Background results Tetrads of Lines References Tetrads of Lines Spanning PG ( 7 , 2 ) Hans Havlicek Research Group Differential Geometry and Geometric Structures Institute of Discrete Mathematics and Geometry Finite Geometries, Fourth Irsee Conference, September 15, 2014 Joint work with Ronald Shaw and Neil Gordon
Background results Tetrads of Lines References The Segre variety S 1 , 1 , 1 ( 2 ) Let V k , k ∈ { 1 , 2 , 3 } , be two-dimensional vector spaces over F 2 = GF ( 2 ) .
Background results Tetrads of Lines References The Segre variety S 1 , 1 , 1 ( 2 ) Let V k , k ∈ { 1 , 2 , 3 } , be two-dimensional vector spaces over F 2 = GF ( 2 ) . P ( V k ) = PG ( 1 , 2 ) are projective lines over F 2 .
Background results Tetrads of Lines References The Segre variety S 1 , 1 , 1 ( 2 ) Let V k , k ∈ { 1 , 2 , 3 } , be two-dimensional vector spaces over F 2 = GF ( 2 ) . P ( V k ) = PG ( 1 , 2 ) are projective lines over F 2 . The non-zero decomposable tensors of � 3 k = 1 V k determine the Segre variety � � S 1 , 1 , 1 ( 2 ) = a 1 ⊗ a 2 ⊗ a 3 | a k ∈ V k \ { 0 } �� 3 � with ambient projective space P k = 1 V k = PG ( 7 , 2 ) . (Over F 2 we identify projective points and non-zero vectors.)
Background results Tetrads of Lines References Orbits The ambient PG ( 7 , 2 ) of the Segre S 1 , 1 , 1 ( 2 ) =: S has 255 points that fall into five orbits O 1 , O 2 , . . . , O 5 under the subgroup G S < GL ( 8 , 2 ) stabilising S . S has 27 points and contains 27 lines.
Background results Tetrads of Lines References Orbits The ambient PG ( 7 , 2 ) of the Segre S 1 , 1 , 1 ( 2 ) =: S has 255 points that fall into five orbits O 1 , O 2 , . . . , O 5 under the subgroup G S < GL ( 8 , 2 ) stabilising S . O 5 : 27 points of the Segre S , S has 27 points and contains 27 lines.
Background results Tetrads of Lines References Orbits The ambient PG ( 7 , 2 ) of the Segre S 1 , 1 , 1 ( 2 ) =: S has 255 points that fall into five orbits O 1 , O 2 , . . . , O 5 under the subgroup G S < GL ( 8 , 2 ) stabilising S . O 5 : 27 points of the Segre S , O 2 : 54 points on bisecants (sums of two points of S at distance 2), S has 27 points and contains 27 lines.
Background results Tetrads of Lines References Orbits The ambient PG ( 7 , 2 ) of the Segre S 1 , 1 , 1 ( 2 ) =: S has 255 points that fall into five orbits O 1 , O 2 , . . . , O 5 under the subgroup G S < GL ( 8 , 2 ) stabilising S . O 5 : 27 points of the Segre S , O 2 : 54 points on bisecants (sums of two points of S at distance 2), O 4 : 54 points on the 27 distinguished tangents of S , S has 27 points and contains 27 lines.
Background results Tetrads of Lines References Orbits The ambient PG ( 7 , 2 ) of the Segre S 1 , 1 , 1 ( 2 ) =: S has 255 points that fall into five orbits O 1 , O 2 , . . . , O 5 under the subgroup G S < GL ( 8 , 2 ) stabilising S . O 5 : 27 points of the Segre S , O 2 : 54 points on bisecants (sums of two points of S at distance 2), O 4 : 54 points on the 27 distinguished tangents of S , O 3 : 108 points on bisecants (sums of two points of S at distance 3), S has 27 points and contains 27 lines.
Background results Tetrads of Lines References Orbits The ambient PG ( 7 , 2 ) of the Segre S 1 , 1 , 1 ( 2 ) =: S has 255 points that fall into five orbits O 1 , O 2 , . . . , O 5 under the subgroup G S < GL ( 8 , 2 ) stabilising S . O 5 : 27 points of the Segre S , O 2 : 54 points on bisecants (sums of two points of S at distance 2), O 4 : 54 points on the 27 distinguished tangents of S , O 3 : 108 points on bisecants (sums of two points of S at distance 3), S has 27 points and O 1 : 12 points (sums of triads of S at contains 27 lines. distance 3).
Background results Tetrads of Lines References Orbits (cont.) The results from the previous slide and generalisations thereof were established by various authors: D. Glynn, T. A. Gulliver, J. G. Maks, and M. K. Gupta (2006) [2]. B. Odehnal, M. Saniga, and H. H. (2012) [3]. R. Shaw, N. Gordon, and H. H. (2012) [5]. M. R. Bremner and St. G. Stavrou (2013) [1]. M. Lavrauw and J. Sheekey (2014) [4].
Background results Tetrads of Lines References Orbits (cont.) Two sets deserve special mention: The union O 2 ∪ O 4 ∪ O 5 (135 points) is a hyperbolic quadric H 7 of PG ( 7 , 2 ) .
Background results Tetrads of Lines References Orbits (cont.) Two sets deserve special mention: The union O 2 ∪ O 4 ∪ O 5 (135 points) is a hyperbolic quadric H 7 of PG ( 7 , 2 ) . The orbit O 1 (12 points) comprises a tetrad of lines spanning PG ( 7 , 2 ) .
Background results Tetrads of Lines References Basic assumption We start out from a(ny) direct sum decomposition V 8 = V a ⊕ V b ⊕ V c ⊕ V d of V 8 := V ( 8 , 2 ) into 2-dimensional spaces V a , V b , V c , V d . So we obtain the tetrad of lines L 4 := { L a , L b , L c , L d } , where L h := P ( V h ) , h ∈ { a , b , c , d } ; P ( V 8 ) = PG ( 7 , 2 ) is the span of this tetrad of lines.
Background results Tetrads of Lines References The stabiliser group G ( L 4 ) Let G ( L 4 ) be that subgroup of GL ( 8 , 2 ) which preserves the foregoing tetrad L 4 of lines.
Background results Tetrads of Lines References The stabiliser group G ( L 4 ) Let G ( L 4 ) be that subgroup of GL ( 8 , 2 ) which preserves the foregoing tetrad L 4 of lines. The group G ( L 4 ) has the semi-direct product structure G ( L 4 ) = N ⋊ Sym ( 4 ) , where N := GL ( V a ) × GL ( V b ) × GL ( V c ) × GL ( V d ) , and where Sym ( 4 ) = Sym ( { a , b , c , d } ) .
Background results Tetrads of Lines References Line weight Let us define the line-weight lw ( p ) of a point p ∈ PG ( 7 , 2 ) as follows: Write p = v a + v b + v c + v d with v h ∈ V h , h ∈ { a , b , c , d } . Then lw ( p ) = r whenever precisely r of the vectors v a , v b , v c , v d are non-zero.
Background results Tetrads of Lines References Orbits The 255 points of PG ( 7 , 2 ) fall into just four G ( L 4 ) -orbits ω 1 , ω 2 , ω 3 , ω 4 , where ω r = { p ∈ PG ( 7 , 2 ) | lw ( p ) = r } . The lengths of these orbits are accordingly × 3 2 = 54 , � 4 � | ω 1 | = 12 , | ω 2 | = 2 × 3 3 = 108 , | ω 4 | = 3 4 = 81 . � 4 � | ω 3 | = 3
Background results Tetrads of Lines References The symplectic form of L 4 There is a unique symplectic form B on V 8 such that the subspaces V a , V b , V c , V d are hyperbolic 2-dimensional spaces which are pairwise orthogonal.
Background results Tetrads of Lines References The quadric of L 4 The tetrad L 4 also determines a particular non-degenerate quadric Q in PG ( 7 , 2 ) . Such a quadric Q is uniquely determined by the two conditions (i) it has equation Q ( x ) = 0 such that the quadratic form Q polarises to give the foregoing symplectic form B ;
Background results Tetrads of Lines References The quadric of L 4 The tetrad L 4 also determines a particular non-degenerate quadric Q in PG ( 7 , 2 ) . Such a quadric Q is uniquely determined by the two conditions (i) it has equation Q ( x ) = 0 such that the quadratic form Q polarises to give the foregoing symplectic form B ; (ii) the 12-set of points ω 1 = L a ∪ L b ∪ L c ∪ L d ⊂ PG ( 7 , 2 ) supporting the tetrad L 4 is external to Q .
Background results Tetrads of Lines References The quadric of L 4 The tetrad L 4 also determines a particular non-degenerate quadric Q in PG ( 7 , 2 ) . Such a quadric Q is uniquely determined by the two conditions (i) it has equation Q ( x ) = 0 such that the quadratic form Q polarises to give the foregoing symplectic form B ; (ii) the 12-set of points ω 1 = L a ∪ L b ∪ L c ∪ L d ⊂ PG ( 7 , 2 ) supporting the tetrad L 4 is external to Q . The quadric Q is seen to be ω 2 ∪ ω 4 (54 + 81 = 135 points), and so it is hyperbolic.
Background results Tetrads of Lines References The normal subgroup G 81 of G ( L 4 ) For each h ∈ { a , b , c , d } let us choose an element ζ h ∈ GL ( V h ) of order 3 that effects a cyclic permutation of the points of L h .
Background results Tetrads of Lines References The normal subgroup G 81 of G ( L 4 ) For each h ∈ { a , b , c , d } let us choose an element ζ h ∈ GL ( V h ) of order 3 that effects a cyclic permutation of the points of L h . We define A ijkl := ( ζ a ) i ⊕ ( ζ b ) j ⊕ ( ζ c ) k ⊕ ( ζ d ) l for i , j , k , l ∈ { 0 , 1 , 2 } .
Background results Tetrads of Lines References The normal subgroup G 81 of G ( L 4 ) For each h ∈ { a , b , c , d } let us choose an element ζ h ∈ GL ( V h ) of order 3 that effects a cyclic permutation of the points of L h . We define A ijkl := ( ζ a ) i ⊕ ( ζ b ) j ⊕ ( ζ c ) k ⊕ ( ζ d ) l for i , j , k , l ∈ { 0 , 1 , 2 } . Then � � G 81 := A ijkl | i , j , k , l ∈ { 0 , 1 , 2 } is a normal subgroup of G ( L 4 ) .
Background results Tetrads of Lines References The normal subgroup G 81 of G ( L 4 ) For each h ∈ { a , b , c , d } let us choose an element ζ h ∈ GL ( V h ) of order 3 that effects a cyclic permutation of the points of L h . We define A ijkl := ( ζ a ) i ⊕ ( ζ b ) j ⊕ ( ζ c ) k ⊕ ( ζ d ) l for i , j , k , l ∈ { 0 , 1 , 2 } . Then � � G 81 := A ijkl | i , j , k , l ∈ { 0 , 1 , 2 } is a normal subgroup of G ( L 4 ) . Observe that ω 4 is a single G 81 -orbit.
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