The Complement of Binary Klein Quadric as a Combinatorial Grassmannian

Abstract

Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (28 6 , 56 3 )-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G 2 (8). It is also pointed out that a set of seven points of G 2 (8) whose labels share a mark corresponds to a Conwell heptad of PG(5, 2). Gradual removal of Conwell heptads from the (28 6 , 56 3 )-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888).Keywords: combinatorial Grassmannian; binary Klein quadric; Conwell heptad; three-qubit Pauli group Let Q + (5, 2) be a hyperbolic quadric in a five-dimensional projective space PG(5, 2). As it is well known (see, e.g., [1,2]), there are 28 points lying off this quadric as well as 56 lines skew (or, external) to it. If the equation of the quadric is taken in a canonical form Q 0 : x 1 x 2 + x 3 x 4 + x 5 x 6 = 0, then the 28 off-quadric points are those listed in Table 1 and the 56 external lines are those given in Table 2. In Table 2, the "+" symbol indicates which point lies on a given line; for example, line 1 consists of points 1, 4 and 9. As it is obvious from this table, each line has three points and through each point there are six lines; hence, these points and lines form a (28 6 , 56 3 )-configuration. Next, a combinatorial Grassmannian G k (|X|) (see, e.g., [3,4]), where k is a positive integer and X is a finite set, |X| = N , is a point-line incidence structure whose points are all k-element subsets of X

Quantum computation and measurements from an exotic space-time R4

“…In the second case, the permutation group is P = A 8 and the geometry is the configuration [28 6 , 56 3 ] on 28 points and 56 lines of size 3. In [18], it was shown that the geometry in question corresponds to the combinatorial Grassmannian of type Gr(2, 8), alias the configuration obtained from the points off the hyperbolic quadric Q + (5, 2) in the complex projective space P G(5, 2). Interestingly, Gr(2, 8) can be nested by gradual removal of a so-called 'Conwell heptad' and be identified to the tail of the sequence of 1 A different enlightning of the projective space P G(3, 2) appears in [17] with the connection to the Kirkman schoolgirl problem.…”

“…Recall that ∂W is the boundary of Akbulut cork W . The 28-letter permutation group P has two generators as follows P = 28|g 1 , g 2 with g 1 = (2, 4, 8, 6, 3)(5, 10, 15, 13, 9) (11,12,18,25,17) (14,20,19,24,21) (16,22,26,28,23), g 2 = (1, 2, 5, 11, 6, 7, 3)(4, 8, 12, 19, 22, 14, 9) (10,16,24,27,21,26,17) (13,20,18,25,28,23,15).…”

“…Using the method described in Appendix 2, one derives the configuration [28 6 , 56 3 ] on 28 points and 56 lines. As shown in [Table 4] [18], the configuration is isomorphic to the combinatorial Grassmannian Gr (2,8) and nested by a sequence of binomial configurations isomorphic to Gr(2, i), i ≤ 8, associated with Cayley-Dickson algebras. This statement is checked by listing the 56 lines on the 28 points of the configuration as follows More precisely, the distinguished configuration [21 5 , 35 3 ] isomorphic to Gr(2, 7) in the list above is stabilized thanks to the subgroup of P isomorphic to A 7 .…”

A Combinatorial Grassmannian Representation of the Magic Three-Qubit Veldkamp Line

“…In V (G 2 (7)), there exists a distinguished symplectic polar space W(5, 2) whose lines comprise three orbits of lines of type (α, α, α), (α, β, β), and (α, β, γ)-that is, the orbits whose cores feature an odd number of points of G 2 (7). Other prominent geometrical objects of V (G 2 (7)) are: a hyperbolic quadric Q + 0 (5, 2) ∈ W(5, 2) formed by 35 points of type α and 105 lines of type (α, α, α); a combinatorial Grassmannian G 2 (7) formed by 21 points of type β and 35 lines of type (β, β, β); and a Conwell heptad with respect to the above-defined Q + 0 (5, 2) represented by seven points of type γ (see also [17]). Figure 2.…”

“…Another prominent geometrical objects of V(G 2 (7)) are: a hyperbolic quadric Q + 0 (5, 2) ∈ W(5, 2) formed by 35 points of type α and Table 2: The seven different types of lines of V(G 2 (7)). Type Form Core composition Number abcd:ef g (α, α, α) abef :cdg three mutually non-collinear points (ab,cd, and ef ) 105 cdef :abg abcd:ef g (α, α, β) abce:df g a line (abc) and a point (f g) 210 abcf g:de abc:def g (α, α, γ) def :abcg two disjoint lines (abc and def ) 70 abcdef :g abcd:ef g (α, β, β) ab:cdef g a line (ef g) and two non-collinear points (ab and cd) 105 cd:abef g abcd:ef g (α, β, γ) abcde:f g a Pasch configuration (abcd) and a point (f g) 105 abcdf g:e abcde:f g (β, β, β) abcdf :eg a Pasch configuration (abcd) 35 abcdg:ef abcde:f g (β, γ, γ) abcdef :g a Desargues configuration (abcde) 21 abcdeg:f 105 lines of type (α, α, α); a combinatorial Grassmannian G 2 (7) formed 21 points of type β and 35 lines of type (β, β, β); and a Conwell heptad with respect to the above-defined Q + 0 (5, 2) represented by seven points of type γ (see also [14]).…”