Finite 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 of a

Polar spaces

Geometry morphisms

Generalized polygons

Jan De Beule Finite in GAP 21 November 2018 1/55 I , geometry of finite simple groups. I many others ....

Some history on (finite) incidence geometry

I Beniamino Segre, classical .

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) over K of 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 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 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 ∈ S, t(x) is called its type, I the 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 |T|, 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 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 = {1, 2, 3, 4, 5, 6, 7} ∪ {{1, 2, 3}, {1, 4, 5}, {1, 6, 7}, {2, 4, 6}, {3, 4, 7}, {2, 5, 7}, {3, 5, 6}}, I T = {“integer”,“set of integers”}, 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 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 . . .

...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 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 P is a point 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 of semilinear bijective maps of the underlying vector space.

A finite projective space PG(n, K), K = Fq, will be denoted PG(n, q).

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 31/55 Groups and projective groups

G, acting on G/Sc(n + 1, K), acting name V(n + 1, K) on PG(n, K) ΓL(n + 1, K) PΓL(n + 1, K) collineations GL(n + 1, K) PGL(n + 1, K) (n + 1, K) PSL(n + 1, K) special homographies

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 32/55 Projective groups in FinInG

file: projectivegroups.g

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 33/55 Groups and actions

I OnProjSubspaces: action function for elements of a FinInG projective group and a subspace of a projective space. I Compatible with generic gap functions for groups, actions, orbits and stabilizers. I Particular implementations: FiningOrbits, FiningStabiliser, FiningSetwiseStabiliser, and variants.

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 34/55 Non Desarguesian projective planes

I There exist many non Desarguesian finite projective planes. I From the algebraic point of view, the coordinatizing structure is intereseting. This leads to the study of e.g. semi fields and other algebraic structures. I The so called finite translation planes can be constructed from a spread of a finite three dimensional projective space.

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 35/55 Collineations

Definition A of an incidence geometry is an incidence and type preserving of the set of elements.

file: incstructureagain.g

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 36/55 Quadratic and sesquilinear forms

Definition Let K be a field. A sesquilinear form on the vector space V(n, K) is a map f : V × V → K such that I f(x + y, z) = f(x, z) + f(y, z), and f(x, y + z) = f(x, y) + f(x, z), ∀x, y, z ∈ V, I f(αx, y) = αf(x, y), ∀x, y ∈ V and ∀α ∈ K, and, I f(x, αy) = αθf(x, y), ∀x, y ∈ V and ∀α ∈ K, for a fixed of K.

When θ = 1K, then f is bilinear.

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 37/55 Quadratic and sesquilinear forms

Definition Let K be a field. A quadratic form on the vector space V(n, K) is a map Q : V → K such that

Q(αv) = α2v, ∀α ∈ K, ∀v ∈ V .

Definition Let Q be a quadratic form on the vector space V. Then the associated bilinear form is the bilinear form

fQ : V × V → K : fQ(v, w) := Q(v + w) − Q(v) − Q(w) .

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 38/55 isotropic subspaces

Definition Let f be a sesquilinear form on a vector space V = V(n, K). Then I a vector v ∈ V is isotropic if f(v, v) = 0, I Rad(f) = {v ∈ V|f(v, w) = 0 , ∀w ∈ V}

Definition Let f be a sesquilinear form on a vector space V = V(n, K). Then a subspace S ⊂ V is totally isotropic if

f(v, w) = 0 , ∀v, w ∈ S .

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 39/55 singular subspaces

Definition Let Q be a quadratic form on a vector space V. Then I a vector v ∈ V is singular if Q(v) = 0,

I Rad(Q) = {v ∈ V|Q(v) = 0} ∩ Rad(fQ).

Definition Let Q be a quadratic form on a vector space V. Then a subspace S ⊂ V is totally singular if S and totally isotropic with relation to fQ and all vectors v ∈ S are singular with relation to Q.

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 40/55 Fields of odd characteristic

Lemma Let f be a bilinear form on the vector space V(n, K), K a field of odd characteristic. Then f determines a unique quadratic form Q such that f = fQ.

Proof. 1 Define Q : V → K : v 7→ 2 f(v, v)

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 41/55 Fields of even characteristic

Lemma Let Q be a quadratic form on the vector space V = V(n, K), K a field of even characteristic. Then the associated bilinear form fQ is alternating.

Proof.

fQ(v, v) := Q(2v) − 2Q(v) = 0, ∀v ∈ V

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 42/55 Finite classical polar spaces

Definition A finite classical polar space is the geometry consisting of the totally isotropic or, respectively, totally singular subspaces with relation to a sesquilinear or, respectively quadratic, form on a vector space V(n, K), K a finite field.

I From the definition it follows that a finite classical polar space is naturally embedded in a finite projective space. I The rank of the polar space as incidence geometry is the Witt index of the underlying form.

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 43/55 Maps on formed spaces

Definition Let f be a sesquilinear form on the vector space V = V(n, K). Let φ : V → V be a map on V. We call φ a I semi-similarity if f(v, w) = λf(φ(v), φ(w))θ ∀v, w ∈ V, for a fixed λ ∈ K and θ a fixed automorphism of K. I a similarity f(v, w) = λf(φ(v), φ(w)) for all v, w ∈ V, for a fixed λ ∈ K, I an isometry f(v, w) = f(φ(v), φ(w)).

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 44/55 Classical theorems

Theorem Let S be a finite classical polar space with underlying sesquilinear or quadratic form f. All collineations of S are induced by the bijective semi-similarities on the formed vector space (V, f).

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 45/55 Axiomatic polar spaces

I Foundations by Veldkamp, Freudenthal, Tits, Buekenhout, Shult, and others, see e.g. [2]. I We restrict to finite generalized quadrangles.

Definition An finite generalized quadrangle is a point-line geometry S satisfying the following axioms. I Each point is incident with 1 + t lines, t ≥ 1, and two points are incident with at most one line. I Each line is incident with 1 + s lines, s ≥ 1, and two lines are incident with at most one line. I If the point P is not on the line l, then P is collinear with exactly one point of l.

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 46/55 Classical theorems

Definition The rank of a polar space is one more than the dimension of the subspaces of maximal dimension.

Theorem A polar space of rank at least three is classical, i.e. it is determined by a form on a vector space over a skewfield.

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 47/55 Groups of collineations of fcps

W(d, q) Q+(d, q) Q−(d, q) Q(d, q) H(d, q2) SI PSp(d, q) PSO+(d, q) PSO−(d, q) PSO(d, q) PSU(d, q2) I PSp(d, q) PGO+(d, q) PGO−(d, q) PGO(d, q) PGU(d, q2) Sim PGSp(d, q) P∆O+(d, q) P∆O−(d, q) PGO(d, q) PGU(d, q2) Col PΓSp(d, q) PΓO+(d, q) PΓO−(d, q) PΓO(d, q) PΓU(d, q2)

I SI: special isometry group, I: isometry group, Sim: similarity group, Col: collineation group. I W(d, q), Q+(d, q), Q−(d, q): d must be odd. I Q(d, q): d must be even.

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 48/55 Projective finite classical groups

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 49/55 Geometry morphisms

Definition Let S = (S, T, t, I) and S0 = (S0, T0, t0, I0) be two incidence geometries. A geometry morphism is a map φ : S → S0, such that

∀A, B ∈ S : t(A) = t(B) =⇒ t0(φ(A)) = t0(φ(B)) ,

and ∀A, B ∈ S, φ(A) I0 φ(B) =⇒ AIB, ∀A, B ∈ S : t0(φ(A)) 6= t0(φ(B)) =⇒ (AIB =⇒ φ(A) I0 φ(B)) .

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 50/55 Examples of geometry morphisms

I IsomorphismPolarSpaces I NaturalEmbeddingBySubspace I KleinCorrespondence I NaturalEmbeddingByFieldReduction

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 51/55 Generalized d-gons

Definition A finite generalized d-gon is a point-line geometry of which the incidence graph has d and 2d.

Theorem If Γ is a finite thick generalized d-gon, then d ∈ {3, 4, 6, 8}.

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 52/55 Separate class of geometries

I Finite classical polar spaces of rank 2 are the finite classical generalized quadrangles. I Split Cayley hexagon and Triality hexagon are available. I Generic function to construct GPs from elements (of e.g. a projective space).

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 53/55 and friends

http://summerschool.fining.org

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 54/55 References

A. Beutelspacher and U. Rosenbaum. Projective geometry: from foundations to applications. Cambridge University Press, Cambridge, 1998. F. Buekenhout and A. M. Cohen. Diagram geometry, volume 57 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Heidelberg, 2013. Related to classical groups and buildings.

Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 55/55