Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Geometric techniques in representation theory

  • 34 Accesses

  • 1 Citations

Abstract

A discussion of results and conjectures, focussed around the extension of the modular representation theory for finite Lie-type groups to more general groups that act on building-like geometries.

This is a preview of subscription content, log in to check access.

References

  1. [A]

    M.G. Aschbacher, Overgroups of Sylow subgroups in sporadic groups. Memoirs A.M.S. no. 343. Providence RI, Amer. Math. Soc., 1986.

  2. [Bu]

    F. Buekenhout, Diagrams for geometries and groups. J. Comb. Th. A (1979), 121–151.

  3. [Cu]

    C.W. Curtis, Modular representations of finite groups with split (B,N)-pairs.. Lecture Notes in Math. 131 New York-Berlin, Springer-Verlag, 1970.

  4. [DGS]

    A. Delgado, D. Goldschmidt and B. Stellmacher, Groups and Graphs. Basel: Birkhäuser, 1985.

  5. [G]

    D. Goldschmidt, Automorphisms of trivalent graphs, Annals of Math. 111 (1980), 377–404.

  6. [K]

    W. Kantor, Some exceptional 2-adic buildings. J. Algebra 92 (1984), 208–223.

  7. [R]

    M. Ronan, Embeddings and hyperplanes of discrete geometries. To appear. (Preprint, University of Illinois, Chicago, 1985).

  8. [RS 1]

    M. Ronan and S. Smith, 2-Local geometries for some sporadic groups. Pp. 283–289 in Santa Cruz Proceedings: AMS. Symposia in Pure Math. no. 37 (eds. Cooperstein and Mason). Providence RI: Amer. Math. Soc., 1980.

  9. [RS 2]

    M. Ronan and S. Smith, Sheaves on buildings and modular representations of Chevalley groups. J. Algebra 96 (1985), 319–346.

  10. [RS 3]

    M. Ronan and S. Smith, Universal presheaves on group geometries, and modular representations. To appear in J. Algebra. (Preprint, University of Illinois, Chicago, 1984).

  11. [RS 4]

    M. Ronan and S. Smith, Computation of 2-modular sheaves and homology for L 4(2), A 7, 3S 6, and M 24. Preprint, University of Illinois, Chicago, 1986.

  12. [RSt]

    M. Ronan and G. Stroth, Minimal parabolic geometries for the sporadic groups. Europ. J. Combinatorics 5 (1984), 59–92.

  13. [Sm]

    S. Smith, Irreducible modules and parabolic subgroups. J. Algebra 75 (1982), 286–289.

  14. [Stb]

    R. Steinberg, Lectures on Chevalley groups. Mimeograph notes, Yale University, 1967.

  15. [Str]

    G. Stroth, Chamber systems, geometries and parabolic systems whose diagram contains only bonds of strength 1 and 2. Preprint No. 205, F.B. Mathematik, Freie Univ. Berlin, 1985.

  16. [Tm 1]

    F.G. Timmesfeld, Tits geometries and parabolic systems in finitely generated groups. Math. Z. 184 (1983), 377–396, 449–487.

  17. [Tm 2]

    F.G. Timmesfeld, Classical locally finite Tits chamber systems of rank 3 and characteristic 2. Preprint, Univ. Giessen, 1984.

  18. [Ti 1]

    J. Tits, Buildings of spherical type and finite BN-pairs. Springer Lecture Notes in Math. vol. 386; Springer-Verlag: Berlin-New York, 1974.

  19. [Ti 2]

    J. Tits, A local approach to buildings. Pp. 517–547 in “The Geometric Vein” (Coxeter Festschrift). Eds. Davis, Grunbaum, Sherk. Berlin-New York, Springer-Verlag, 1981.

Download references

Author information

Additional information

Based on a lecture given at the conference “Groups and Geometries-Finite and Algebraic,” on 26 March 1986 at Noordwijkerhout, Netherlands (a NATO Advanced Research Workshop).

Partially supported by NSF Grant MCS 83-00855.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Smith, S.D. Geometric techniques in representation theory. Geom Dedicata 25, 355–373 (1988). https://doi.org/10.1007/BF00191932

Download citation

Keywords

  • General Group
  • Representation Theory
  • Modular Representation
  • Geometric Technique
  • Modular Representation Theory