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.