2014/07/15 by Stefaan De Winter, De Winter, Stefaan, Sara Rottey +3
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO
paper · pdf · doi:10.48550/arxiv.1407.3953
arxiv created 2014/07/15 · arxiv updated 2014/07/16
The linear representation Tn^*(K) of a point set K in a hyperplane of PG(n+1,q) is a point-line geometry embedded in this projective space. In this paper, we will determine the isomorphisms between two linear representations Tn^*(K) and Tn^*(K'), under a few conditions on K and K'. First, we prove that an isomorphism between Tn^*(K) and Tn^*(K') is induced by an isomorphism between the two linear representations Tn^*(K) and Tn^*(K') of their closures K and K'. This allows us to focus on the automorphism group of a linear representation Tn^*(S) of a subgeometry S\congPG(n,q) embedded in a hyperplane of the projective space PG(n+1,qt). To this end we introduce a geometry X(n,t,q) and determine its automorphism group. The geometry X(n,t,q) is a straightforward generalization of Hqn+2 which is known to be isomorphic to the linear representation of a Baer subgeometry. By providing an elegant algebraic description of X(n,t,q) as a coset geometry we extend this result and prove that X(n,t,q) and Tn^*(S) are isomorphic. Finally, we compare the full automorphism group of T^*n(S) with the "natural" group of automorphisms that is induced by the collineation group of its ambient space.