Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry

In this paper we show that if $\theta$ is a $T$-design of an association scheme $(\Omega, \mathcal{R})$, and the Krein parameters $q_{i,j}^h$ vanish for some $h \not \in T$ and all $i, j \not \in T$ ($i, j, h \neq 0$), then $\theta$ consists of precisely half of the vertices of $(\Omega, \mathcal{R})$ or it is a $T'$-design, where $|T'|>|T|$. We then apply this result to various problems in finite geometry. In particular, we show for the first time that nontrivial $m$-ovoids of generalised octagons of order $(s, s^2)$ do not exist. We give short proofs of similar results for (i) partial geometries with certain order conditions; (ii) thick generalised quadrangles of order $(s,s^2)$; (iii) the dual polar spaces $\mathsf{DQ}(2d, q)$, $\mathsf{DW}(2d-1,q)$ and $\mathsf{DH}(2d-1,q^2)$, for $d \ge 3$; (iv) the Penttila-Williford scheme. In the process of (iv), we also consider a natural generalisation of the Penttila-Williford scheme in $\mathsf{Q}^-(2n-1, q)$, $n\geqslant 3$.


Introduction
It is well known that vanishing Krein parameters of association schemes have important consequences in combinatorics, for example in determining the feasibility of parameter sets for distance-regular graphs, and in placing bounds upon the orders of generalised polygons. Vanishing Krein parameters may also be used to say something about subsets of the vertices of an association scheme. The key result of this paper is a simple observation (Theorem 1.3) which states that given certain vanishing Krein parameters, either the possible sums of eigenspaces containing a Delsarte design are constrained, or else the Delsarte design consists of exactly half of the vertices of the association scheme. This leads to various interesting consequences which we explore in this paper. We will assume the reader is familiar with the basic theory of association schemes, but we refer to [34,Chapter 30] for background and notation.
The starting point for this work is the following theorem (Theorem 1.1) of Cameron, Goethals and Seidel [6]. A short proof of it was given by Martin [21]. It states that the Schur product of two vectors, (1) each lying in precisely one eigenspace, projects trivially to a third eigenspace if the corresponding Krein parameter vanishes. In particular, this gives us combinatorial meaning, since if the two vectors in question are characteristic vectors of subsets, then their Schur product indicates the intersection of the two sets. Cameron, Goethals, and Seidel [7] proved that strongly regular graphs with either q 1 11 = 0 or q 2 22 = 0 have strongly regular subconstituents about every vertex. Moreover, they characterised such graphs, when connected, as being a pentagon, a Smith graph (cf. [27]), or the complement of a Smith graph. Similar results have been investigated for distance-regular graphs with slightly larger diameter [17,18]. There is also an important relation between vanishing Krein parameters and the triple intersection numbers. Given x, y, z ∈ V (Γ), we define the triple intersection numbers Q ri Q sj Q th p x,y,z r,s,t = 0, for all x, y, z ∈ Ω.
Theorem 1.2 has been used to show that distance-regular graphs with certain intersection arrays do not exist [10,19,32,36].
We now reach the main theorem of this paper. Theorem 1.3 shows, that given certain vanishing Krein parameters, Delsarte designs must be constrained to a smaller subset of the eigenspaces in which they lie, or else consist of exactly half of the vertices. Theorem 1.3. Let (Ω, R) be an association scheme, and let θ be a subset of the vertices, such that its characteristic vector χ θ satisfies or |θ| = 1 2 |Ω|. The proof of this result will appear in Section 3. We will order the simultaneous eigenspaces in the natural cometric ordering. We have (c) All of the 4-subsets containing one element. In this case χ θ ∈ V 0 ⊥ V 1 and |θ| = 35 = |Ω| 2 . (d) A set (2) of half the elements of Ω, with stabiliser S 5 in S 8 . In this case Reformulated in the language of T -designs, Theorem 1.3 states that for S = {1, . . . , d}\T if q h i,j vanishes for some h ∈ S and all i, j ∈ S then either |θ| = |Ω|/2 or θ is a (T ∪ {h})-design.
In order to use Theorem 1.3 we require at a minimum that q h hh = 0. Moreover, the smallest such setting is where S = {h}. A subset θ ∈ Ω is called an intriguing set of type h (cf. [3]) if χ θ ∈ V 0 ⊥ V h . Hence an intriguing set of type i is a {1, . . . , d}\{h}design. Intriguing sets are often known by other names in different settings. For example, Cameron-Liebler line classes (cf. [8,12,13,22,25]) and Boolean degree 1 functions (cf. [16]) are all variations of intriguing sets. We have the following special case of Theorem 1.3 for intriguing sets of type h. Corollary 1.5. If q h hh = 0, then a nontrivial intriguing set of type h contains exactly half of the vertex set. Corollary 1.5 yields alternative proofs of three results in the literature: (i) a nontrivial m-ovoid of a generalised quadrangle of order (s, s 2 ) is a hemisystem [7] (see also Corollary 5.3); (ii) a nontrivial m-ovoid of a dual polar space of the form DQ(2d, q), is a hemisystem [1, Theorem 1.1] (see also Theorem 6.1); (iii) a nontrivial relative m-ovoid of a generalised quadrangle of order (q, q 2 ), containing a doubly subtended quadrangle of order (q, q), is a relative hemisystem [2] (see also Theorem 7.2).

Some background and notational conventions
We remind the reader that we can calculate Krein parameters of an association scheme (Ω, R) from the matrix of eigenvalues P , or the matrix of dual eigenvalues Q.
Lemma 2.1 (cf. [5,Theorem 2.3.2]). In an association scheme (Ω, R), let the k ℓ be the valencies, and let the m h be the multiplicities (of the simultaneous eigenspaces). Then There is a bijection between the power-set of the vertices and the set of all {0, 1}vectors by taking the characteristic vector of a set. We write χ S for the characteristic vector of a subset S of a domain that is clear from the context. It is clear that Schur multiplication of two {0, 1}-vectors produces another {0, 1}-vector with a one in an entry if and only if the corresponding entry is one in both of the original vectors. Thus intersection of two vertex subsets is described by Schur multiplication of their characteristic vectors. That is, χ S1∩S2 = χ S1 • χ S2 for all subsets S 1 , S 2 of our domain. Moreover, the all-ones vector 1 is the identity under Schur multiplication, and 1 − χ S = χ S c , where S c is the complement of S in our domain.
We will need one more item of notation. Recall that the eigenspaces of the association scheme in Theorem 1.3 are denoted V i . Each V i has a linear projection map E i ; a minimal idempotent for the association scheme. Recall that the E i have the following property: A partial linear space is a (nonempty) incidence geometry of points and lines such that any two distinct points are contained in at most one line, and every line contains at least two points. An m-ovoid of a partial linear space is a subset S of the points such that every line is incident with m elements of S. If every line has s + 1 points incident with it, then an s+1 2 -ovoid is a hemisystem. The trivial m-ovoids are the empty set and the whole point set, that is, m is 0 or s + 1.

Vanishing Krein parameters in cometric, Q-bipartite and Q-antipodal schemes
q i+j ij ̸ = 0 for all i and j such that i + j ⩽ d. Thus such schemes have a large proportion of vanishing Krein parameters. Indeed, Terwilliger [30] was interested in cometric schemes for this reason. A scheme is called metric if the equivalent conditions hold when q h ij is replaced by p h ij . Many of the classical association schemes are both metric and cometric. Schemes which are cometric but not metric are relatively rare, hence the interest in the Penttila-Williford scheme. We consider this scheme further in Section 7. Moreover, most association schemes arising from geometries are cometric, such as the dual polar spaces, for example.
Algebraic Combinatorics, Vol. 6 #1 (2023) Corollary 4.1. Let (Ω, R) be a cometric association scheme, and let θ be a subset of the vertices, such that An association scheme is primitive if all of its associate graphs, Γ i , are connected. Otherwise it is called imprimitive. The following conjecture, due to Bannai and Ito, if true, would imply that, for sufficiently large diameter, most of the Krein parameters of a primitive distance-regular graph vanish.

Conjecture 4.2 (Bannai and Ito [4, p. 312]). For sufficiently large diameter, a primitive association scheme is metric if and only if it is cometric.
There is no guarantee that q i ii = 0 for some i, even in a cometric scheme. However, we can say a bit more, particularly in the case of imprimitive cometric association schemes. For a cometric association scheme, define . For d ⩾ 2, a distance-regular graph is called almost Qbipartite if a * i = 0 for i < d and a * d > 0. An almost Q-bipartite distance-regular graph is either the halved (2d + 1)-cube, the folded (2d + 1)-cube, or the collinearity graph of the dual polar space DH(2d − 1, q 2 ). Since a * 1 = q 1 11 = 0, it follows from Corollary 1.5 that an intriguing set of type 1 consists of half the vertex set.
An imprimitive metric association scheme other than a cycle is bipartite, or antipodal, or both [5, Theorem 4.2.1]. Suzuki [28] proved that an imprimitive cometric association scheme is Q-bipartite, or Q-antipodal, or both Q-bipartite and Q-antipodal, or has either four or six classes. Cerzo and Suzuki [9] later showed that the exceptional four-class case does not exist, followed by a similar non-existence result by Tanaka and Tanaka for the exceptional six-class case [29]. Therefore, an imprimitive cometric association scheme other than a cycle is Q-bipartite, or Q-antipodal, or both.
The Q-bipartite condition is very strong, leading to the following result for constraining Delsarte designs.
by Theorem 1.3. Repeating this argument eventually constrains χ θ to lie in V 0 = ⟨1⟩, but this is a contradiction, since θ is a nontrivial proper subset of the vertices.

Partial geometries and generalised quadrangles
Partial geometries are a class of partial linear spaces having strongly regular collinearity graphs. To be more precise, a partial geometry pg(s, t, α) is a partial linear space such that every line contains s + 1 points, every point is incident with t + 1 lines and for a point P and line ℓ which are not incident, there are α points on ℓ collinear with P . A partial geometry gives rise to a strongly regular graph with parameters simply by considering the vertices to be the points, and adjacency to be collinearity of points. By default, the association scheme arising from the partial geometry will be that arising from the strongly regular collinearity graph, and the minimal idempotents will have the natural ordering. (3) A partial geometry is thick if s, t > 1. Bruck nets and generalised quadrangles occur naturally as partial geometries. Indeed, a pg(s, t, α) is a generalised quadrangle precisely when α = 1.
and t ⩾ s 2 . Moreover, in the positive case, α = 1 if and only if t = s 2 .
Remark 6.2. The conditions of Theorem 1.3 being satisfied does not imply the existence of a T -design, θ, such that |θ| = 1 2 |Ω|. Such a T -design may or may not exist. By way of example, the dual polar space DW(5, 3) has Krein parameter q 3 33 = 0, which implies that every m-ovoid of this space is a hemisystem. However, it has no hemisystems. While the dual polar space DQ(6, 3) also has Krein parameter q 3 33 = 0, and does have hemisystems (see [1]).

The Penttila-Williford scheme
Let Q be a generalised quadrangle of order (s, t) and let Q ′ be a generalised subquadrangle of Q of order (s, t ′ ). Let P be a point of Q not in Q ′ . Then P ⊥ ∩ Q ′ is an ovoid O P of Q ′ subtended by P . If there is exactly one other point P ′ such that O P is also subtended by P ′ , then we say O P is doubly subtended. If every ovoid of Q ′ is doubly subtended, then we say that Q ′ is doubly subtended in Q.
Let Q be a generalised quadrangle of order (s, s 2 ), where s > 2, and let Q ′ be a doubly subtended generalised quadrangle of order (s, s) contained in Q. For example, we could take the classical generalised quadrangle Q − (5, q) and the subquadrangle Q(4, q). Let Ω be the set of points of Q ∖ Q ′ , and define the following relations on Ω × Ω: • R 0 is the identity relation; i=0 form a primitive association scheme (see [24]), which we call the Penttila-Williford scheme. The Penttila-Williford scheme is Q-bipartite (see [24]), and so we have the following immediate consequence of Corollary 4.4. The following result of Bamberg and Lee follows as a corollary of Theorem 7.1 and the fact that a relative m-cover is an intriguing set of type 1 [2, Corollary 3.4].
Theorem 7.2 ([2]). Given a generalised quadrangle of order (q, q 2 ) containing a doubly subtended quadrangle of order (q, q), a nontrivial relative m-ovoid is a relative hemisystem. If it exists, q is even.
We now look at a generalisation of the Penttila-Williford scheme which, although a natural extension, does not appear to be in the literature. We state it explicitly and consider some of its properties by drawing on tools from Cossidente and Pavese who studied the generalisation of relative m-ovoids [11]. Let Ω be the set of points of Q − (2n − 1, q) not contained in a fixed nondegenerate hyperplane Π. So |Ω| = q n−1 (q n−1 − 1). We define σ : Ω → Ω as follows. Let P be a point of Ω. Then the line joining Π ⊥ and P is a hyperbolic line and so contains a unique second point P ′ of Ω. Let σ be the central collineation of PG(2n − 1, q) having axis Π and centre Π ⊥ , mapping P to P ′ . Then σ commutes with the polarity ⊥ and so stabilises Ω. Moreover, since P P ′ has only two points of Q − (2n − 1, q) on it, we have σ(P ′ ) = P and hence σ 2 is the identity. It turns out that σ is independent of the choice of P . We write X ∼ Y as a shorthand notation for the 'collinear and not equal' relation.
Algebraic Combinatorics, Vol. 6 #1 (2023) 205 Theorem 7.3. Let q be a prime power, greater than 2, and let Ω be the set of points of Q − (2n − 1, q) not contained in a fixed nondegenerate hyperplane Π. Define the following relations on Ω: is an association scheme with matrix of eigenvalues: Proof. We will sketch the proof since it is similar to the proof of [24, Theorem 1]. First, one verifies that the relations R i are symmetric and the intersection numbers p h ij are well-defined. Next, the parameters p h ij can be computed geometrically and we obtain the intersection matrices L 1 , L 2 , L 3 , and L 4 , which we list in Appendix A. The eigenvectors for L 1 give the matrix of eigenvalues as described above. □ Lemma 7.4. The association scheme described in Theorem 7.3 is not metric. It is cometric precisely when n = 3, in which case there is a single cometric ordering.
Proof. The dual intersection matrices L * i have been computed using Lemma 2.1 and are given in Appendix B. Observe that L * 1 is tridiagonal when n = 3, and hence the association scheme is cometric with respect to the given ordering of eigenspaces. When n > 3, there are 10, 6, 10, and 6 entries of L * 1 , L * 2 , L * 3 , and L * 4 , respectively, which are non-zero and not on the diagonal. Any ordering of the eigenspaces induces a permutation σ ∈ S 4 on the indices. Now, since (L * i ) jh = q h ij , it follows that (L σ(i) ) jh = q σ(h) σ(i)σ(j) , and so L σ(i) = T ⊤ L i T , where T is the permutation matrix induced by σ. Note that the diagonal of L i is fixed by T , and so the rows and columns of L * 1 and L * 3 clearly cannot be permuted to a tridiagonal form, as they have too many nondiagonal, non-zero entries. Similarly, L * 2 and L * 3 have too few entries; even if they were permuted into a tridiagonal form, it would result in b * x and c * y being zero for some x, y ∈ {1, 2, 3, 4}, which is not possible. Hence the association scheme is not cometric for n > 3. Since the number of non-diagonal, non-zero entries of L * 2 , L * 3 , and L * 4 are constant for n ⩾ 3, the same argument shows that only the given ordering is a cometric ordering for n = 3. A similar treatment shows that none of the intersection matrices (see Appendix A) can be expressed in tridiagonal form, for n ⩾ 3, and so the association scheme is not metric. □ Theorem 7.5. Let θ be a nontrivial intriguing set of type 1, intriguing set of type 3, or a {2, 4}-design of the association scheme described in Theorem 7.3. Then |θ| = |Ω| 2 . If n = 3 and θ is a nontrivial {2, 3}-design, then either θ is a {1, 2, 3}-design or |θ| = |Ω| 2 . Proof. Recall that q h ij = (L * i ) jh . By inspecting the dual intersection matrices (Appendix B), we see that the following Krein parameters always vanish: q 1 11 , q 3 33 , q 1 13 , q 1 31 , q 1 33 , q 3 11 , q 3 13 , q 3 31 .

Implications of vanishing Krein parameters on Delsarte designs
Additionally, when n = 3, the following also vanish: 2n−1, q), with respect to a nondegenerate hyperplane Π, is a subset R of points of Q − (2n − 1, q) ∖ Π such that every generator of Q − (2n − 1, q) not contained in Π meets R in m points. A relative m-ovoid is said to be nontrivial or proper if it is nonempty and not the entire set of points of Q − (2n − 1, q) ∖ Π. It is a relative hemisystem if it comprises half of the points of Q − (2n − 1, q) ∖ Π. Now the points of Q − (2n − 1, q) ∖ Π in a generator of Q − (2n − 1, q) (not contained in Π) form a clique of size q n−2 for the R 3 -relation and so has inner distribution vector a = (1, 0, 0, q n−2 − 1, 0). Now and so a relative m-ovoid is an intriguing set of type 1 (by [26, 3.3 Relative hemisystems of Q − (4n + 1, q) are known to exist for q even and n ⩾ 2 [11,Theorem 3.3]. Other than for Q − (7, 2), where no relative hemisystem exists [11,Remark 3.5], the question of their existence is open for Q − (4n − 1, q), q even and n ⩾ 2.

Generalised octagons
A finite generalised octagon is a partial linear space such that (i) there are no ordinary n-gons in the geometry with n < 8, and (ii) there exists an ordinary octagon in the geometry. If there are parameters (s, t) such that every line is incident with s + 1 points and every point is incident with t + 1 lines, then we say that the generalised octagon has order (s, t). If a generalised octagon of order (s, t) has s, t ⩾ 3, then we say it is thick. The dual incidence structure of a generalised octagon of order (s, t) (whereby points and lines are interchanged) is again a generalised octagon, but with order (t, s). The only known examples of finite thick generalised octagons are the Ree-Tits octagons of order (q, q 2 ) and their duals, where q is an odd power of 2. Indeed, they are the natural geometries for the exceptional groups of Lie type 2 F 4 (q). Now it is customary in the theory of generalised polygons with gonality at least 6 to use the term distance-j-ovoid for a maximum (4) set of points mutually at distance j, and reserve the term ovoid for a maximum set of pairwise opposite points. However, for convenient notation, we will still stipulate that an m-ovoid of a generalised polygon is a set of points such that every line meets it in m points; in the original vein of [31, §3]. The authors believe the following result is new. and hence with intersection matrix The eigenvalues of L are s − 1 , s − 1 − √ 2st, s − 1 + √ 2st, −(t + 1), and s(1 + t), from which we can compute the eigenvectors of L via standard sequences and the eigenvalue multiplicities via Biggs' formula (cf. [33, pp. 13-4]). After scaling by the eigenvalue multiplicities, the eigenvectors form the columns of the matrix of dual eigenvalues, Q.
With the aid of Mathematica [37], we computed the matrix of eigenvalues to be Consider the point-set of a line. It has size s + 1 and is a clique with respect to the first relation. Hence it has inner distribution vector a = (1, s, 0, 0, 0). Considering the MacWilliams transform (see [ The numerator of this sum, and hence q 2 22 , is clearly zero when s 2 = t or s = 1. Thus by Theorem 1.3 a non-trivial m-ovoid is a hemisystem when s 2 = t. However, 2st = 2s 3 is a perfect square by Feit-Higman [15], so 2 divides s resulting in an odd number of points on a line. A hemisystem must contain exactly half of the points on every line and so an m-ovoid must be trivial.