Variations on the Dembowski-Wagner theorem

Let D be a symmetric design admitting a null polarity such that either all singular lines, or all nonsingular lines, have size (v − λ)/(k − λ); assume that this number is greater than λ in the case of singular lines. Then D is either a projective space or an orthogonal symmetric design. © 2003 Elsevier Ltd. All rights reserved. The Dembowski–Wagner Theorem [3] characterizes projective spaces as the only symmetric designs such that all lines have size (v − λ)/(k − λ). This note contains two analogous characterization results concerning a symmetric design admitting a null polarity. Whereas the proof in [3] depends on classical axioms for projective spaces [8], the results below follow very easily from far more difficult theorems due to Buekenhout and Shult [2] and Hall [5] concerning polar spaces. Recall that, for any symmetric design D, if x and y are distinct points then their line xy is defined to be the intersection of all blocks containing x and y. Two points are on exactly one line, and a line of size (v−λ)/(k −λ) has nonempty intersection with every block [3]. Assume that D is equipped with a null polarity x → x⊥ (thus, x ∈ x⊥ for all points x). We call a line singular if it contains distinct points x, y such that y ∈ x⊥, and nonsingular otherwise. If x = z ∈ xy ⊆ x⊥ then x, z ⊆ z⊥ and hence y ∈ xy = xz ⊆ z⊥: a line cannot be both singular and nonsingular. In addition to the projective geometries PG(d, q) we will need to consider Higman’s “orthogonal” symmetric designs [6], having the same parameters; its points are the singular points x of a d-dimensional orthogonal GF(q)-space with q and d odd, and its blocks correspond to the hyperplanes x⊥. Theorem 1. Let D be a symmetric design admitting a null polarity such that all singular lines have size (v − λ)/(k − λ) > λ. Then D is either a projective space or an orthogonal design. E-mail address: [email protected] (W.M. Kantor). 0195-6698/$ see front matter © 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0195-6698(03)00109-4 240 W.M. Kantor / European Journal of Combinatorics 25 (2004) 239–241 Proof. Consider the geometry G consisting of the points of D and the singular lines. By hypothesis, every block z⊥ has nonempty intersection with every singular line L; and if z⊥ contains two points of L then it contains L. This is just the Buekenhout–Shult property for G. Clearly no point is perpendicular to all others; and the hypothesis (v−λ)/(k−λ) > λ implies that there are three noncollinear pairwise perpendicular points. It follows from [2] that G arises from a polar space. The polar space determines the blocks of D since x⊥ consists of the points collinear with x (with respect to G); and hence G produces a symmetric design if and only if |x⊥ ∩ y⊥| is the same whenever x and y are distinct. It is easy to check all polar spaces in order to see that only symplectic and odd-dimensional orthogonal ones produce symmetric designs. The designs in the symplectic case are projective spaces. Theorem 2. Let D be a symmetric design admitting a null polarity such that all nonsingular lines have size (v − λ)/(k − λ). Then D is a projective space. Proof. This time consider the geometry G′ consisting of the points of D and the nonsingular lines. Every block z⊥ has nonempty intersection with each nonsingular line L. This time, if z⊥ ⊇ L then no line zu, u ∈ L, is in G′, while if z⊥ ∩ L is a pointw then zu is a line of G′ for each u ∈ L−{w}. Thus, G′ is a “copolar space” and is “indecomposable” and “reduced” in the sense of [5]. All possibilities for G′ were determined in [5, Theorem 2]. Each possible G′ determines the blocks of D since x⊥ consists of x and the points not collinear with x (with respect to G′); and, as above, G′ produces a symmetric design if and only if |x⊥ ∩ y⊥| is the same same whenever x and y are distinct. It is straightforward to check the geometries listed in [5, Theorem 2] in order to see that the only ones arising from symmetric designs are the geometries of points and nonsingular lines of symplectic geometries. Once again the corresponding designs are projective spaces. Remark 1. It is natural to wonder about theorems of the following sort: if a symmetric design has sufficiently many lines of size (v − λ)/(k − λ) then it must be a projective space. How many is “sufficiently many” ? One might conjecture that it is enough to have more than half the number L(d, q) = (qd+1 − 1)(qd − 1)/(q2 − 1)(q − 1) of such lines that PG(d, q) has. However, elementary examples obtained using the method in [7] show that there are symmetric designs with the same parameters as PG(d, q), d ≥ 3, other than PG(d, q), having at least (qd−1 − qd−2)qd−1 = L(d, q)(1 − o(1/q)) lines of size q + 1. Remark 2. These theorems were motivated by an application to a family of symmetric designs related to the groups G2(q) [4]. In memory. The involvement of Jaap Seidel in [2] was described at length in [1], including the following: “Seidel laid much pressure on me with his unique insight and skill to coach. He insisted that I got in touch with Shult and I did so. Eventually the story led to the Buekenhout–Shult 1974 theory. The team was built by Seidel”. These skills are familiar to all who knew Jaap.