arXiv:1001.0659v2 [math-ph] 16 Feb 2010 On the Veldkamp Space of GQ(4,2) Metod Saniga Astronomical Institute, Slovak Academy of Sciences SK-05960 Tatransk´ a Lomnica, Slovak Republic (msaniga@astro.sk) (16 February 2010) Abstract The Veldkamp space, in the sense of Buekenhout and Cohen, of the generalized quadrangle GQ(4, 2) is shown not to be a (partial) linear space by simply giving several examples of Veldkamp lines (V-lines) having two or even three Veldkamp points (V-points) in common. Alongside the ordinary V-lines of size five, one also finds V-lines of cardinality three and two. There, however, exists a subspace of the Veldkamp space isomorphic to PG(3,4) having 45 perps and 40 plane ovoids as its 85 V-points, with its 357 V-lines being of four distinct types. A V-line of the first type consists of five perps on a common line (altogether 27 of them), the second type features three perps and two ovoids sharing a tricentric triad (240 members), whilst the third and fourth type each comprises a perp and four ovoids in the rosette centered at the (common) center of the perp (90). It is also pointed out that 160 non-plane ovoids (tripods) fall into two distinct orbits — of sizes 40 and 120 — with respect to the stabilizer group of a copy of GQ(2, 2); a tripod of the first/second orbit sharing with the GQ(2,2) a tricentric/unicentric triad, respectively. Finally, three remarkable subconfig- urations of V-lines represented by fans of ovoids through a fixed ovoid are examined in some detail. Keywords: GQ(4, 2) — Geometric Hyperplane — Veldkamp Space — PG(3, 4) 1 Introduction Generalized quadrangles of types GQ(2,t), with t = 1, 2, and 4, have recently been found to play a prominent role in quantum information and black hole physics; the first type for grasping the geometrical nature of the so-called Mermin squares [1, 2], the second for underlying commutation properties between the elements of two-qubit Pauli group [1, 2, 3], and the third one for fully encoding the E 6(6) symmetric entropy formula describing black holes and black strings in D =5 [4]. Whereas GQ(2, 2) is isomorphic to its point-line dual, this is not the case with the remaining two geometries; the dual of GQ(2, 1) being GQ(1, 2), that of GQ(2, 4) GQ(4, 2) [5]. These two duals, strangely, did not appear in the above-mentioned physical contexts. It is, therefore, natural to ask why this is so. We shall try to shed light on this matter by invoking the concept of the Veldkamp space of a point line incidence structure [6]. The Veldkamp space of GQ(2,4) was shown to be a linear space isomorphic to PG(5, 2) [7]. Here, we shall demonstrate that the Veldkamp space of GQ(4,2), due to the existence of two different kinds of ovoids in GQ(4, 2), is not even a partial linear space, though it contains a (linear) subspace isomorphic to PG(3, 4) in which GQ(4, 2) lives as a non-degenerate Hermitian variety. 2 Rudiments of the Theory of Finite Generalized Quadran- gles and Veldkamp Spaces To make our exposition as self-contained as possible, and for the reader’s convenience as well, we will first gather all essential information about finite generalized quadrangles [5], then introduce the concept of a geometric hyperplane [8] and, finally, that of the Veldkamp space of a point-line incidence geometry [6]. A finite generalized quadrangle of order (s, t), usually denoted GQ(s, t), is an incidence structure S =(P,B, I), where P and B are disjoint (non-empty) sets of objects, called respectively points and lines, and where I is a symmetric point-line incidence relation satisfying the following axioms [5]: (i) each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point; and (iii) if x is a point and L is a line not incident with x, then there 1