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A Finite Geometry Package

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A Finite Geometry Package Desargues Desargues Outline Finite ... a finite geometry package Geometry Examples of Problems John Bamberg Features of Desargues Anton Betten Jan De Beule Maska Law Another example Max Neunh¨offer Michael Pauley Sven Reichard Desargues Outline Outline Finite Geometry Examples of 1 Outline Problems Features of Desargues Another 2 Finite Geometry example 3 Examples of Problems 4 Features of Desargues 5 Another example • Polar Spaces • Generalised Polygons Incidence Geometry Consists of: • Objects (points, lines, planes, etc) • Incidence relation (anti-reflexive and symmetric) • A maximal flag contains an object of each type. The rank is the number of types of object. Desargues Finite Geometry Outline Finite Geometry Examples of • Projective Geometry/Affine Geometry Problems Features of Desargues Another example • Generalised Polygons Incidence Geometry Consists of: • Objects (points, lines, planes, etc) • Incidence relation (anti-reflexive and symmetric) • A maximal flag contains an object of each type. The rank is the number of types of object. Desargues Finite Geometry Outline Finite Geometry Examples of • Projective Geometry/Affine Geometry Problems Features of • Polar Spaces Desargues Another example Incidence Geometry Consists of: • Objects (points, lines, planes, etc) • Incidence relation (anti-reflexive and symmetric) • A maximal flag contains an object of each type. The rank is the number of types of object. Desargues Finite Geometry Outline Finite Geometry Examples of • Projective Geometry/Affine Geometry Problems Features of • Polar Spaces Desargues • Generalised Polygons Another example Desargues Finite Geometry Outline Finite Geometry Examples of • Projective Geometry/Affine Geometry Problems Features of • Polar Spaces Desargues • Generalised Polygons Another example Incidence Geometry Consists of: • Objects (points, lines, planes, etc) • Incidence relation (anti-reflexive and symmetric) • A maximal flag contains an object of each type. The rank is the number of types of object. • Incidence Relation: A ⊂ B or B ⊂ A • Types 1, 2, 3,..., d − 1 Theorem A projective space or polar space of rank at least 3 is classical, that is, it comes from a vector space. Desargues Projective Space Outline Start with a vector space V (d, (q)) Finite GF Geometry • Objects Examples of Problems Points: 1-dim subspaces Features of Lines: 2-dim subspaces Desargues Planes: 3-dim subspaces, etc... Another example • Types 1, 2, 3,..., d − 1 Theorem A projective space or polar space of rank at least 3 is classical, that is, it comes from a vector space. Desargues Projective Space Outline Start with a vector space V (d, (q)) Finite GF Geometry • Objects Examples of Problems Points: 1-dim subspaces Features of Lines: 2-dim subspaces Desargues Planes: 3-dim subspaces, etc... Another example • Incidence Relation: A ⊂ B or B ⊂ A Theorem A projective space or polar space of rank at least 3 is classical, that is, it comes from a vector space. Desargues Projective Space Outline Start with a vector space V (d, (q)) Finite GF Geometry • Objects Examples of Problems Points: 1-dim subspaces Features of Lines: 2-dim subspaces Desargues Planes: 3-dim subspaces, etc... Another example • Incidence Relation: A ⊂ B or B ⊂ A • Types 1, 2, 3,..., d − 1 Desargues Projective Space Outline Start with a vector space V (d, (q)) Finite GF Geometry • Objects Examples of Problems Points: 1-dim subspaces Features of Lines: 2-dim subspaces Desargues Planes: 3-dim subspaces, etc... Another example • Incidence Relation: A ⊂ B or B ⊂ A • Types 1, 2, 3,..., d − 1 Theorem A projective space or polar space of rank at least 3 is classical, that is, it comes from a vector space. Points: All one-dim subspaces. 2 q6−1 Lines: (q + 1) q−1 two-dim subspaces. 3 q4−1 Planes: (q + 1) q−1 three-dim subspaces. Spreads A spread of W(5, q) is a set of q3 + 1 planes which form a partition of the set of points. Desargues Spreads of W(5, q) Outline Finite Geometry W(5, q) Examples of Consider V (6, q) equipped with an alternating form: Problems Features of Desargues hu, vi = u1v2 − u2v1 + u3v4 − u4v3 + u5v6 − u6v5. Another example We get a polar geometry consisting of Spreads A spread of W(5, q) is a set of q3 + 1 planes which form a partition of the set of points. Desargues Spreads of W(5, q) Outline Finite Geometry W(5, q) Examples of Consider V (6, q) equipped with an alternating form: Problems Features of Desargues hu, vi = u1v2 − u2v1 + u3v4 − u4v3 + u5v6 − u6v5. Another example We get a polar geometry consisting of Points: All one-dim subspaces. 2 q6−1 Lines: (q + 1) q−1 two-dim subspaces. 3 q4−1 Planes: (q + 1) q−1 three-dim subspaces. Desargues Spreads of W(5, q) Outline Finite Geometry W(5, q) Examples of Consider V (6, q) equipped with an alternating form: Problems Features of Desargues hu, vi = u1v2 − u2v1 + u3v4 − u4v3 + u5v6 − u6v5. Another example We get a polar geometry consisting of Points: All one-dim subspaces. 2 q6−1 Lines: (q + 1) q−1 two-dim subspaces. 3 q4−1 Planes: (q + 1) q−1 three-dim subspaces. Spreads A spread of W(5, q) is a set of q3 + 1 planes which form a partition of the set of points. What we need... 6 Points: Easy, all one-dim subspaces of GF(3) . Planes: Take orbit of one plane gap> sp := Sp(6,3);; gap> plane := [[1,0,0,1,0,0], [0,1,0,0,1,0],[0,0,1,0,0,1]]*Z(3)^0;; gap> planes := Orbit(sp, plane, OnSubspacesByCanonicalBasis);; Group: Sylow 13-subgroup of Sp(6, 3) gap> syl13 := SylowSubgroup(sp, 13); <group of 6x6 matrices of size 13 in characteristic 3> Solution: Stitch together orbits on planes Desargues Problem Find all spreads of W(5, 3) which have automorphism group of Outline Finite order divisible by 13. Geometry Examples of Problems Features of Desargues Another example Planes: Take orbit of one plane gap> sp := Sp(6,3);; gap> plane := [[1,0,0,1,0,0], [0,1,0,0,1,0],[0,0,1,0,0,1]]*Z(3)^0;; gap> planes := Orbit(sp, plane, OnSubspacesByCanonicalBasis);; Group: Sylow 13-subgroup of Sp(6, 3) gap> syl13 := SylowSubgroup(sp, 13); <group of 6x6 matrices of size 13 in characteristic 3> Solution: Stitch together orbits on planes Desargues Problem Find all spreads of W(5, 3) which have automorphism group of Outline Finite order divisible by 13. Geometry Examples of What we need... Problems 6 Features of Points: Easy, all one-dim subspaces of GF(3) . Desargues Another example gap> sp := Sp(6,3);; gap> plane := [[1,0,0,1,0,0], [0,1,0,0,1,0],[0,0,1,0,0,1]]*Z(3)^0;; gap> planes := Orbit(sp, plane, OnSubspacesByCanonicalBasis);; Group: Sylow 13-subgroup of Sp(6, 3) gap> syl13 := SylowSubgroup(sp, 13); <group of 6x6 matrices of size 13 in characteristic 3> Solution: Stitch together orbits on planes Desargues Problem Find all spreads of W(5, 3) which have automorphism group of Outline Finite order divisible by 13. Geometry Examples of What we need... Problems 6 Features of Points: Easy, all one-dim subspaces of GF(3) . Desargues Another Planes: Take orbit of one plane example Group: Sylow 13-subgroup of Sp(6, 3) gap> syl13 := SylowSubgroup(sp, 13); <group of 6x6 matrices of size 13 in characteristic 3> Solution: Stitch together orbits on planes Desargues Problem Find all spreads of W(5, 3) which have automorphism group of Outline Finite order divisible by 13. Geometry Examples of What we need... Problems 6 Features of Points: Easy, all one-dim subspaces of GF(3) . Desargues Another Planes: Take orbit of one plane example gap> sp := Sp(6,3);; gap> plane := [[1,0,0,1,0,0], [0,1,0,0,1,0],[0,0,1,0,0,1]]*Z(3)^0;; gap> planes := Orbit(sp, plane, OnSubspacesByCanonicalBasis);; Desargues Problem Find all spreads of W(5, 3) which have automorphism group of Outline Finite order divisible by 13. Geometry Examples of What we need... Problems 6 Features of Points: Easy, all one-dim subspaces of GF(3) . Desargues Another Planes: Take orbit of one plane example gap> sp := Sp(6,3);; gap> plane := [[1,0,0,1,0,0], [0,1,0,0,1,0],[0,0,1,0,0,1]]*Z(3)^0;; gap> planes := Orbit(sp, plane, OnSubspacesByCanonicalBasis);; Group: Sylow 13-subgroup of Sp(6, 3) gap> syl13 := SylowSubgroup(sp, 13); <group of 6x6 matrices of size 13 in characteristic 3> Solution: Stitch together orbits on planes gap> syl := SylowSubgroup(sp, 13); <projective group with Frobenius of size 13> gap> planes := Planes( w ); <planes of W(5, 3)> gap> planes := AsList( planes );; gap> orbits := Orbits(syl, planes , OnLieVarieties);; gap> Collected( List( orbits, Size )); [ [ 1, 2 ], [ 13, 86 ] ] gap> IsPartialSpread := s -> ForAll( Combinations(s,2), c -> ProjectiveDimension( Meet(c[1], c[2]) ) = -1 );; gap> partialspreads := Filtered(orbits, IsPartialSpread);; gap> 13s := Filtered(partialspreads, i -> Size(i) = 13);; gap> 26s := List(Combinations(13s,2), Union);; gap> two := Union(Filtered(partialspreads, i -> Size(i) = 1));; gap> 28s := List(26s, x -> Union(x, two) );; gap> spreads := Filtered( 28s, IsPartialSpread);; gap> Size(spreads); 5 Desargues In Desargues... Outline gap> w := SymplecticSpace(5, 3); Finite W(5, 3) Geometry gap> sp := IsometryGroup( w ); PSp(6,3) Examples of Problems Features of Desargues Another example gap> orbits := Orbits(syl, planes , OnLieVarieties);; gap> Collected( List( orbits, Size )); [ [ 1, 2 ], [ 13, 86 ] ] gap> IsPartialSpread := s -> ForAll( Combinations(s,2), c -> ProjectiveDimension( Meet(c[1], c[2]) ) = -1 );; gap> partialspreads := Filtered(orbits, IsPartialSpread);; gap> 13s := Filtered(partialspreads, i -> Size(i) = 13);; gap> 26s := List(Combinations(13s,2), Union);; gap> two := Union(Filtered(partialspreads, i -> Size(i) = 1));; gap> 28s := List(26s, x -> Union(x, two) );; gap> spreads := Filtered( 28s, IsPartialSpread);; gap> Size(spreads); 5 Desargues In Desargues..
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