSciNet  MATH  Google Scholar 

  34. W.M. Kantor, Some exceptional 2-adic buildings. J. Algebra 92, 208–223 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  35. W.M. Kantor, Generalized polygons, scabs and gabs, in Buildings and the Geometry of Diagrams, Proc. C.I.M.E. Session, Como 1984. Springer Lecture Notes, vol. 1181 (Springer, Berlin, 1986), pp. 79–158

    Chapter  Google Scholar 

  36. W.M. Kantor, R. Liebler, J. Tits, On discrete chamber-transitive automorphism groups of affine buildings. Bull. Am. Math. Soc. 16, 129–133 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  37. W.M. Kantor, T. Meixner, M. Wester, Two exceptional 3-adic affine buildings. Geom. Dedic. 33, 1–11 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  38. P. Köhler, T. Meixner, M. Wester, The affine building of type \(\widetilde{\mathrm{A}}_{2}\) over a local field of characteristic two. Arch. Math. (Basel) 42, 400–407 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  39. P. Libois, Quelques espaces linéaires. Bull. Soc. Math. Belg. 16, 13–22 (1964)

    MathSciNet  MATH  Google Scholar 

  40. T. Meixner, A note on three rank four geometries for \(P\varOmega_{6}^{-}(3)\). J. Geom. 54, 84–90 (1995)

    MathSciNet  MATH  Google Scholar 

  41. T. Meixner, Buekenhout geometries of rank 3 which involve the Petersen graph. J. Math. Soc. Jpn. 48, 467–491 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  42. T. Meixner, F. Timmesfeld, Chamber systems with string diagrams. Geom. Dedic. 15, 115–123 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  43. A. Neumaier, Some sporadic geometries related to PG(3, 2). Arch. Math. (Basel) 42, 89–96 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Pasini, H. van Maldeghem, Some constructions and embeddings of the tilde geometry. Note Mat. 21, 1–33 (2002/2003)

    Google Scholar 

  45. S.E. Payne, J.A. Thas, Finite Generalized Quadrangles, 2nd edn. EMS Series of Lectures in Mathematics (European Mathematical Society (EMS), Zürich, 2009)

    Book  MATH  Google Scholar 

  46. G. Pickert, Projektive Ebenen. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol. LXXX (Springer, Berlin, 1955)

    Book  MATH  Google Scholar 

  47. M.A. Ronan, Triangle geometries. J. Comb. Theory 37, 294–319 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  48. R.D. Schafer, An Introduction to Nonassociative Algebras. Pure and Applied Mathematics, vol. 22 (Academic Press, New York, 1966)

    MATH  Google Scholar 

  49. T.A. Springer, F.D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups. Springer Monographs in Mathematics (Springer, Berlin, 2000)

    MATH  Google Scholar 

  50. R.P. Stanley, Enumerative Combinatorics. Volume 1, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 49 (Cambridge University Press, Cambridge, 2012)

    Google Scholar 

  51. K. Tent, Free polygons, twin trees and cat(1)-spaces. Pure Appl. Math. Q. 7, 1037–1052 (2011)

    MathSciNet  MATH  Google Scholar 

  52. J. Tits, Les groupes de Lie exceptionnels et leur interprétation géométrique. Bull. Soc. Math. Belg. 8, 48–81 (1956)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  54. J. Tits, Buildings of Spherical Type and Finite BN-Pairs. Lecture Notes in Mathematics, vol. 386 (Springer, Berlin, 1974)

    MATH  Google Scholar 

  55. J. Tits, R.M. Weiss, Moufang Polygons. Springer Monographs in Mathematics (Springer, Berlin, 2002)

    MATH  Google Scholar 

  56. H. van Maldeghem, Generalized Polygons. Monographs in Mathematics, vol. 93 (Birkhäuser, Basel, 1998)

    Book  MATH  Google Scholar 

  57. H. van Maldeghem, A geometric characterization of the perfect Ree-Tits generalized octagons. Proc. Lond. Math. Soc. (3) 76, 203–256 (1998)

    Article  Google Scholar 

  58. G.M. Weetman, A construction of locally homogeneous graphs. J. Lond. Math. Soc. (2) 50, 68–86 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  59. G.M. Weetman, Diameter bounds for graph extensions. J. Lond. Math. Soc. (2) 50, 209–221 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  60. R.M. Weiss, The Structure of Affine Buildings. Annals of Mathematics Studies, vol. 168 (Princeton University Press, Princeton, 2009)

    MATH  Google Scholar 

  61. M. Wester, Endliche fahnentransitive Tits-Geometrien und ihre universellen Überlagerungen. Mitt. Math. Sem. Giess. 170, 143 (1985)

    MathSciNet  Google Scholar 

  62. S. Yoshiara, The flag-transitive C 3-geometries of finite order. J. Algebr. Comb. 5, 251–284 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dépt. Mathématique, Université Libre de Bruxelles, Académie royale de Belgique, Bruxelles, Belgium

    Francis Buekenhout

  2. Dept. Mathematics & Computer Science, Eindhoven University of Technology, Eindhoven, Netherlands

    Arjeh M. Cohen

Authors
  1. Francis Buekenhout
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  2. Arjeh M. Cohen
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Buekenhout, F., Cohen, A.M. (2013). Diagrams. In: Diagram Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 57. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34453-4_2

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