Project title

"ABELIAN AND NON-ABELIAN REPRESENTATIONS
OF CLASSICAL AND SPORADIC GEOMETRIES

EPSRC grant reference number

GR/L16606

Principal investigator name

Dr. A.A. Ivanov

Objectives

  • To determine the abelian and non-abelian universal representations
    of the symplectic and unitary dual polar spaces of GF(2)-type.
  • To develop uniform methods for studying representations of classical
    and sporadic geometries based on the principle of simple connectedness.
  • To study the universal natural representations of the dual parapolar
    space for the Monster group.

    Summary of results

  • We have proved a reduction theorem for the
    universal abelian representations of the symplectic and unitary dual
    polar spaces of GF(2)-type. This theorem shows that the universality
    problem for the known representations is equivalent to a specific
    question about the structure of the permutational GF(2)-modules of
    the groups L_n(4) and L_n(2) acting respectively on 1- and 2-dimensional
    subspaces in their natural modules. Our reduction theorem was used by
    P.McClurg and P.Li to complete the proof of the Brouwer conjecture about
    the dimension of the universal abelian representation of the dual
    polar spaces of symplectic type.
  • We have proved that the universal non-abelian representation groups of
    the GF(3)-type extended dual polar spaces of the largest Fischer
    3-transposition group Fi_24 and of the Monster group M are
    isomorphic respectively to 3.Fi_24' x 3.Fi_24'and M x M. Within this
    project we have developed a uniform machinery for calculating the
    universal non-abelian representations using simple connectedness
    results for the geometries under consideration. We have also achieved
    a characterization of the Petersen-type geometry of the McLaughlin
    sporadic group McL and checked that its universal abelian representation
    of the geometry is supported by the Leech lattice modulo 2.
  • We have realized an important step in studying the c-extensions
    ofthe F_4-geometries of GF(2)-type by characterizing the
    c.F_4(1)-geometries associated with the Fischer group Fi_22 and
    to its triple cover cover 3.Fi_22. In this characterization
    we did not assume any non-trivial automorphisms of the geometries but
    rather a certain combinatorial condition of regularity.

    EPSRC support is gladly acknowledged

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