Linear representations of subgeometries

Abstract

The linear representation \(T_n^*(\mathcal {K})\) of a point set \(\mathcal {K}\) in a hyperplane of \(\mathrm {PG}(n+1,q)\) is a point-line geometry embedded in this projective space. In this paper, we will determine the isomorphisms between two linear representations \(T_n^*(\mathcal {K})\) and \(T_n^*(\mathcal {K}')\), under a few conditions on \(\mathcal {K}\) and \(\mathcal {K}'\). First, we prove that an isomorphism between \(T_n^*(\mathcal {K})\) and \(T_n^*(\mathcal {K}')\) is induced by an isomorphism between the two linear representations \(T_n^*(\overline{\mathcal {K}})\) and \(T_n^*(\overline{\mathcal {K}'})\) of their closures \(\overline{\mathcal {K}}\) and \(\overline{\mathcal {K}'}\). This allows us to focus on the automorphism group of a linear representation \(T_n^*(\mathcal {S})\) of a subgeometry \(\mathcal {S}\cong \mathrm {PG}(n,q)\) embedded in a hyperplane of the projective space \(\mathrm {PG}(n+1,q^t)\). 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 \(H_{q}^{n+2}\) 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 \(T_n^*(\mathcal {S})\) are isomorphic. Finally, we compare the full automorphism group of \(T^*_n(\mathcal {S})\) with the “natural” group of automorphisms that is induced by the collineation group of its ambient space.