Relative hemisystems on the Hermitian surface

J Algebr Comb (2013) 38:275–284 DOI 10.1007/s10801-012-0403-z Relative hemisystems on the Hermitian surface Antonio Cossidente Received: 25 January 2012 / Accepted: 2 October 2012 / Published online: 13 October 2012 © Springer Science+Business Media New York 2012 Abstract Let S be a generalized quadrangle of order (q 2 , q) containing a subquadrangle S  of order (q, q). Then any line of S either meets S  in q + 1 points or is disjoint from S  . After Penttila and Williford (J. Comb. Theory, Ser. A 118:502– 509, 2011), we call a subset H of the lines disjoint from S  a relative hemisystem of S with respect to S  , provided that for each point x of S \ S  , exactly half of the lines through x disjoint from S  lie in H . A new infinite family of relative hemisystems on the generalized quadrangle H(3, q 2 ) admitting the linear group PSL(2, q) as an automorphism group is constructed. The association schemes arising from our construction are not equivalent to those arising from the Penttila–Williford relative hemisystems. Keywords Generalized quadrangle · Relative hemisystem · Association scheme 1 Introduction and motivation A generalized quadrangle of order (s, t) is an incidence structure of points and lines with the properties that any two points (lines) are incident with at most one line (point), every point is incident with t + 1 lines, every line is incident with s + 1 points, and for any point P and line l that are not incident, there is a unique point on l collinear with P . The standard reference is [12]. One of the classical generalized quadrangles is H(3, q 2 ), the incidence structure of all points and lines of a nonsingular Hermitian surface in PG(3, q 2 ). It is a generalized quadrangle of order (q 2 , q) with automorphism group PΓ U(4, q 2 ). The dual of H(3, q 2 ) is the generalized quadrangle Q− (5, q), the incidence structure of all points A. Cossidente () Department of Mathematics and Computer Sciences, University of Basilicata, Contrada Macchia Romana, 85100 Potenza, Italy e-mail: 276 J Algebr Comb (2013) 38:275–284 and lines of an elliptic quadric in PG(5, q), a generalized quadrangle of order (q, q 2 ), with automorphism group PΓ O− (6, q), [12, Theorem 3.2.3]. In his celebrated paper [14], Beniamino Segre introduced the notion of regular system on H(3, q 2 ). A regular system of order m on H(3, q 2 ) is a set R of lines of H(3, q 2 ) with the property that every point lies on exactly m lines of R, 0 < m < q + 1. Segre proved that, if q is odd, such a system must necessarily have m = (q + 1)/2 and called a regular system on H(3, q 2 ) of order (q + 1)/2 a hemisystem on H(3, q 2 ). He also constructed a hemisystem on H(3, 9) admitting the linear group PSL(3, 4) as an automorphism group. In [3], the nonexistence of regular systems on H(3, q 2 ) for q even was established. A simple proof that a regular system on H(3, q 2 ) is a hemisystem (and so q is odd) was also given by Thas [15]. Cameron, Goethals, and Seidel [6] adopted a more general approach defining a hemisystem on a generalized quadrangle of order (s, s 2 ), s odd, to be a set of points meeting every line in (s + 1)/2 points. In 1995, Thas [16] conjectured the nonexistence of hemisystems on H(3, q 2 ) for q > 3. In 2005, exactly forty years after Segre’s paper appeared, Penttila and Cossidente [7] presented counterexamples to this conjecture. They proved that the generalized quadrangle H(3, q 2 ), q odd, has a hemisystem admitting PΩ − (4, q) and giving Segre’s example for q = 3. Also they constructed a “sporadic” example on H(3, 25) admitting 3.A7 .2. The interest in constructing hemisystems on generalized quadrangles is motivated, for instance, by the study of the so-called partial quadrangles. These were introduced by Cameron [5]. A partial quadrangle PQ(s, t, μ) is an incidence structure of points and lines with the properties that any two points are incident with at most one line, every point is incident with t + 1 lines, every line is incident with s + 1 points, any two noncollinear points are jointly collinear with exactly μ points, and for any point P and line l that are not incident, there is at most one point Q on l collinear with P . However, there are not many constructions of partial quadrangles known, and most of them arise from a generalized quadrangle of order (s, s 2 ) by deleting a point, all lines on that point, and all points collinear with that point; this gives a PQ(s − 1, s 2 , s 2 − s). The preceding results imply that a hemisystem on a generalized quadrangle of order (s, s 2 ) gives a partial quadrangle PQ((s − 1)/2, s 2 , (s − 1)2 /2) (the points of the partial quadrangle being the points of the hemisystem, and the lines of the partial quadrangle being the lines of the generalized quadrangle). Hemisystems on H(3, q 2 ) are also related to strongly regular graphs. Indeed, a hemisystem yields a strongly regular decomposition of the collinearity graph of Q− (5, q) in the sense of [9]. Indeed, in [11], strongly regular graphs whose vertices can be partitioned into two subsets of equal size on each of which a strongly regular subgraph is induced are investigated, and it is shown that such graphs must belong either to the two parameter family of Smith graphs, see [6, Sect. 6], or to a special one parameter family of graphs. Notice that the collinearity graph of Q− (5, q) is a Smith graph. Finally, the construction of hemisystems in [7] was extended in a very clever way in the beautiful paper [1] of Bamberg, Giudici, and Royle by proving that every flock generalized quadrangle of order (s 2 , s), s odd, contains a hemisystem. J Algebr Comb (2013) 38:275–284 277 As we have already observed, regular systems on H(3, q 2 ) for even q cannot exist. Is there an analogue of hemisystems on H(3, q 2 ), q even? The answer is affirmative and is contained in a recent paper by Penttila and Williford [13]. Let S be a generalized quadrangle of order (q 2 , q) containing a subquadrangle S  of order (q, q). Then any line of S either meets S  in q +1 points or is disjoint from S  . After [13], we call a subset H of the lines disjoint from S  a relative hemisystem of S with respect to S  , provided that for each point x of S \ S  , exactly half of the lines through x disjoint from S  lie in H . Of course, q must be even for this to be possible, since every point of S \ S  is on q lines disjoint from S  . It happens that the symplectic generalized quadrangle W(3, q) that has order (q, q) can always be embedded in the generalized quadrangle H(3, q 2 ) as we will show later (Sect. 2.1) and a relative hemisystem of H(3, q 2 ) with respect to W(3, q) produces certain association schemes. Indeed, Penttila and Williford [13] proved that a relative hemisystem of H(3, q 2 ) always gives rise to a primitive 3-class cometric association scheme, which is not metric. Theorem 1.1 [13, Theorem 4] If H(3, q 2 ), q > 2 has a relative hemisystem with respect to W(3, q), then a primit (...truncated)