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Finite Incidence Geometry in GAP GAP in Algebraic Research Jan De Beule 21 November 2018 Contents Introduction Projective spaces Incidence Geometry Projective geometry over (skew)fields Groups of collineations of a projective space Polar spaces Geometry morphisms Generalized polygons Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 1/55 I Jacques Tits, geometry of finite simple groups. I many others .... Some history on (finite) incidence geometry I Beniamino Segre, classical algebraic geometry. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 2/55 I many others .... Some history on (finite) incidence geometry I Beniamino Segre, classical algebraic geometry. I Jacques Tits, geometry of finite simple groups. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 2/55 Some history on (finite) incidence geometry I Beniamino Segre, classical algebraic geometry. I Jacques Tits, geometry of finite simple groups. I many others .... Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 2/55 FinInG I Finite Incidence Geometry in GAP. I Development started in 2005. I Authors: John Bamberg, Anton Betten, Philippe Cara, Jan De Beule, Michel Lavrauw, and Max Neunhöffer I http://www.fining.org I https://bitbucket.org/jdebeule/fining I https://www.gap-system.org/Packages/ fining.html I support ’at’ fining.org Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 3/55 Projective spaces I Let K be a (skew)field. I Let V(n + 1; K) be a (left) vector space over K of dimension n + 1. Definition A projective space of dimension n over the skewfield K is the geometry consisting of the sub vector spaces of V(n + 1; K), with incidence being symmetrized set theoretic containment. We denote this projective space as PG(n; K). The points, lines, planes, ..., hyperplanes, respectively, are the 1-, 2-, 3-, ..., n-dimensional subspaces of V(n + 1; K). Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 4/55 My first projective space in GAP Figure: Load FinFinG Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 5/55 I Points I Lines I List I in My first projective space in GAP I GF I ProjectiveSpace, alternatively PG Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 6/55 I in My first projective space in GAP I GF I ProjectiveSpace, alternatively PG I Points I Lines I List Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 6/55 My first projective space in GAP I GF I ProjectiveSpace, alternatively PG I Points I Lines I List I in Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 6/55 My first projective space in GAP Figure: Fano plane in FinInG, file: myfirstpg.g Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 7/55 Incidence geometry Definition (based on e.g. [2]) An incidence structure is a tuple (S; T; t; I), such that I S a non empty set called the set of elements, I the set T is a finite, non empty set, called the set of types, I t : S! T, is a function and for x 2 S, t(x) is called its type, I the relation I is a symmetric relation on S, called the incidence relation, I two elements of the same type are never incident. The rank of an incidence structure (S; T; t; I) is jTj, the cardinality of T. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 8/55 Incidence geometry Definition A flag of an incidence structure (S; T; I) is a set F ⊂ S such that every two elements of F are incident. From the definition it follows immediately that a flag contains at most one element of a given type. Definition A flag F is maximal if it is not contained in a larger flag. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 9/55 Incidence geometry Definition An incidence structure is called an incidence geometry if every maximal flag contains an element of each type. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 10/55 A projective space as incidence geometry I Rank I TypesOfElementsOfIncidenceStructure I ElementsOfIncidenceStructure I Points, Lines, Planes, Solids, Hyperplanes I IsIncident Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 11/55 A projective space as incidence geometry file: pgincidence1.g Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 12/55 A projective space as incidence geometry file: pgincidence2.g Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 13/55 Some examples I The projective space PG(n; K) is an incidence geometry of rank n. I Define I S = f1; 2; 3; 4; 5; 6; 7g [ ff1; 2; 3g; f1; 4; 5g; f1; 6; 7g; f2; 4; 6g; f3; 4; 7g; f2; 5; 7g; f3; 5; 6gg, I T = f“integer”,“set of integers”g, I t is the obvious function from S to T, I the incidence relation I is symmetrized containment. Then (S; T; t; I) is an incidence geometry of rank 2. We could have called elements of type “integer” points as well, and elements of type “set of integers” lines as well, and vice versa. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 14/55 Graph theoretical point of view I An finite incidence structure of rank r is a multipartite graph (r partitions), together with a type function on the set of vertices and where the incidence relation is simply the adjacency relation of the graph. I Given a finite incidence structure, it is easy to create this incidence graph. I The flags are the cliques of the graph. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 15/55 An incidence structure in FinInG file: incstructure.g Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 16/55 Elements and objects I An element of an incidence geometry is represented by an object. I VectorSpaceToElement and UnderlyingObject. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 17/55 Elements and objects file: vectorspacetoelement.g Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 18/55 Elements and shadows Definition The shadow of type i of an element, respectively flag, is the set of elements of type i incident with the element, respectively with all elements of the flag. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 19/55 Elements and shadows file: shadow.g Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 20/55 More examples of incidence geometries . ...available in FinInG. I Coset geometries I Affine spaces. I Finite classical polar spaces. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 21/55 Desarguesian projective spaces Theorem Let K be a skewfield and n ≥ 2. The projective geometry PG(n; K) is Desarguesian. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 22/55 Desarguesian projective spaces Figure: The theorem of Desargues, figure taken from [1] . Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 23/55 Pappian projective spaces Theorem Let K be a field and n ≥ 2. The projective geometry PG(n; K) is Pappian. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 24/55 Pappian projective spaces Figure: The theorem of Pappus, figure taken from [1] . Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 25/55 Axiomatic projective planes Definition An axiomatic projective plane P is a point line geometry satisfying the following axioms I Any two points determine a unique line. I Any two lines meet in exactly one point. I There are four points of which no three are collinear. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 26/55 Axiomatic projective spaces Definition An axiomatic projective space P is a point line geometry satisfying the following axioms I Any two points determine a unique line. I (Veblen-Younga) Let A, B, C, and D be four points such that the line AB intersects the line CD. Then the line AC also intersects the line BD. I Any line is incident with at least three points. I There are at least two lines. asometimes also called the axiom of Pasch Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 27/55 Axiomatic projective spaces Figure: The axiom of Veblen-Young, figure taken from [1] . Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 28/55 Classical theorems Theorem A projective space P is Desarguesian if and only if P = PG(n; K) for some n ≥ 2 and K a skewfield. A projective space P of dimension at least 3 is Desarguesian. Theorem A projective space P is Pappian if and only if P = PG(n; K) for some n ≥ 2 and K a field. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 29/55 One more classical theorem Theorem (Hessenberg) If the theorem of Pappus hold in the projective space P, then also the theorem of Desargues. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 30/55 Classical theorems Corollary (by Wedderburn) A finite projective space P is Pappian if and only if it is Desarguesian. Theorem (Fundamental theorem of projective geom- etry) All the collineations of PG(n; K) are induced by the group of semilinear bijective maps of the underlying vector space. A finite projective space PG(n; K), K = Fq, will be denoted PG(n; q).
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