# The Complement of Binary Klein Quadric as a Combinatorial Grassmannian

^{1}

^{2}

Next Article in Journal

Next Article in Special Issue

Next Article in Special Issue

Previous Article in Journal

Previous Article in Special Issue

Previous Article in Special Issue

Institute for Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstraße 8–10, A-1040 Vienna, Austria

Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic

Academic Editor: Palle E.T. Jorgensen

Received: 9 May 2015 / Accepted: 5 June 2015 / Published: 8 June 2015

(This article belongs to the Special Issue Mathematical physics)

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).

Let
${\mathcal{Q}}^{+}(5,2)$ 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
${\mathcal{Q}}_{0}:{x}_{1}{x}_{2}+{x}_{3}{x}_{4}+{x}_{5}{x}_{6}=0$, 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 and whose lines are all (k+ 1)-element subsets of X, incidence being inclusion. Obviously, G_{k}(N) is a
$\left({(\begin{array}{c}N\\ k\end{array})}_{N-k},{(\begin{array}{c}N\\ k+1\end{array})}_{k+1}\right)$-configuration; hence, G_{2}(8) is another (28_{6}, 56_{3})-configuration.

It is straightforward to see that the two (28_{6}, 56_{3})-configurations are, in fact, isomorphic. To this end, one simply employs the bijection between the 28 off-quadric points and the 28 points of G_{2}(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 G_{2}(8); thus, line 1 of PG(5, 2) corresponds to the line {1, 4, 6} of G_{2}(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
${\mathcal{Q}}^{+}(5,2)$ of PG(5, 2), a Conwell heptad (in the modern language also known as a maximal exterior set of
${\mathcal{Q}}^{+}(5,2)$, 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
${\mathcal{Q}}^{+}(5,2)$. 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 G_{2}(8) whose representatives have mark “8” in common. Clearly, the remaining seven heptads correspond to those septuples of points of G_{2}(8) that share one of the remaining seven marks each. Finally, we observe that by removing from our off-quadric (28_{6}, 56_{3})-configuration the seven points of a Conwell heptad and all the 21 lines defined by pairs of them one gets a (21_{5}, 35_{3})-configuration isomorphic to G_{2}(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
${\mathcal{Q}}^{+}(5,2)$ is isomorphic to the combinatorial Grassmannian G_{2}(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 K_{7} from G_{2}(8) one obtains G_{2}(7), and so on, was shown in a more general (“G_{(}_{n}_{+1)} minus K_{n}”) setting in [10] (see also [11]). Finally, it is worth pointing out that the group of automorphisms of the (28_{6}, 56_{3})-configuration is isomorphic to S_{8} ≅ SL_{4}(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).

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.

The author declares no conflict of interest.

- Hirschfeld, J.W.P. Finite Projective Spaces of Three Dimensions; Oxford University Press: Oxford, UK, 1985. [Google Scholar]
- Hirschfeld, J.W.P.; Thas, J.A. General Galois Geometries; Oxford University Press: Oxford, UK, 1991. [Google Scholar]
- Prażmowska, M. Multiplied perspectives and generalizations of Desargues configuration. Demonstratio Math.
**2006**, 39, 887–906. [Google Scholar] - Owsiejczuk, A.; Prażmowska, M. Combinatorial generalizations of generalized quadrangles of order (2, 2). Des. Codes Cryptogr.
**2009**, 53, 45–57. [Google Scholar] - Conwell, G.M. The 3-space PG(3, 2) and its group. Ann. Math.
**1910**, 11, 60–76. [Google Scholar] - Thas, J.A. Maximal exterior sets of hyperbolic quadrics: The complete classification. J. Combin. Theory Ser. A
**1991**, 56, 303–308. [Google Scholar] - Saniga, M.; Holweck, F.; Pracna, P. Cayley-Dickson algebras and finite geometry. Discrete Comput. Geom. 2014. arXiv:1405.6888. Available online: http://arxiv.org/abs/1405.6888 accessed on 3 September 2014.
- Havlicek, H.; Odehnal, B.; Saniga, M. Factor-group-generated polar spaces and (multi-)qudits. SIGMA
**2009**, 5. [Google Scholar] [CrossRef] - Thas, K. The geometry of generalized Pauli operators of N-qudit Hilbert space, and an application to MUBs. EPL
**2009**, 86. [Google Scholar] [CrossRef] - Prażmowska, M.; Prażmowski, K. Binomial partial Steiner triple systems containing complete graphs. 2014. arXiv:1404.4064 Available online: http://arxiv.org/abs/1404.4064 accessed on 4 June 2015.
- Petelczyc, K.; Prażmowska, M; Prażmowski, K. Complete classication of the (15
_{4}20_{3})-congurations with at least three K_{5}-graphs. Discret. Math.**2015**, 338, 1243–1251. [Google Scholar]

No. | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} |
---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 0 | 0 | 0 |

2 | 1 | 1 | 0 | 0 | 1 | 0 |

3 | 1 | 1 | 0 | 0 | 0 | 1 |

4 | 1 | 1 | 0 | 1 | 0 | 0 |

5 | 1 | 1 | 1 | 0 | 1 | 0 |

6 | 1 | 1 | 1 | 0 | 0 | 1 |

7 | 1 | 1 | 0 | 1 | 1 | 0 |

8 | 1 | 1 | 0 | 1 | 0 | 1 |

9 | 0 | 0 | 1 | 1 | 0 | 0 |

10 | 0 | 0 | 1 | 1 | 1 | 0 |

11 | 0 | 0 | 1 | 1 | 0 | 1 |

12 | 0 | 1 | 1 | 1 | 0 | 0 |

13 | 1 | 0 | 1 | 1 | 1 | 0 |

14 | 1 | 0 | 1 | 1 | 0 | 1 |

15 | 0 | 0 | 0 | 0 | 1 | 1 |

16 | 1 | 0 | 0 | 0 | 1 | 1 |

17 | 0 | 0 | 1 | 0 | 1 | 1 |

18 | 0 | 0 | 0 | 1 | 1 | 1 |

19 | 0 | 1 | 1 | 0 | 1 | 1 |

20 | 0 | 1 | 0 | 1 | 1 | 1 |

21 | 1 | 1 | 1 | 1 | 1 | 1 |

22 | 1 | 1 | 0 | 0 | 0 | 0 |

23 | 1 | 0 | 1 | 1 | 0 | 0 |

24 | 0 | 1 | 1 | 1 | 1 | 0 |

25 | 0 | 1 | 1 | 1 | 0 | 1 |

26 | 0 | 1 | 0 | 0 | 1 | 1 |

27 | 1 | 0 | 1 | 0 | 1 | 1 |

28 | 1 | 0 | 0 | 1 | 1 | 1 |

No. | 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 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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 | + | + | + |

off- ${\mathcal{Q}}_{0}$ | G_{2}(8) | off- ${\mathcal{Q}}_{0}$ | G_{2}(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} |

# of Heptads Removed | Configuration | CG | Remark |
---|---|---|---|

0 | (28_{6}, 56_{3}) | G_{2}(8) | |

1 | (21_{5}, 35_{3}) | G_{2}(7) | |

2 | (15_{4}, 20_{3}) | G_{2}(6) | Cayley-Salmon |

3 | (10_{3}, 10_{3}) | G_{2}(5) | Desargues |

4 | (6_{2}, 4_{3}) | G_{2}(4) | Pasch |

5 | (3_{1}, 1_{3}) | G_{2}(3) | single line |

6 | (1_{0}, 0_{3}) | G_{2}(2) | single point |

7 | empty set |

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).