The isomorphism problem for linear representations and their graphs The isomorphism problem for linear representations and their graphs

Abstract

Abstract A linear representation T*n(K) of a point set K is a point-line geometry, embedded in a projective space PG(n+1; q), where K is contained in a hyperplane. We put constraints on K which ensure that every automorphism of T*n(K) is induced by a collineation of the ambient projective space. This allows us to show that, under certain conditions, two linear representations T*n(K) and T*n(K′) are isomorphic if and only if the point sets K and K′ are PΓL-equivalent. We also deal with the slightly more general problem of isomorphic incidence graphs of linear representations. In the last part of this paper, we give an explicit description of the group of automorphisms of T*n(K) that are induced by collineations of PG(n + 1; q).