Linear representations of subgeometries
| العنوان: | Linear representations of subgeometries |
|---|---|
| المؤلفون: | Sara Rottey, Geertrui Van de Voorde, Stefaan De Winter |
| المساهمون: | Mathematics-TW, Mathematics, Algebra |
| بيانات النشر: | Springer Netherlands, 2014. |
| سنة النشر: | 2014 |
| مصطلحات موضوعية: | Discrete mathematics, Collineation, Group (mathematics), Applied Mathematics, automorphism group, Automorphism, linear representation, coset geometry, Computer Science Applications, Combinatorics, Hyperplane, subgeometry, FOS: Mathematics, Mathematics - Combinatorics, Coset, Projective space, Isomorphism, Combinatorics (math.CO), Algebraic number, Mathematics |
| 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. |
| اللغة: | English |
| URL الوصول: | https://explore.openaire.eu/search/publication?articleId=doi_dedup___::69b790283e392d1a23c89dd031626422 https://hdl.handle.net/20.500.14017/f348d261-bcd2-40dd-b148-0e0d1141a1aa |
| Rights: | OPEN |
| رقم الانضمام: | edsair.doi.dedup.....69b790283e392d1a23c89dd031626422 |
| قاعدة البيانات: | OpenAIRE |
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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. 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