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 (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. The 28 points lying off the quadric ${\mathcal{Q}}_0$.

No.	x1	x2	x3	x4	x5	x6
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

and the 56 external lines are those given in

Table 2. The 56 lines having no points in common with the quadric ${\mathcal{Q}}_0$.

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																					+		+			+

. 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 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, 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. A bijection between the 28 off-quadric points and the 28 points of G₂(8).

off- ${\mathcal{Q}}_0$	G2(8)	off- ${\mathcal{Q}}_0$	G2(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}

(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 ${\mathcal{Q}}^{+}(5,2)$ 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 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. A nested sequence of configurations located in the complement of a hyperbolic quadric of PG(5, 2).

# of Heptads Removed	Configuration	CG	Remark
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
7			empty set

. 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 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 K_n”) 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 S₈ ≅ SL₄(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).