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GAP 4 Package Fining GAP 4 Package FinInG Finite Incidence Geometry 1.4.1 31-Mar-2018 John Bamberg, Anton Betten, Philippe Cara, Jan De Beule, Michel Lavrauw, Max Neunhoeffer. John Bamberg, Anton Betten, Philippe Cara, Jan De Beule, Michel Lavrauw, Max Neunhoeffer. Email: [email protected] Homepage: http://www.fining.org GAP 4 Package FinInG 2 Copyright © 2014-2018 by the authors This package may be distributed under the terms and conditions of the GNU Public License Version 2 or higher. The development group of FinInG welcomes contact with users. In case you have obtained the package as a deposited package part of archive during the installation of GAP, we call on your beneficence to register at http://www.fining.org when you use FinInG, or to tell us by sending an e-mail to [email protected]. Please also tell us about the use of FinInG in your research or teaching. We are very interested in results obtained using FinInG and we might refer to your work in the future. If your work is published, we ask you to cite FinInG like a journal article or book. We provide the necessary BibTex and LaTeX code in Section 1.2. Acknowledgements The development phase of FinInG started roughly in 2005. The idea to write a package for projective geometry in GAP had emerged before, and resulted in pg, a relic that still can be found in the undeposited packages section of the GAP website. One of the main objectives was to develop the new package was to create a tighter connection between finite geometry and the group theoretical functions present in GAP. The authors thank Michael Pauley, Maska Law and Sven Reichard, for their contributions during the early days of the project. Jan De Beule and Michel Lavrauw have been supported by the Research Foundation Flanders – Belgium (FWO), and John Bamberg has been supported by a Marie Curie grant and an ARC grant during almost the whole development phase of FinInG. The authors are grateful for this support. John Bamberg, Philippe Cara and Jan De Beule have spent several weeks in Vicenza while developing FinInG. We acknowledge the hospitality of Michel Lavrauw and Corrado Zanella. Our stays in Vicenza were always fruitful and very enjoyable. During or daily morning and afternoon coffee breaks, we discussed several topics, while enjoying coffee in Caffe Pigaffeta. Although one cannot acknowledge everybody, such as the bakery who provided us with the sandwiches, we acknowledge very much the hospitality of Carla and Luigi, who introduced us to many different kinds of coffees from all over the world. Contents 1 Introduction 6 1.1 Philosophy.......................................6 1.2 How to cite FinInG ...................................6 1.3 Overview of this manual................................7 1.4 Getting and installing FinInG ..............................7 1.5 The Development Team................................. 13 2 Examples 16 2.1 Elementary examples.................................. 16 2.2 Some objects with interesting combinatorial properties................ 21 2.3 Geometry morphisms.................................. 25 2.4 Some geometrical objects................................ 30 2.5 Some particular incidence geometries......................... 31 2.6 Elation generalised quadrangles............................ 35 2.7 Algebraic varieties................................... 37 3 Incidence Geometry 40 3.1 Incidence structures................................... 40 3.2 Elements of incidence structures............................ 45 3.3 Flags of incidence structures.............................. 51 3.4 Shadow of elements................................... 55 3.5 Enumerating elements of an incidence structure.................... 57 3.6 Lie geometries..................................... 64 3.7 Elements of Lie geometries............................... 66 3.8 Changing the ambient geometry of elements of a Lie geometry............ 68 4 Projective Spaces 70 4.1 Projective Spaces and basic operations......................... 70 4.2 Subspaces of projective spaces............................. 72 4.3 Shadows of Projective Subspaces............................ 82 4.4 Enumerating subspaces of a projective space...................... 84 5 Projective Groups 86 5.1 Projectivities, collineations and correlations of projective spaces............ 87 5.2 Construction of projectivities, collineations and correlations.............. 90 5.3 Basic operations for projectivities, collineations and correlations of projective spaces 93 5.4 The groups PGL, PGL, and PSL in FinInG ...................... 96 3 GAP 4 Package FinInG 4 5.5 Basic operations for projective groups......................... 99 5.6 Natural embedding of a collineation group in a correlationcollineation group..... 99 5.7 Basic action of projective group elements....................... 100 5.8 Projective group actions................................. 100 5.9 Special subgroups of the projectivity group...................... 102 5.10 Nice Monomorphisms................................. 105 6 Polarities of Projective Spaces 108 6.1 Creating polarities of projective spaces......................... 108 6.2 Operations, attributes and properties for polarities of projective spaces........ 110 6.3 Polarities, absolute points, totally isotropic elements and finite classical polar spaces 113 6.4 Commuting polarities.................................. 115 7 Finite Classical Polar Spaces 116 7.1 Finite Classical Polar Spaces.............................. 116 7.2 Canonical and standard Polar Spaces.......................... 118 7.3 Basic operations for finite classical polar spaces.................... 124 7.4 Subspaces of finite classical polar spaces........................ 128 7.5 Basic operations for polar spaces and subspaces of projective spaces......... 130 7.6 Shadow of elements................................... 135 7.7 Projective Orthogonal/Unitary/Symplectic groups in FinInG ............. 137 7.8 Enumerating subspaces of polar spaces......................... 140 8 Orbits, stabilisers and actions 142 8.1 Orbits.......................................... 142 8.2 Stabilisers........................................ 145 8.3 Actions and nice monomorphisms revisited...................... 151 9 Affine Spaces 155 9.1 Affine spaces and basic operations........................... 155 9.2 Subspaces of affine spaces............................... 157 9.3 Shadows of Affine Subspaces.............................. 162 9.4 Iterators and enumerators................................ 163 9.5 Affine groups...................................... 164 9.6 Low level operations.................................. 167 10 Geometry Morphisms 168 10.1 Geometry morphisms in FinInG............................ 168 10.2 Type preserving bijective geometry morphisms.................... 170 10.3 Klein correspondence and derived dualities...................... 171 10.4 Embeddings of projective spaces............................ 176 10.5 Embeddings of polar spaces.............................. 180 10.6 Projections....................................... 187 10.7 Projective completion.................................. 187 GAP 4 Package FinInG 5 11 Algebraic Varieties 189 11.1 Algebraic Varieties................................... 189 11.2 Projective Varieties................................... 191 11.3 Quadrics and Hermitian varieties............................ 192 11.4 Affine Varieties..................................... 196 11.5 Geometry maps..................................... 196 11.6 Segre Varieties..................................... 197 11.7 Veronese Varieties.................................... 198 11.8 Grassmann Varieties.................................. 200 12 Generalised Polygons 202 12.1 Categories........................................ 202 12.2 Generic functions to create generalised polygons................... 204 12.3 Attributes and operations for generalised polygons.................. 207 12.4 Elements of generalised polygons........................... 216 12.5 The classical generalised hexagons........................... 221 12.6 Elation generalised quadrangles............................ 227 13 Coset Geometries and Diagrams 241 13.1 Coset Geometries.................................... 241 13.2 Automorphisms, Correlations and Isomorphisms................... 253 13.3 Diagrams........................................ 255 A The structure of FinInG 262 A.1 The different components................................ 262 A.2 The complete inventory................................. 262 A.3 The filter graph(s).................................... 297 B The finite classical groups in FinInG 298 B.1 Standard forms used to produce the finite classical groups............... 298 B.2 Direct commands to construct the projective classical groups in FinInG ....... 300 B.3 Basis of the collineation groups............................. 304 C Low level functions for morphisms 305 C.1 Field reduction and vector spaces............................ 305 C.2 Field reduction and forms................................ 306 C.3 Low level functions................................... 307 References 310 Index 311 Chapter 1 Introduction 1.1 Philosophy FinInG (pronounciation: [fInIN]) is a package for computation in Finite Incidence Geometry. It pro- vides users with the basic tools to work in various areas of finite geometry from the realms of pro- jective spaces to the flat lands of generalised polygons. The algebraic power of GAP is employed, particularly in its facility with matrix and permutation groups. 1.2 How to cite FinInG The development group of
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