Advertisement

Journal of Algebraic Combinatorics

, Volume 32, Issue 1, pp 1–14 | Cite as

The local recognition of reflection graphs of spherical Coxeter groups

  • Ralf Gramlich
  • Jonathan I. Hall
  • Armin Straub
Article

Abstract

Based on the third author’s thesis (arXiv:0805.2403) in this article we complete the local recognition of commuting reflection graphs of spherical Coxeter groups arising from irreducible crystallographic root systems.

Keywords

Local recognition of graphs Coxeter groups 

References

  1. 1.
    Altmann, K.: Centralisers of fundamental subgroups. PhD thesis, Technische Universität Darmstadt (2007) Google Scholar
  2. 2.
    Bennett, C.D., Gramlich, R., Hoffman, C., Shpectorov, S.: Odd-dimensional orthogonal groups as amalgams of unitary groups, part 1: general simple connectedness. J. Algebra 312, 426–444 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brown, M., Connelly, R.: On graphs with a constant link, II. Discrete Math. 11, 199–232 (1975) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Buekenhout, F., Hubaut, X.: Locally polar spaces and related rank 3 groups. J. Algebra 45, 391–434 (1977) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bourbaki, N.: Elements of Mathematics. Lie Groups and Lie Algebras: Chapters 4–6. Springer, Berlin (2002) zbMATHGoogle Scholar
  6. 6.
    Cohen, A.M.: Local recognition of graphs, buildings, and related geometries. In: Kantor W.M., Liebler R.A., Payne S.E., Shult E.E. (eds.) Finite Geometries, Buildings, and Related Topics, pp. 85–94. Oxford Science Publications, The Clarendon Press, New York (1990) Google Scholar
  7. 7.
    Cohen, A.M., Cuypers, H., Gramlich, R.: Local recognition of non-incident point-hyperplane graphs. Combinatorica 25, 271–296 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cohen, A.M., Shult, E.E.: Affine polar spaces. Geom. Dedicata 35, 43–76 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cuypers, H., Pasini, A.: Locally polar geometries with affine planes. European J. Combin. 13, 39–57 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gorenstein, D., Lyons, R., Solomon, R.: The Classification of the Finite Simple Groups. AMS, Providence (1994) zbMATHGoogle Scholar
  11. 11.
    Gramlich, R.: Developments in finite Phan theory. Innov. Incidence Geom. To appear Google Scholar
  12. 12.
    Gramlich, R., Hoffman, C., Shpectorov, S.: A Phan-type theorem for Sp(2n,q). J. Algebra 264, 358–384 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gramlich, R., Witzel, S.: The sphericity of the Phan geometries of type B n and C n and the Phan-type theorem of type F 4. Submitted. arXiv:0901.1156
  14. 14.
    Hall, J.I.: Locally Petersen graphs. J. Graph Theory 4, 173–187 (1980) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hall, J.I.: Graphs with constant link and small degree or order. J. Graph Theory 8, 419–444 (1985) CrossRefGoogle Scholar
  16. 16.
    Hall, J.I.: A local characterization of the Johnson scheme. Combinatorica 7, 77–85 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Harary, F.: Graph Theory. Westview Press (1994) Google Scholar
  18. 18.
    Hall, J.I., Shult, E.E.: Locally cotriangular graphs. Geom. Dedicata 18, 113–159 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Humphreys, J.E.: Remarks on “A theorem on special linear groups”. J. Algebra 22, 316–318 (1972) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1992) zbMATHGoogle Scholar
  21. 21.
    Mullineux, G.: A characterization of A n by centralizers of short involutions. Quart. J. Math. Oxford Ser. 29, 213–220 (1978) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Pasechnik, D.V.: Geometric characterization of the sporadic groups Fi22, Fi23, and Fi24. J. Combin. Theory Ser. A 68, 100–114 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Phan, K.-W.: A theorem on special linear groups. J. Algebra 16, 509–518 (1970) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Phan, K.-W.: On groups genererated by three-dimensional special unitary groups, I. J. Austral. Math. Soc. Ser. A 23, 67–77 (1977) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Phan, K.-W.: On groups genererated by three-dimensional special unitary groups, II. J. Austral. Math. Soc. Ser. A 23, 129–146 (1977) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    SAGE Mathematical Software: Version 2.8. http://www.sagemath.org (2007)
  27. 27.
    Shult, E.E.: Groups, polar spaces and related structures. In: Proc. Advanced Study Inst., Breukelen, 1975. Math. Centre Tracts, vol. 57, pp. 130–161. Math. Centrum, Amsterdam (1974) Google Scholar
  28. 28.
    Straub, A.: Local recognition of reflection graphs on Coxeter groups. Master’s thesis, Technische Universität Darmstadt. arXiv:0805.2403 (2008)
  29. 29.
    Timmesfeld, F.G.: The Curtis-Tits presentation. Adv. Math. 189, 38–67 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Weetman, G.: A construction of locally homogeneous graphs. J. London Math. Soc. 50, 68–86 (1994) zbMATHMathSciNetGoogle Scholar
  31. 31.
    Weetman, G.: Diameter bounds for graphs extensions. J. London Math. Soc. 50, 209–221 (1994) zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Ralf Gramlich
    • 1
    • 2
  • Jonathan I. Hall
    • 3
  • Armin Straub
    • 4
  1. 1.FB MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.School of MathematicsUniversity of BirminghamBirminghamUK
  3. 3.Department of MathematicsMichigan State UniversityEast LansingUSA
  4. 4.Department of MathematicsTulane UniversityNew OrleansUSA

Personalised recommendations