GEOMETRIC OBJECTS WITH A MORE COMBINATORIAL FLAVOR

This research is a synergy between algebra, geometry and combinatorics. We are studying geometric objects, defined over finite fields, with a more combinatorial flavor and present the results of the investigation of classification problems in geometry and combinatorics. Objects called BLT-sets, living in a vector space over a finite field, are of great interest to finite geometry, as they provide access to most of the objects that have been studied for a long time (translation planes, generalized quadrangles, flocks). On the other hand, there are objects that are invariant under a finite group. An example are the root systems and have been classified. The groups act as permutations on the roots and Coxeter groups turn out as symmetry groups of BLT-sets. For example, the automorphism group of BLT sets in characteristic 23 and 47 is a Coxeter group of type F4 or order 1152 (or closely related to this group). Similar behavior can be found with other examples. In terms of geometry, we consider an example of a transitive BLT-set in the finite field of order 67 and present the results of two pairs of elements in the 2-dimensional projective linear groups corresponding to the generators for the groups of order 17 and 4, respectively.