The local recognition of reflection graphs of spherical Coxeter groups
Ralf Gramlich
, Jonathan I. Hall
and Armin Straub
DOI: 10.1007/s10801-009-0201-4
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.
Pages: 1–14
Keywords: keywords local recognition of graphs; Coxeter groups
Full Text: PDF
References
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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)
3. Brown, M., Connelly, R.: On graphs with a constant link, II. Discrete Math. 11, 199-232 (1975)
4. Buekenhout, F., Hubaut, X.: Locally polar spaces and related rank 3 groups. J. Algebra 45, 391-434 (1977)
5. Bourbaki, N.: Elements of Mathematics. Lie Groups and Lie Algebras: Chapters 4-6. Springer, Berlin (2002)
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)
7. Cohen, A.M., Cuypers, H., Gramlich, R.: Local recognition of non-incident point-hyperplane graphs. Combinatorica 25, 271-296 (2005)
8. Cohen, A.M., Shult, E.E.: Affine polar spaces. Geom. Dedicata 35, 43-76 (1990)
9. Cuypers, H., Pasini, A.: Locally polar geometries with affine planes. European J. Combin. 13, 39-57 (1992)
10. Gorenstein, D., Lyons, R., Solomon, R.: The Classification of the Finite Simple Groups. AMS, Providence (1994)
11. Gramlich, R.: Developments in finite Phan theory. Innov. Incidence Geom. To appear
12. Gramlich, R., Hoffman, C., Shpectorov, S.: A Phan-type theorem for Sp(2n, q). J. Algebra 264, 358- 384 (2003)
13. Gramlich, R., Witzel, S.: The sphericity of the Phan geometries of type Bn and Cn and the Phan-type theorem of type F4. Submitted.
14. Hall, J.I.: Locally Petersen graphs. J. Graph Theory 4, 173-187 (1980)
15. Hall, J.I.: Graphs with constant link and small degree or order. J. Graph Theory 8, 419-444 (1985)
16. Hall, J.I.: A local characterization of the Johnson scheme. Combinatorica 7, 77-85 (1987)
17. Harary, F.: Graph Theory. Westview Press (1994)
18. Hall, J.I., Shult, E.E.: Locally cotriangular graphs. Geom. Dedicata 18, 113-159 (1985)
19. Humphreys, J.E.: Remarks on “A theorem on special linear groups”. J. Algebra 22, 316-318 (1972)
20. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1992)
21. Mullineux, G.: A characterization of An by centralizers of short involutions. Quart. J. Math. Oxford Ser. 29, 213-220 (1978)
22. Pasechnik, D.V.: Geometric characterization of the sporadic groups Fi22, Fi23, and Fi24. J. Combin. Theory Ser. A 68, 100-114 (1994)
23. Phan, K.-W.: A theorem on special linear groups. J. Algebra 16, 509-518 (1970)
24. Phan, K.-W.: On groups genererated by three-dimensional special unitary groups, I. J. Austral. Math. Soc. Ser. A 23, 67-77 (1977)
25. Phan, K.-W.: On groups genererated by three-dimensional special unitary groups, II. J. Austral. Math. Soc. Ser. A 23, 129-146 (1977) J Algebr Comb (2010) 32: 1-14
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