Relative hemisystems on the Hermitian surface

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 $\mathcal{H}(3,q^{2})$ 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.

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 $\mathcal{H}(3,q^{2})$, 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 $\mathrm{P\varGamma U}(4,q^{2})$. The dual of $\mathcal{H}(3,q^{2})$ is the generalized quadrangle $\mathcal{Q}^{-}(5,q)$, the incidence structure of all points and lines of an elliptic quadric in PG(5,q), a generalized quadrangle of order (q,q ²), with automorphism group PΓO⁻(6,q), [12, Theorem 3.2.3].

In his celebrated paper [14], Beniamino Segre introduced the notion of regular system on $\mathcal{H}(3,q^{2})$. A regular system of order m on $\mathcal{H}(3,q^{2})$ is a set $\mathcal{R}$ of lines of $\mathcal{H}(3,q^{2})$ with the property that every point lies on exactly m lines of $\mathcal{R}$, 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 $\mathcal{H}(3,q^{2})$ of order (q+1)/2 a hemisystem on $\mathcal{H}(3,q^{2})$. He also constructed a hemisystem on $\mathcal{H}(3,9)$ admitting the linear group PSL(3,4) as an automorphism group.

In [3], the nonexistence of regular systems on $\mathcal {H}(3,q^{2})$ for q even was established. A simple proof that a regular system on $\mathcal{H}(3,q^{2})$ 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 $\mathcal{H}(3,q^{2})$ 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 $\mathcal{H}(3,q^{2})$, q odd, has a hemisystem admitting PΩ ⁻(4,q) and giving Segre’s example for q=3. Also they constructed a “sporadic” example on $\mathcal{H}(3,25)$ admitting 3.A ₇.2. 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 $\mathcal{H}(3,q^{2})$ are also related to strongly regular graphs. Indeed, a hemisystem yields a strongly regular decomposition of the collinearity graph of $\mathcal{Q}^{-}(5,q)$ 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 $\mathcal{Q}^{-}(5,q)$ 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.

As we have already observed, regular systems on $\mathcal{H}(3,q^{2})$ for even q cannot exist. Is there an analogue of hemisystems on $\mathcal{H}(3,q^{2})$, 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 $\mathcal {W}(3,q)$ that has order (q,q) can always be embedded in the generalized quadrangle $\mathcal{H}(3,q^{2})$ as we will show later (Sect. 2.1) and a relative hemisystem of $\mathcal{H}(3,q^{2})$ with respect to $\mathcal{W}(3,q)$ produces certain association schemes. Indeed, Penttila and Williford [13] proved that a relative hemisystem of $\mathcal{H}(3,q^{2})$ always gives rise to a primitive 3-class cometric association scheme, which is not metric.

Theorem 1.1

[13, Theorem 4]

If $\mathcal{H}(3,q^{2})$, q>2 has a relative hemisystem with respect to $\mathcal{W}(3,q)$, then a primitive Q-polynomial 3-class scheme can be constructed on the lines of the relative hemisystem with the following relations:

• R ₁: We have (l,m)∈R ₁ if and only if l and m are not concurrent and |O _l ∩O _m |=1;

• R ₂: We have (l,m)∈R ₂ if and only if l and m are not concurrent and |O _l ∩O _m |=q+1;

• R ₃: We have (l,m)∈R ₃ if and only if l and m are concurrent;

where for a fixed line l which does not meet $\mathcal{W}(3,q)$, O _l denotes the set of lines meeting both l and $\mathcal{W}(3,q)$.

Also, they showed that $\mathcal{H}(3,q^{2})$, q even, has a relative hemisystem with respect to an embedded $\mathcal{W}(3,q)$ admitting the orthogonal group PΩ ⁻(4,q) [13, Theorem 5] and left as an open question whether or not $\mathcal{H}(3,q^{2})$ has any nonisomorphic relative hemisystems. Notice that the Penttila–Williford relative hemisystem could be considered the analogue of the Cossidente–Penttila hemisystem just because they have the same automorphism group.

In this paper we will construct a new infinite family of relative hemisystems of $\mathcal{H}(3,q^{2})$, q=2^m, m>2, admitting the linear group PSL(2,q) as an automorphism group. Our approach is elementary in nature. We start from the two Penttila–Williford relative hemisystems arising from the action of the group PΩ ⁻(4,q) on generators of $\mathcal{H}(3,q^{2})$ disjoint from a given subquadrangle $\mathcal{W}(3,q)$. This means that an elliptic quadric $\mathcal{Q}^{-}(3,q)$ of $\mathcal{W}(3,q)$ has been fixed. Then, we consider a conic section of $\mathcal{Q}^{-}(3,q)$ and its group that is isomorphic to PSL(2,q). Of course, each of the two Penttila–Williford relative hemisystems splits into orbits under the action of PSL(2,q). The idea is to consider a mild perturbation of the Penttila–Williford relative hemisystems suitably “interchanging” certain PSL(2,q)-orbits.

2 The generalized quadrangle $\mathcal{H}(3,q^{2})$

In this section we give some general information on the generalized quadrangle $\mathcal{H}(3,q^{2})$.

In PG(3,q ²) a nonsingular Hermitian surface is defined to be the set of all absolute points of a nondegenerate unitary polarity and is denoted by $\mathcal{H}(3,q^{2})$.

A Hermitian surface $\mathcal{H}(3,q^{2})$ has the following properties, for which [14], [10, Chap. 19] are excellent sources.

• 1. The number of points on $\mathcal{H}(3,q^{2})$ is (q ²+1)(q ³+1).

• 2. Any line of Σ=PG(3,q ²) meets $\mathcal {H}(3,q^{2})$ in either 1 or q+1 or q ²+1 points. The latter lines are the generators of $\mathcal {H}(3,q^{2})$, and there are (q+1)(q ³+1) such lines. The intersections of size q+1 are Baer sublines, whereas lines meeting $\mathcal {H}(3,q^{2})$ in one point are called tangent lines.

• 3. Through every point P of $\mathcal{H}(3,q^{2})$ there pass exactly q+1 generators, and these generators are coplanar. The plane containing these generators, say π _P , is the polar plane of P with respect to the unitary polarity defining $\mathcal{H}(3,q^{2})$. The tangent lines through P are precisely the remaining q ²−q lines of π _P incident with P, and π _P is called the tangent plane to $\mathcal{H}$ at P.

• 4. Every plane of Σ which is not a tangent plane to $\mathcal{H}(3,q^{2})$ meets $\mathcal{H}(3,q^{2})$ in a nondegenerate Hermitian curve $\mathcal {H}(2,q^{2})$.

2.1 $\mathcal{W}(3,q)$ inside $\mathcal{H}(3,q^{2})$

The symplectic group PSp(4,q) is embedded in PΓU(4,q ²) as a subfield subgroup, stabilizing a subquadrangle of $\mathcal{H}(3,q^{2})$ isomorphic to $\mathcal{W}(3,q)$. In terms of coordinates, assuming that $\mathcal{H}(3,q^{2})$ has the equation $X_{1}^{q+1}+X_{2}^{q+1}+X_{3}^{q+1}+X_{4}^{q+1}=0$, where X ₁,…,X ₄ are homogeneous projective coordinates in PG(3,q ²), q even, the set {(x,x ^q,y,y ^q):x,y∈GF(q ²)} is the point set of a symplectic Baer subgeometry $\mathcal{W}(3,q)$ embedded in $\mathcal{H}(3,q^{2})$ [14]. The stabilizer of $\mathcal{W}(3,q)$ in PΓU(4,q ²) has exactly two orbits on points of $\mathcal{H}(3,q^{2})$, $\mathcal {W}(3,q)$ and its complement. In particular, every generator of $\mathcal{H}(3,q^{2})$ either meets $\mathcal{W}(3,q)$ in a totally isotropic line of $\mathcal{W}(3,q)$ or is disjoint from it. This means that a point of $\mathcal{H}(3,q^{2})\setminus\mathcal {W}(3,q)$ lies on a unique totally isotropic line of $\mathcal{W}(3,q)$. The group PSp(4,q) has two conjugacy classes of subgroups isomorphic to PSL(2,q ²)≃PΩ ⁻(4,q). A group in the first class stabilizes a regular spread of $\mathcal{W}(3,q)$ (i.e., a partition of the point set of $\mathcal{W}(3,q)$ into q ²+1 totally isotropic lines). A group in the second class stabilizes an elliptic quadric $\mathcal{Q}^{-}(3,q)$ that is an ovoid of $\mathcal {W}(3,q)$ (i.e., a subset E of q ²+1 points of $\mathcal{W}(3,q)$ such that every totally isotropic line of $\mathcal{W}(3,q)$ meets E at exactly one point). We are interested in this second class. Fix an elliptic quadric $\mathcal{Q}^{-}(3,q)$ of $\mathcal{W}(3,q)$ and its group PΩ ⁻(4,q). Let us fix a conic section, say $\mathcal{C}$, of $\mathcal{Q}^{-}(3,q)$ obtained by intersecting $\mathcal{Q}^{-}(3,q)$ with a plane, say π. Since q is even, $\mathcal{C}$ has a nucleus N. It turns out that $\pi=N^{\mathcal{A}}$, where $\mathcal{A}$ is the symplectic polarity fixed by PSp(4,q). There are q+1 totally isotropic lines on N in π, and each of them is tangent to $\mathcal{C}$. By Witt’s theorem, every isometry of $\mathcal{A}$ extends to a semisimilarity of $\mathcal{U}$, where $\mathcal{U}$ is the unitary polarity fixed by PΓU(4,q ²). It follows that the GF(q ²)-span of π is the tangent plane $\pi^{\mathcal{U}}$ to $\mathcal{H}(3,q^{2})$ at N. The stabilizer of $\mathcal{C}$ in PΩ ⁻(4,q) is a group isomorphic to G=PSL(2,q). The group G stabilizes q/2 elliptic quadrics of $\mathcal{W}(3,q)$, all sharing the conic $\mathcal{C}$ [4, Type 3g(ii), p. 262]. Also, let t be the unique nonlinear involutory automorphism fixing every point of $\mathcal {W}(3,q)$. It is easily seen that t commutes with G.

3 The Penttila–Williford relative hemisystem

Here, we briefly recall the construction of the infinite family of relative hemisystems of $\mathcal{H}(3,q^{2})$ due to Penttila and Williford [13]. We have already observed that the stabilizer of $\mathcal{W}(3,q)$ in PΓU(4,q ²) has exactly two orbits on points of $\mathcal{H}(3,q^{2})$, namely $\mathcal{W}(3,q)$ and its complement. Also, it acts transitively on totally isotropic lines of $\mathcal{W}(3,q)$. Using the action of the normalizer of a Singer cyclic group of PΩ ⁻(4,q), Penttila and Williford proved that PΩ ⁻(4,q), q>2, has exactly two orbits on generators of $\mathcal{H}(3,q^{2})$ disjoint from $\mathcal{W}(3,q)$ [13, Theorem 5]. It turns out that each of these orbits, say H₁ and H₂, is a relative hemisystem with respect to $\mathcal{W}(3,q)$. By construction, the involution t introduced above switches the two relative hemisystems H₁ and H₂ [13, Theorem 5].

4 The new family

We adopt the geometric setting of the Penttila–Williford construction. In particular, we start from the two relative hemisystems H₁ and H₂ described in Sect. 3. From now on we assume that q>2.

From our discussion in Sect. 2.1 it follows that the group G permutes the q+1 generators of $\mathcal{H}(3,q^{2})$ on N. Let g be one of such generators. The stabilizer K of g in G has order q(q−1). The subgroup S of K of order q, which is elementary abelian, fixes g pointwise. Hence, each involution in S has g as an axis [8]. A subgroup B of K of order q−1 partitions the point set of $g\setminus\{N, g\cap\mathcal {C}\}$ into q+1 orbits of size q−1, q of which consist of imaginary points of g, and exactly one of them lies on $\mathcal {W}(3,q)$. Call the B-orbits on imaginary points of g, Σ ₁,…,Σ _q . Since G permutes the generators on N, each orbit Σ _i gives rise to a G-orbit of size q ²−1, say $\overline{\varSigma}_{i}$, i=1,…,q, lying on $N^{\mathcal{U}}$. If $P\in\overline{\varSigma}_{i}$, the stabilizer of P in G is S.

Lemma 4.1

The 2-group S acts on the q generators of $\mathcal {H}(3,q^{2})$ on P distinct from g with two orbits of size q/2.

Proof

Since G is a subgroup of the group PΩ ⁻(4,q) stabilizing a Penttila–Williford relative hemisystem, any PΩ ⁻(4,q)-orbit on generators through P is the union of G-orbits. Under the action of PΩ ⁻(4,q), the set of q generators on P distinct from g splits into two orbits of size q/2. On the other hand, the action of S on points of $\mathcal{H}(3,q^{2})\setminus\{g\}$ is semiregular, and hence all S-orbits in $P^{\mathcal{U}}\setminus\{g\}$ have size q. We want to prove that each S-orbit in $P^{\mathcal{U}}\setminus\{g\}$ is an arc.

Let us choose d∈GF(q) so that the polynomial x ²+dxy+y ² is irreducible over GF(q). Define Q:GF(q)⁴→GF(q) by Q(x,y,z,w)=xw+y ²+dyz+z ². Then Q is a quadratic form of minus type. Let $\mathcal{Q}^{-}(3,q)$ be the set of zeroes of Q, so that $\mathcal{Q}^{-}(3,q)$ is the elliptic quadric {(1,s,t,s ²+dst+t ²):s,t∈GF(q)}∪{(0,0,0,1)}∈PG(3,q). Let

It is easy to see that M _λ is an isometry of Q. Let S={M _λ :λ∈GF(q)}. Then S is an elementary abelian group of order q. The group S fixes the conic $\mathcal{C}=\{(1,s,0,s^{2}):s \in\mathrm{GF}(q)\}\cup\{(0,0,0,1)\}$ obtained by sectioning $\mathcal{Q}^{-}(3,q)$ with the plane z=0. The conic $\mathcal{C}$ has nucleus (0,1,0,0). Let us consider now the Hermitian form H((x,y,z,w),(x′,y′,z′,w′))=xw′^q+wx′^q+dyz′^q+dzy′^q, and let $\mathcal{H}(3,q^{2})$ be the associated Hermitian surface. Let g={(0,y,0,w):y,z∈GF(q ²)}. Then g is a generator of $\mathcal{H}(3,q^{2})$ on the point {(0,0,0,1)}. Let P=(0,1,0,a) with a∈GF(q ²)∖GF(q). Then $P^{\mathcal{U}}$ ($\mathcal{U}$ is the Hermitian polarity associated with $\mathcal {H}(3,q^{2})$) has the equation z=(a ^q/d)x. Restricting S to $P^{\mathcal{U}}=\{(x,y,(a^{q}/d)x,w):x,y,w\in\mathrm{GF}(q^{2})\}$ shows that M _λ acts as

on the coordinates x,y,w, which has each orbit on $P^{\mathcal{U}}$ not on g, a q-arc. Indeed, the determinant of the matrix

with λ ₁,λ ₂≠0 and λ ₁≠λ ₂ is $x^{3}(\lambda_{1}\lambda_{2}^{2}+\lambda_{1}^{2}\lambda_{2})$, which is zero if and only if the point (x,y,w) lies on g or λ ₁=λ ₂.

Since the isometry group of $\mathcal{Q}^{-}(3,q)$ is transitive on conic sections of $\mathcal{Q}^{-}(3,q)$, we can choose $\mathcal{C}$ as we did. Since the group of $\mathcal{C}$ is transitive on points of $\mathcal {C}$, we can always assume that the generator g contains the point (0,0,0,1). Finally, $\mathcal{Q}^{-}(3,q)$ determines $\mathcal{H}(3,q^{2})$. Notice that the group S acts on $P^{\mathcal{U}}$ as an elation group of order q whose centers lie on g. Let A be an S-orbit in $P^{\mathcal{U}}\setminus\{g\}$. Let X∈A, and let ι be an involution in S. Then ι(X)∈A, and the line Xι(X) meets g at one point, which is the center of ι. Since A is an arc, the centers of the involutions of S are all distinct. In particular, there is exactly one involution in S (M _λ for λ=a+a ^q) fixing each of the q generators on P distinct from g. Hence, the points of A lie on q/2 generators, A meets each of the q/2 generators lying on an orbit of the stabilizer of P in $\mathrm{P}\varOmega_{4}^{-}(q)$ exactly twice, and the q/2 generators on P in the stabilizer of P in PΩ ⁻(4,q) are also an orbit of the stabilizer of P in G. □

It follows that G has 2q orbits of size (q ³−q)/2 on generators meeting $\overline{\varSigma}_{i}$, i=1,…,q (distinct from the generators on N). Of course, all these q ²(q ²−1) generators are disjoint from $\mathcal{W}(3,q)$. The number of generators of $\mathcal{H}(3,q^{2})$ disjoint from $\mathcal{W}(3,q)$ is (q+1)(q ³−q ²)=q ²(q ²−1), and hence the 2q orbits above cover all generators of $\mathcal{H}(3,q^{2})$ disjoint from $\mathcal{W}(3,q)$.

We are ready to construct our new relative hemisystem by perturbating a Penttila–Williford relative hemisystem.

The involution t fixing $\mathcal{W}(3,q)$ fixes no point outside $\mathcal{W}(3,q)$ and hence partitions the set {Σ ₁,…,Σ _q } into pairs {Σ _i ,t(Σ _i )}. We recall that the involution t interchanges H₁ and H₂. Each such pair produces four G-orbits on generators of size (q ³−q)/2, say X _i and t(Y _i ) (arising from $\overline{\varSigma }_{i}$) and Y _i and t(X _i ) (arising from $t(\overline{\varSigma_{i}})$), where we can assume that X _i and Y _i are in H ₁ and hence that t(X _i ) and t(Y _i ) are in H ₂. If P(X) denotes the point set of $\mathcal{H}(3,q^{2})$ covered by the generators in the set X, we have that P(X _i )∪P(Y _i ) is the same as P(t(X _i ))∪P(t(Y _i )), and both sets are t-invariant. Indeed, X _i is a G-orbit whose generators meet $\overline{\varSigma}_{i}$. Then P(X _i ) is the union of point G-orbits, and one of them is certainly $\overline{\varSigma }_{i}$. All the other point G-orbits have full size q(q ²−1) (if R is a point of $\mathcal{H}(3,q^{2})\setminus\mathcal{W}(3,q)$, then it can be proved that Stab _G (R) acts as the identity on $\mathcal {H}(3,q^{2})\setminus\mathcal{W}(3,q)$). The involution t sends $\overline{\varSigma}_{i}$ to $t(\overline{\varSigma}_{i})$ and either fixes a point G-orbit on P(X _i ) or moves a point G-orbit O on P(X _i ) to another point G-orbit, say t(O), of course of the same size. Since the generators of H₁ cover the whole of $\mathcal{H}(3,q^{2})\setminus\mathcal{W}(3,q)$, t(O) is a point G-orbit on generators belonging again to H₁. Such a point G-orbit must be on P(Y _i ) since t sends $\overline{\varSigma }_{i}$ to $t(\overline{\varSigma}_{i})$. On the other hand, t interchanges H₁ and H₂, and hence we have the following lemma.

Lemma 4.2

The sets P(X _i )∪P(Y _i ) and P(t(X _i ))∪P(t(Y _i )) are both t-invariant and cover the same point set of $\mathcal{H}(3,q^{2})$

We have the following theorem.

Theorem 4.3

There exists an infinite family of relative hemisystems of $\mathcal {H}(3,q^{2})$, q even, q>4, admitting PSL(2,q) as an automorphism group.

Proof

The key idea is to consider a mild perturbation of one of the two relative hemisystems of Penttila and Williford arising from a given elliptic quadric on $\mathcal{C}$ fixed by G=PSL(2,q), by deleting certain G-orbits from H₁ and by adding their images under t belonging to H₂ as in Lemma 4.2. Since q>4, there exist (apart from H₁ and H₂) at least q other relative hemisystems fixed by G:

On the other hand, the number of G-invariant relative hemisystems of Penttila and Williford (H₁ and H₂ included) is exactly q. It follows that there exists a relative hemisystem admitting G as an automorphism group. □

Remark 4.4

From the Penttila–Williford construction, once fixed a symplectic subquadrangle $\mathcal{W}(3,q)$ of $\mathcal{H}(3,q^{2})$, it follows that any elliptic quadric on the conic $\mathcal{C}$ gives rise to two Penttila–Williford relative hemisystems. Since the number of elliptic quadrics in $\mathcal{W}(3,q)$ on $\mathcal{C}$ stabilized by G is q/2, certainly there exist q relative hemisystems of Penttila–Williford type on $\mathcal{H}(3,q^{2})$. Assume that q=4. In this case we have eight G-orbits on generators of $\mathcal {H}(3,16)$ disjoint from $\mathcal{W}(3,4)$, each of size 60, say X ₁, t(X ₁), Y ₁, t(Y ₁), Z ₁, t(Z ₁), U ₁, t(U ₁). Since there are two elliptic quadrics of $\mathcal{W}(3,4)$ on $\mathcal{C}$, there are four Penttila–Williford relative hemisystems:

Let us fix an elliptic quadric, say E ₁, of $\mathcal{W}(3,4)$ on $\mathcal{C}$. We get two Penttila–Williford relative hemisystems, suppose H ₁ and H ₂. From our construction, if we choose, for instance, X ₁ and Y ₁ and replace them with t(X ₁) and t(Y ₁), we get the relative hemisystem H₄ whose complement is the relative hemisystem H₃, and these two relative hemisystems are of Penttila–Williford type arising from the second elliptic quadric, say E ₂, of $\mathcal{W}(3,q)$ on $\mathcal{C}$. Of course, no other replacement is possible. This is why q>4 in the theorem is required. In other words, any PSL₂(4)-relative hemisystem is of Penttila–Williford type.

Remark 4.5

The smallest Hermitian surface $\mathcal{H}(3,4)$ contains 45 points and 27 lines. Fix a symplectic Baer subgeometry $\mathcal {W}(3,2)$ of $\mathcal{H}(3,4)$. The symplectic group PSp₄(2)≃S₆ stabilizing $\mathcal{W}(3,2)$ has two orbits on points of $\mathcal{H}(3,4)$, $\mathcal{W}(3,2)$, and its complement in $\mathcal{H}(3,4)$, and the subgroup A₆ of PSp(4,2) has three orbits on generators: an orbit of size 15 (the totally isotropic lines of $\mathcal{W}(3,2)$) and two orbits of size 6 on generators disjoint from $\mathcal{W}(3,2)$ forming a double-six. Of course, each orbit of size 6 is trivially a relative hemisystem of $\mathcal{H}(3,4)$ with respect to $\mathcal{W}(3,2)$ admitting A₆ as an automorphism group.

5 The equivalence problem

Our relative hemisystems are not equivalent to those constructed by Penttila and Williford due to the fact that they have nonisomorphic automorphism groups. As already observed, a relative hemisystem of $\mathcal{H}(3,q^{2})$ gives rise to a primitive 3-class cometric association scheme, which is not metric. In this section we show that the association schemes arising from our construction are not equivalent to those arising from the Penttila–Williford relative hemisystems. First of all, we need to introduce some definitions.

An ovoid of a generalized quadrangle is a set of points meeting each line in exactly one point. If a generalized quadrangle S of order (s,t) contains a subquadrangle S′ of order (s,t′), then it can be seen that x ^⊥∩S′ is an ovoid of S′ for all points x∈S∖S′. Following [2], we say that this ovoid is subtended by the point x and denote it by $\mathcal{O}_{x}$. An ovoid of S′ is said to be doubly subtended if it is subtended by exactly two points of S∖S′. We say that the generalized quadrangle S′ is doubly subtended in S if every subtended ovoid of S′ is doubly subtended. We will denote by x′ the other point subtending $\mathcal{O}_{x}$ and refer to x and x′ as antipodes. In [2], Brown introduced the notion of doubly subtended subquadrangles S′ of order (r,r) in a generalized quadrangle of order (r,r ²). This is, for instance, the case of $\mathcal{Q}(4,q)$, which is the dual of $\mathcal{W}(3,q)$, inside $\mathcal{Q}^{-}(5,q)$. Brown proved that S′ is doubly subtended in S if and only if S has an involutory automorphism that fixes S′ pointwise. This automorphism simply interchanges antipodes while leaving the points of S′ fixed. In other words, the point set of S∖S′ is an algebraic 2-fold cover of the strongly regular graph formed by taking all subtended ovoids of S′ with two ovoids adjacent when they intersect in one point. This geometric context enabled Penttila and Williford to form a primitive Q-polynomial scheme. In particular, utilizing a slight modification of the argument of Thas in [15], Penttila and Williford showed that if $S=\mathcal {Q}^{-}(5,q)$ and $S'=\mathcal{Q}(4,q)$, the set X ₁ of half of the points of S∖S′ such that every line of S∖S′ meets X ₁ in q/2 points forms a primitive Q-polynomial subscheme of the Q-polynomial Q-bipartite scheme on the points of S∖S′, see [13, Theorems 3 and 5] for more details.

In the sequel we will prefer to adopt the dual setting, and hence we will work on the elliptic quadric $\mathcal{Q}^{-}(5,q)$ instead of $\mathcal{H}(3,q^{2})$.

Let $\mathcal{Q} = \mathcal{Q}^{-}(5,q)$, and let H₁,H₂ and G₁,G₂ be two relative hemisystems of $\mathcal{Q}$ with respect to the subquadrangle $\mathcal{Q}'=\mathcal{Q}(4,q)$. For each vertex x∈H₁, let x′ denote its antipode in H₂, and for each vertex x∈G₁, let x″ denote its antipode in G₂. Let α be the involution that fixes $\mathcal{Q}'$ pointwise. Since α swaps antipodes of all sets that are relative hemisystems with respect to $\mathcal{Q}'$, this map satisfies α(H₁)=H₂ and α(G₁)=G₂. Now suppose that the association schemes afforded by H₁ and G₁ are isomorphic. This is equivalent to saying that there is a map β from H₁ to G₁ such that for all x,y∈H₁, we have that β(x) is collinear with β(y) if and only if x is collinear to y and that β(x) is collinear with β(y)″ if and only if x is collinear to y′. We will show that β lifts to an automorphism of $\mathcal{Q}$.

We first extend β to a map from H₁∪H₂ to G₁∪G₂ by defining β(x)=α(β(α(x))) for all x∈H₂. Note that this implies that β commutes with the restriction of α to $\mathcal{Q} \backslash\mathcal{Q}'$. Then β(x) is collinear with β(y) if and only if x is collinear to y for all x,y∈H₂ as well. Now let x∈H₁ and y′∈H₂. We have that x,y′ are collinear if and only if β(x),β(y)″ are collinear, which occurs if and only if β(x) is collinear with α(β(y))=β(α(y))=β(y′). Therefore all points of $\mathcal{Q}\backslash\mathcal{Q}'$ are collinear if and only if their images under β are collinear.

We now extend β to all of $\mathcal{Q}$ by defining it on $\mathcal{Q}'$. We call a collection of q ²−q lines in $\mathcal{Q} \backslash\mathcal{Q}'$ a bundle if the lines meet in a single point in $\mathcal{Q}'$. For each bundle B, let p _B be the point of $\mathcal{Q}'$ where all of the lines meet. Bundles are precisely the sets of q ²−q lines in $\mathcal{Q} \backslash\mathcal{Q}'$ which do not meet in $\mathcal{Q} \backslash \mathcal{Q}'$ and with the property that no line in $\mathcal{Q} \backslash\mathcal{Q}'$ meets two lines of the set. This implies that bundles can be defined using only the incidence structure $\mathcal{Q} \backslash\mathcal{Q}'$, and so β must map bundles to other bundles. This yields a natural extension of β to $\mathcal{Q}'$. Let $p_{B} \in\mathcal{Q}'$ and define β(p _B )=p _β(B). Clearly, p _B is collinear with $x \in\mathcal{Q} \backslash\mathcal{Q}'$ if and only if x∈B, which occurs if and only if β(x)∈β(B) if and only if β(p _B ) is collinear with β(x).

Lastly, let $p_{C} \in\mathcal{Q}'$, where C is a bundle of $\mathcal {Q} \backslash\mathcal{Q}'$. We have that p _B is collinear with p _C if and only if B is disjoint from C, which occurs if and only if β(B) is disjoint from β(C). The latter occurs if and only if p _β(B)=β(p _B ) is collinear with p _β(C)=β(p _C ). All cases considered, β is an automorphism of $\mathcal {Q}$ which maps H₁ to G₁, so H₁ and G₁ are equivalent.