On the Generation of Some Embeddable GF(2) Geometries
B.N. Cooperstein
DOI: 10.1023/A:1008719600293
Abstract
The generating rank is determined for several GF(2)-embeddable geometries and it is demonstrated that their generating and embedding ranks are equal. Specifically, we prove that each of the two generalized hexagons of order (2, 2) has generating rank 14, that the central involution geometry of the Hall-Janko sporadic group has generating rank 28, and that the dual polar space DU(6,2) has generating rank 22. We also include a survey of all instances in which either the generating or embedding rank of an embeddable GF(2) geometry is known.
Pages: 15–28
Keywords: point-line geometry; embeddable geometry; embedding rank; generating rank
Full Text: PDF
References
1. M.K. Bardoe, “On the universal embedding of the near-hexagon for U4(3),” Geometriae Dedicata 56 (1995), 7-17. COOPERSTEIN
2. M.K. Bardoe, “The universal embedding for the for U4(3) involution geometry,” Journal of Algebra 186 (1996), 368-383.
3. M.K. Bardoe, “The universal embedding for the involution geometry of the Suzuki Sporadic group,” Journal of Algebra 186 (1996), 447-460.
4. M.K. Bardoe, “The universal embedding for the involution geometry of Co1,” Journal of Algebra 217 (1999), 555-572.
5. R.J. Blok and A.E. Brouwer, “Spanning point-line geometries in buildings of spherical type,” Journal of Geometry 62 (1998), 26-35.
6. A.E. Brouwer, personal communication.
7. A.E. Brouwer, A.M. Cohen, A.M. Hall, and H. Wilbrink, “Near polygons and fischer spaces,” Geometriae Dedicata 49 (1994), 349-368.
8. F. Buekenhout (Ed.), Handbook of Incidence Geometry, North Holland, Amsterdam, 1995.
9. A. Cohen and J. Tits, “On generalized hexagons and a near octagon whose lines have three points,” European Journal of Combinatorics 6 (1985), 13-27.
10. J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
11. B.N. Cooperstein, “The geometry of root subgroups in exceptional groups, I,” Geometriae Dedicata 8 (1979), 317-381.
12. B.N. Cooperstein, “On the generation of dual polar spaces of symplectic type over GF(2),” European Journal of Combinatorics 18 (1997), 741-749.
13. B.N. Cooperstein, “On the generation of dual polar spaces of unitary type over finite fields,” European Journal of Combinatorics 18 (1997), 849-856.
14. B.N. Cooperstein, “Generating long root subgroup geometries of classical groups over finite prime fields,” Bulletin of the Belgium Mathematics Society 5 (1998), 531-548.
15. B.N. Cooperstein and E.E. Shult, “Combinatorial construction of some near polygons,” Journal of Combinatorial Theory, Ser. A 78 (1997), 120-140.
16. B.N. Cooperstein and E.E. Shult, “Frames and bases of lie incidence geometries,” Journal of Geometry 60 (1997), 17-46.
17. D. Frohardt and P. Johnson, “Geometric hyperplanes in generalized hexagons of order (2, 2),” Communications in Algebra 22 (1994), 773-797.
18. D. Frohardt and S.D. Smith, “Universal embedding for the 3D4(2) hexagon and J2 near-octagon,” European Journal of Combinatorics 13 (1992), 455-472.
19. J.I. Hall, “Linear representations of a cotriangular space,” Linear Algebra and its Applications 49 (1983), 257-273.
20. S. Heiss, A note on embeddable F2-geometries. Preprint.
21. W.M. Kantor, “Subgroups of classical groups generated by long root elements,” Transactions of the American Mathematical Society 248 (1979), 347-379.
22. S.E. Payne and J.A. Thas, Finite Generalized Hexagons, Pitman, London, 1984.
23. M. Ronan and S.D. Smith, “Sheaves on buildings and modular representations of Chevalley groups,” Journal of Algebra 96 (1985), 319-346.
24. E.E. Shult, “Generalized hexagons as geometric hyperplanes of near hexagons,” In Groups, Combinatorics and Geometry, M.W. Liebeck and J. Saxl (Eds.), London Mathematics Society, 1992, pp. 229-239.
25. E.E. Shult and A. Yanushka,“Near n-gons and line systems,” Geometriae Dedicata 12 (1980), 1-72.
26. J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Springer-Verlag, Berlin, 1974.
27. H. V\ddot olklein, “On the geometry of the adjoint representation of a Chevalley group,” Journal of Algebra 127 (1989), 139-154.
28. A. Wells, “Universal projective embeddings of the Grassmannian, half spinor and dual orthogonal geometries,” Quarterly Journal of Mathematics 34 (1983), 375-386.
29. S. Yoshiara, “Embeddings of flag-transitive classical locally polar geometries of rank 3,” Geometriae Dedicata 43 (1992), 121-165.
2. M.K. Bardoe, “The universal embedding for the for U4(3) involution geometry,” Journal of Algebra 186 (1996), 368-383.
3. M.K. Bardoe, “The universal embedding for the involution geometry of the Suzuki Sporadic group,” Journal of Algebra 186 (1996), 447-460.
4. M.K. Bardoe, “The universal embedding for the involution geometry of Co1,” Journal of Algebra 217 (1999), 555-572.
5. R.J. Blok and A.E. Brouwer, “Spanning point-line geometries in buildings of spherical type,” Journal of Geometry 62 (1998), 26-35.
6. A.E. Brouwer, personal communication.
7. A.E. Brouwer, A.M. Cohen, A.M. Hall, and H. Wilbrink, “Near polygons and fischer spaces,” Geometriae Dedicata 49 (1994), 349-368.
8. F. Buekenhout (Ed.), Handbook of Incidence Geometry, North Holland, Amsterdam, 1995.
9. A. Cohen and J. Tits, “On generalized hexagons and a near octagon whose lines have three points,” European Journal of Combinatorics 6 (1985), 13-27.
10. J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
11. B.N. Cooperstein, “The geometry of root subgroups in exceptional groups, I,” Geometriae Dedicata 8 (1979), 317-381.
12. B.N. Cooperstein, “On the generation of dual polar spaces of symplectic type over GF(2),” European Journal of Combinatorics 18 (1997), 741-749.
13. B.N. Cooperstein, “On the generation of dual polar spaces of unitary type over finite fields,” European Journal of Combinatorics 18 (1997), 849-856.
14. B.N. Cooperstein, “Generating long root subgroup geometries of classical groups over finite prime fields,” Bulletin of the Belgium Mathematics Society 5 (1998), 531-548.
15. B.N. Cooperstein and E.E. Shult, “Combinatorial construction of some near polygons,” Journal of Combinatorial Theory, Ser. A 78 (1997), 120-140.
16. B.N. Cooperstein and E.E. Shult, “Frames and bases of lie incidence geometries,” Journal of Geometry 60 (1997), 17-46.
17. D. Frohardt and P. Johnson, “Geometric hyperplanes in generalized hexagons of order (2, 2),” Communications in Algebra 22 (1994), 773-797.
18. D. Frohardt and S.D. Smith, “Universal embedding for the 3D4(2) hexagon and J2 near-octagon,” European Journal of Combinatorics 13 (1992), 455-472.
19. J.I. Hall, “Linear representations of a cotriangular space,” Linear Algebra and its Applications 49 (1983), 257-273.
20. S. Heiss, A note on embeddable F2-geometries. Preprint.
21. W.M. Kantor, “Subgroups of classical groups generated by long root elements,” Transactions of the American Mathematical Society 248 (1979), 347-379.
22. S.E. Payne and J.A. Thas, Finite Generalized Hexagons, Pitman, London, 1984.
23. M. Ronan and S.D. Smith, “Sheaves on buildings and modular representations of Chevalley groups,” Journal of Algebra 96 (1985), 319-346.
24. E.E. Shult, “Generalized hexagons as geometric hyperplanes of near hexagons,” In Groups, Combinatorics and Geometry, M.W. Liebeck and J. Saxl (Eds.), London Mathematics Society, 1992, pp. 229-239.
25. E.E. Shult and A. Yanushka,“Near n-gons and line systems,” Geometriae Dedicata 12 (1980), 1-72.
26. J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Springer-Verlag, Berlin, 1974.
27. H. V\ddot olklein, “On the geometry of the adjoint representation of a Chevalley group,” Journal of Algebra 127 (1989), 139-154.
28. A. Wells, “Universal projective embeddings of the Grassmannian, half spinor and dual orthogonal geometries,” Quarterly Journal of Mathematics 34 (1983), 375-386.
29. S. Yoshiara, “Embeddings of flag-transitive classical locally polar geometries of rank 3,” Geometriae Dedicata 43 (1992), 121-165.