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Letter

The Complement of Binary Klein Quadric as a Combinatorial Grassmannian

by 1,2
1
Institute for Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstraße 8–10, A-1040 Vienna, Austria
2
Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic 
Academic Editor: Palle E.T. Jorgensen
Mathematics 2015, 3(2), 481-486; https://doi.org/10.3390/math3020481
Received: 9 May 2015 / Accepted: 5 June 2015 / Published: 8 June 2015
(This article belongs to the Special Issue Mathematical physics)

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 (286; 563)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G2(8). It is also pointed out that a set of seven points of G2(8) whose labels share a mark corresponds to a Conwell heptad of PG(5, 2). Gradual removal of Conwell heptads from the (286; 563)-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 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 (286, 563)-configuration.
Next, a combinatorial Grassmannian Gk(|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 and whose lines are all (k+ 1)-element subsets of X, incidence being inclusion. Obviously, Gk(N) is a ( ( N k ) N k , ( N k + 1 ) k + 1 )-configuration; hence, G2(8) is another (286, 563)-configuration.
It is straightforward to see that the two (286, 563)-configurations are, in fact, isomorphic. To this end, one simply employs the bijection between the 28 off-quadric points and the 28 points of G2(8) shown in Table 3 (here, by a slight abuse of notation, X = {1, 2, 3, 4, 5, 6, 7, 8}) and verifies step by step that each of the above-listed 56 lines of PG(5, 2) is also a line of G2(8); thus, line 1 of PG(5, 2) corresponds to the line {1, 4, 6} of G2(8), line 2 to the line {1, 2, 4}, line 3 to {1, 3, 4}, etc.
This isomorphism entails a very interesting property related to so-called Conwell heptads [5]. Given a Q + ( 5 , 2 ) of PG(5, 2), a Conwell heptad (in the modern language also known as a maximal exterior set of Q + ( 5 , 2 ), see, e.g., [6]) is a set of seven off-quadric points such that each line joining two distinct points of the heptad is skew to the Q + ( 5 , 2 ). There are altogether eight such heptads: any two of them have a unique point in common and each of the 28 points off the quadric is contained in two heptads. The points in Table 1 are arranged in such a way that the last seven of them represent a Conwell heptad, as it is obvious from the bottom part of Table 2. From Table 3 we read off that this particular heptad corresponds to those seven points of G2(8) whose representatives have mark “8” in common. Clearly, the remaining seven heptads correspond to those septuples of points of G2(8) that share one of the remaining seven marks each. Finally, we observe that by removing from our off-quadric (286, 563)-configuration the seven points of a Conwell heptad and all the 21 lines defined by pairs of them one gets a (215, 353)-configuration isomorphic to G2(7); gradual removal of additional heptads and the corresponding lines yields a remarkable nested sequence of configurations displayed in Table 4. Interestingly enough, this nested sequence of binomial configurations is identical with part of that found to be associated with Cayley-Dickson algebras [7]. Moreover, given the fact that PG(5, 2) is the natural embedding space for the symplectic polar space W (5, 2) that geometrizes the structure of the three-qubit Pauli group [8,9], this particular sequence of configurations may lead to further intriguing insights into the physical relevance of this group.
To conclude this letter, there are a few facts that deserve a special mention. First, the fact that the complement of Q + ( 5 , 2 ) is isomorphic to the combinatorial Grassmannian G2(8) can be implicitly be traced down even in the original paper of Conwell [5]. As mentioned above, the complement contains eight heptads and each point of the complement can be identified with the (unordered) pair of heptads through it; also the “grassmannian” rule of forming lines on the complement remains valid. After this observation is made, the combinatorial characterization of heptads becomes evident: these are the maximal cliques of the (binary) collinearity. (Clearly, Conwell himself could not formulate his characterization in this combinatorial language.) Second, the fact that removing a complete graph K7 from G2(8) one obtains G2(7), and so on, was shown in a more general (“G(n+1) minus Kn”) setting in [10] (see also [11]). Finally, it is worth pointing out that the group of automorphisms of the (286, 563)-configuration is isomorphic to S8SL4(2):2 (which is the group of collineations and correlations of PG(3, 2), also isomorphic—via the Klein correspondence—to the group of all collineations of PG(5, 2) preserving a hyperbolic quadric).

Acknowledgments

This work was partially supported by the VEGA Grant Agency, Project 2/0003/13, as well as by the Austrian Science Fund (Fonds zur Förderung der Wissenschaftlichen Forschung (FWF)), Research Project M1564-N27 “Finite-Geometrical Aspects of Quantum Theory.” We thank the anonymous referees for a number of constructive remarks and suggestions.

Conflicts of Interest

The author declares no conflict of interest.

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Table 1. The 28 points lying off the quadric Q 0.
Table 1. The 28 points lying off the quadric Q 0.
No.x1x2x3x4x5x6
1111000
2110010
3110001
4110100
5111010
6111001
7110110
8110101
9001100
10001110
11001101
12011100
13101110
14101101
15000011
16100011
17001011
18000111
19011011
20010111
21111111

22110000
23101100
24011110
25011101
26010011
27101011
28100111
Table 2. The 56 lines having no points in common with the quadric Q 0.
Table 2. The 56 lines having no points in common with the quadric Q 0.
No.12345678910111213141516171819202122232425262728
1+++
2+++
3+++
4+++
5+++
6+++
7+++
8+++
9+++
10+++
11+++
12+++
13+++
14+++
15+++
16+++
17+++
18+++
19+++
20+++
21+++
22+++
23+++
24+++
25+++
26+++
27+++
28+++
29+++
30+++
31+++
32+++
33+++
34+++
35+++

36+++
37+++
38+++
39+++
40+++
41+++
42+++
43+++
44+++
45+++
46+++
47+++
48+++
49+++
50+++
51+++
52+++
53+++
54+++
55+++
56+++
Table 3. A bijection between the 28 off-quadric points and the 28 points of G2(8).
Table 3. A bijection between the 28 off-quadric points and the 28 points of G2(8).
off- Q 0G2(8)off- Q 0G2(8)
1{1, 4}15{2, 3}
2{3, 5}16{4, 7}
3{2, 5}17{5, 6}
4{4, 6}18{1, 5}
5{2, 6}19{1, 7}
6{3, 6}20{6, 7}
7{1, 2}21{4, 5}
8{1, 3}22{7, 8}
9{1, 6}23{5, 8}
10{2, 4}24{3, 8}
11{3, 4}25{2, 8}
12{5, 7}26{4, 8}
13{3, 7}27{1, 8}
14{2, 7}28{6, 8}
Table 4. A nested sequence of configurations located in the complement of a hyperbolic quadric of PG(5, 2).
Table 4. A nested sequence of configurations located in the complement of a hyperbolic quadric of PG(5, 2).
# of Heptads RemovedConfigurationCGRemark
0(286, 563)G2(8)
1(215, 353)G2(7)
2(154, 203)G2(6)Cayley-Salmon
3(103, 103)G2(5)Desargues
4(62, 43)G2(4)Pasch
5(31, 13)G2(3)single line
6(10, 03)G2(2)single point
7empty set
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