The Complement of Binary Klein Quadric as a Combinatorial Grassmannian

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 (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 (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. 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).