"ABELIAN AND NON-ABELIAN REPRESENTATIONS
OF CLASSICAL AND SPORADIC GEOMETRIES
EPSRC grant reference number
Principal investigator name
Dr. A.A. Ivanov
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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