### 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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**
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