| الوصف: |
The linear representation $$T_n^*(\mathcal {K})$$Tn?(K) of a point set $$\mathcal {K}$$K in a hyperplane of $$\mathrm {PG}(n+1,q)$$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 $$T_n^*(\mathcal {K})$$Tn?(K) and $$T_n^*(\mathcal {K}')$$Tn?(K?), under a few conditions on $$\mathcal {K}$$K and $$\mathcal {K}'$$K?. First, we prove that an isomorphism between $$T_n^*(\mathcal {K})$$Tn?(K) and $$T_n^*(\mathcal {K}')$$Tn?(K?) is induced by an isomorphism between the two linear representations $$T_n^*(\overline{\mathcal {K}})$$Tn?(K¯) and $$T_n^*(\overline{\mathcal {K}'})$$Tn?(K?¯) of their closures $$\overline{\mathcal {K}}$$K¯ and $$\overline{\mathcal {K}'}$$K?¯. This allows us to focus on the automorphism group of a linear representation $$T_n^*(\mathcal {S})$$Tn?(S) of a subgeometry $$\mathcal {S}\cong \mathrm {PG}(n,q)$$S?PG(n,q) embedded in a hyperplane of the projective space $$\mathrm {PG}(n+1,q^t)$$PG(n+1,qt). To this end we introduce a geometry $$X(n,t,q)$$X(n,t,q) and determine its automorphism group. The geometry $$X(n,t,q)$$X(n,t,q) is a straightforward generalization of $$H_{q}^{n+2}$$Hqn+2 which is known to be isomorphic to the linear representation of a Baer subgeometry. By providing an elegant algebraic description of $$X(n,t,q)$$X(n,t,q) as a coset geometry we extend this result and prove that $$X(n,t,q)$$X(n,t,q) and $$T_n^*(\mathcal {S})$$Tn?(S) are isomorphic. Finally, we compare the full automorphism group of $$T^*_n(\mathcal {S})$$Tn?(S) with the "natural" group of automorphisms that is induced by the collineation group of its ambient space. |