Two-valenced association schemes and the Desargues theorem

The main goal of the paper is to establish a sufficient condition for a two-valenced association scheme to be schurian and separable. To this end, an analog of the Desargues theorem is introduced for a noncommutative geometry defined by the scheme in question. It turns out that if the geometry has enough many Desarguesian configurations, then under a technical condition the scheme is schurian and separable. This result enables us to give short proofs for known statements on the schurity and separability of quasi-thin and pseudocyclic schemes. Moreover, by the same technique we prove a new result: given a prime $p$, any $\{1,p\}$-scheme with thin residue isomorphic to an elementary abelian $p$-group of rank greater than two, is schurian and separable.


Introduction
One of the fundamental problems in theory of association schemes is to determine whether a given scheme is schurian, i.e., comes from a permutation group, and/or separable, i.e., uniquely determined by its intersection number array (for the exact definitions, see Section 2). In the two last decades these two problems are intensively studied for the two-valenced schemes see, e.g., [4,11,9,2]; here an association scheme is said to be two-valenced if the valencies of its basic relations take exactly two values, and if they are 1 and k, the term {1, k}-valenced scheme is also used.
An analysis of the known proofs that certain two-valenced schemes are schurian or separable shows that in all cases the following two properties are significant. The first one is that there are sufficiently many intersection numbers of the scheme in question that are equal to 1; to define this property precisely, we introduce in Section 3 the saturation condition (a special case of it appeared in [2]). The second property expresses the fact that in a noncommutative "affine" geometry determined by the two-valenced scheme, there are sufficiently many Desarguesian configurations, see Section 4.
A two-valenced scheme having the first and second properties is said to be saturated and Desarguesian, respectively. A model example illustrating these two properties is the scheme of a finite affine space that is two-valenced, saturated (except for few cases), and Desarguesian if the dimension of the affine space is at least 3 (see Examples in Sections 3 and 4). The work of the third author was supported by the RAS Program of Fundamental Research "Modern Problems of Theoretical Mathematics". Theorem 1.1. Let X be a two-valenced scheme. Assume that X is saturated and Desarguesian. Then X is schurian and separable.
For the scheme of a finite affine space of dimension at least 3, Theorem 1.1 expresses a well-known fact that in the Desarguesian case, this scheme is reconstructed from its automorphism group and is uniquely determined by its order.
The power of Theorem 1.1 is illustrated by the statements below. The first two of them are known results, but using the theorem we are able to give much shorter and clear proofs than in the original papers.
There exists a function f such that any pseudocyclic scheme of valency k > 1 and degree at least f (k) is schurian and separable.
From the proof of Corollary 1.2 given in Section 6, it follows that the function f satisfies the inequality f (k) ≤ 1 + 3k 6 . A more subtle arguments used in [2] show that f (k) ≤ 1 + 6k(k − 1) 2 . Corollary 1.3. [11,10] Any quasi-thin scheme satisfying the condition n s * s = 2 for all basis relations s, is schurian and separable.
Except for the separability statement, the result of Corollary 1.3 is contained in [11]. On the other hand, the schurian and separable quasi-thin schemes were characterized in [10]. This result can also be deduced from an analog of Theorem 1.1, in which the Desarguesian condition is replaced by a weaker one (namely, the amount of the required Desarguesian configuration is reduced). To keep the text more compact, we do not go into detailed explanation of this topic.
The second main result of the present paper concerns the schurity and separability of a class of meta-thin schemes introduced in [5]. A meta-thin scheme can be thought as an extension of a regular scheme X by another regular scheme. Even if the meta-thin scheme is a {1, p}-scheme, where p is a prime, the schurity and separability problems seem to be very complicated. In a sense, the answer depends on the scheme X , which can be chosen as the thin residue of the scheme in question. For example, if the group associated with X has distributive lattice of normal subgroups, then the scheme in question is schurian and separable [7]. However, there are many non-schurian and non-separable meta-thin {1, p}-schemes for which that group is elementary abelian of order p 2 [6]. The following theorem, which is also deduced from Theorem 1.1, shows that if the group is an elementary abelian p-group of rank greater than two, then the situation is smooth. Theorem 1.4. Given a prime p, any {1, p}-scheme with thin residue isomorphic to an elementary abelian p-group of rank greater than two, is schurian and separable.
The paper is organized as follows. For the reader convenience a background on association schemes and related concepts is given in Section 2. In Sections 3 and 4 we introduce and study the saturated and Desarguesian two-valenced schemes, respectively. The proof of Theorem 1.1 is given in Section 5. In the final Section 6 we prove the corollaries and Theorem 1.4.
For a set S of relations on Ω, we denote by S ∪ the set of all unions of the elements of S, and put S * = {s * : s ∈ S} and αS = ∪ s∈S αs, where α ∈ Ω.
An elementary abelian p-group of order p m is denoted by E p m .

Association schemes
In our presentation of association schemes, we follow papers [10,3] and monograph [12]. All the facts we use, can be found in these sources and references therein.
where Ω is a finite set and S is a partition of Ω × Ω, is called an association scheme or scheme on Ω if the following conditions are satisfied: 1 Ω ∈ S, S * = S, and given r, s, t ∈ S, the number c t rs := |αr ∩ βs * | does not depend on the choice of (α, β) ∈ t.
The elements of Ω, S, S ∪ , and the numbers c t rs are called the points, basis relations, relations and intersection numbers of X , respectively. The numbers |Ω| and |S| are called the degree and the rank of X . A unique basic relation containing a pair (α, β) ∈ Ω × Ω is denoted by r(α, β). Since the mapping r : Ω × Ω → S depends only on X , it should be denoted by r X , but we usually omit the subindex if this does not lead to confusion.
2.2. Complex product. The set S ∪ contains the relation r · s for all r, s ∈ S ∪ . It follows that this relation is the union (possibly empty) of basis relations of X ; the set of these relations is called the complex product of r and s and denoted by rs.
In what follows, for any X, Y ⊆ S, we denote by XY the union of all sets rs with r ∈ X and s ∈ Y . Obviously, (XY )Z = X(Y Z) for all X, Y, Z ⊆ S.

2.3.
Valencies. For any basic relation s ∈ S, the number |αs| with α ∈ Ω equals the intersection number c 1 ss * , and hence does not depend on the choice of the point α. It is called the valency of s and denoted by n s ; we say that s is thin if n s = 1.
For the intersection numbers we have the following well-known identities: (1) c t * r * s * = c t sr and n t c t * rs = n r c r * st = n s c s * tr , r, s, t ∈ S.
The scheme X is said to be regular (respectively, {1, k}-valenced) if n s = 1 (respectively, k > 1 and n s = 1 or k) for all s ∈ S.

Isomorphisms and schurity.
A bijection from the point set of a scheme X to the point set of a scheme X ′ is called an isomorphism from X to X ′ if it induces a bijection between their sets of basis relations. The schemes X and X ′ are said to be isomorphic if there exists an isomorphism from X to X ′ .
An isomorphism from a scheme X to itself is called automorphism if the induced bijection on the basis relations of X is the identity. The set of all automorphisms of a scheme X is a group with respect to composition and will be denoted by Aut(X ). 1 Conversely, let K ≤ Sym(Ω) be a transitive permutation group, and let S denote the set of orbits in the induced action of K on Ω 2 . Then, X = (Ω, S) is a scheme; we say that X is associated with K. A scheme on Ω is said to be schurian if it is associated with some transitive permutation group on Ω. A scheme X is schurian if and only if it is associated with the group Aut(X ).

2.5.
Algebraic isomorphisms and separability. Let X and X ′ be schemes. A bijection ϕ : In this case, X and X ′ are said to be algebraically isomorphic.
Each isomorphism f from X onto X ′ induces an algebraic isomorphism between these schemes. The set of all isomorphisms inducing the algebraic isomorphism ϕ is denoted by Iso(X , X ′ , ϕ). In particular, where id S is the identity mapping on S. A scheme X is said to be separable if for any algebraic isomorphism ϕ : S → S ′ , the set Iso(X , X ′ , ϕ) is not empty.
The algebraic isomorphism ϕ induces a bijection from S ∪ onto (S ′ ) ∪ : the union r ∪ s ∪ · · · of basis relations of X is taken to r ′ ∪ s ′ ∪ · · · . This bijection is also denoted by ϕ.
2.6. Faithful maps. Let X = (Ω, S) and X ′ = (Ω ′ , S ′ ) be schemes, and let ϕ : S → S ′ be an algebraic isomorphism. A bijection f from a subset of Ω to a subset of Ω ′ is said to be ϕ-faithful if A ϕ-faithful map which is ϕ-extendable to every point of Ω, is said to be ϕextendable. From the definitions of schemes and algebraic isomorphisms, it follows that every ϕ-faithful map f with | Dom(f )| ≤ 2 is ϕ-extendable. In these terms, one can give a sufficient condition for schurity and separability of a scheme.

Saturation condition
Throughout this section, k > 1 is an integer and X = (Ω, S) is a scheme. By technical reasons, the basis relations of X are mainly denoted below by x, y, z rather than r, s, t. Our primary goal is to define a graph with vertex set that accumulates an information about intersection numbers equal to 1 (this graph was used implicitly in [9] an explicitly in [2]). The following simple lemma (not formulated but proved in [2]) indicates a way how we do this.
|x * y| = k ⇔ c y xs = 1 for all s ∈ x * y. Proof. We have n x = n x * = n y = n y * = k. By formulas (1), this implies that Since c y xs ≥ 1 for all s ∈ x * y, we are done. Let us define a relation ∼ on S k by setting x ∼ y if the right-or left-hand side in formula (5) holds true. This relation is symmetric, because for all x, y ∈ S k . The (undirected) graph X = X k associated with the scheme X has vertex set S k and adjacency relation ∼. Note that this graph can contain loops.
The scheme X is said to be k-saturated if for any set T ⊆ S with at most four elements, the set is not empty.
In this case, any two vertices of the graph X are connected by a path of length at most two. A k-saturated {1, k}-scheme is said to be saturated and the mention of k is omitted. The following statement immediately follows from the definitions. Lemma 3.3. Any algebraic isomorphism from X to X ′ induces an isomorphism from the graph X to the graph X ′ associated with X ′ . In particular, X is k-saturated if and only if so is X ′ .
Example: schemes of affine spaces. Let A be a finite affine space with point set Ω and line set L, see [1]. Denote by P the set of parallel classes of lines. The lines belonging to a class P ∈ P form a partition of Ω; the corresponding equivalence relation with removed diagonal is denoted by e P . Using the axioms of affine spaces, one can easily verify that the set S = S A of all the e P together with 1 Ω forms a commutative scheme such that where q is the cardinality of a line. We say that X = (Ω, S) is the scheme associated with the affine space A. Formulas (7) and (8) imply that X is a {1, q − 1}-valenced scheme. Moreover, X is a complete graph, i.e., any two distinct vertices form an edge. In particular, the scheme X is (q − 1)-saturated whenever the rank of X is at least 6.
We complete the section by establishing a sufficient condition for a scheme X to be k-saturated in terms of the indistinguishing number c(s) = t∈S c s tt * of the relation s, see [9]. One can see that this number is equal to the cardinality of the set (9) Ω α,β = {γ ∈ Ω : r(γ, α) = r(γ, β)} for any (α, β) ∈ s.

Desarguesian two-valenced schemes
The concept of a Desarguesian scheme comes from a property of a geometry to be Desarguesian. Throughout this section, k > 1 is an integer and X = (Ω, S) is a {1, k}-scheme. We also keep notation of Section 3.
Let us define a noncommutative geometry associated with the scheme X as follows: the points are elements of S, the lines are the sets x * y, x, y ∈ S, and the incidence relation is given by inclusion. Thus the point z ∈ S belongs to the line x * y if and only if z ∈ x * y. The geometry is extremely unusual: the line x * y does not necessarily contain the points x, y, and can be different from y * x. However, in the terms of this geometry, one can define Desarguesian configurations, see below.
Assume that we are given two triangles with vertices x, y, z ∈ S and u, v, w ∈ S, respectively, that are perspective with respect to a point q, i.e., (10) u ∈ x * q, v ∈ y * q, w ∈ z * q, see the configuration depicted in Fig. 1; note that the intersections of lines do not necessarily consist of a unique point, and even may be empty. However if (11) x * z ∩ uw * = {r}, z * y ∩ wv * = {s}, x * y ∩ uv * = {t} for some r, s, t ∈ S, then, as in the case of Desargues' theorem, we would like that the point t would lie on the line rs. When this is true, this configuration is said to be Desarguesian. More precisely, the ten relations in Fig. 1 form a Desarguesian configuration if conditions (10) and (11) are satisfied and t ∈ rs. In what follows we are going to study {1, k}-schemes with sufficiently many Desarguesian configurations. Let x, y, z ∈ S k and r, s ∈ S be basis relations of the scheme X . We say that they form an initial configuration if (12) x ∼ z ∼ y and r ∈ x * z, s ∈ z * y.
In geometric language, this means that the points r and s belong to the lines x * z and z * y, respectively, and each of these lines consists of exactly k points.
Definition 4.1. The relations r and s are said to be linked with respect to (x, y, z) if the initial configuration is contained in a Desarguesian configuration, namely, there exist q ∈ N (x, y, z), u, v, w ∈ S, t ∈ rs, for which conditions (10) and (11) are satisfied, where N (x, y, z) = N ({x, y, z}) (a more compact picture of the linked relations is given in Fig.2).
Assume that the relations r and s are linked with respect to (x, y, z). Then the relation t is uniquely determined by the third of equalities (11). The following statement shows that in this case, t does not depend on the choice of q and u, v, w. Below, we fix a point α ∈ Ω and set (13) r x,y = r ∩ (αx × αy) for all r ∈ S and x, y ∈ S k .  Proof. By formulas (11), we have (15) u x,q · w * q,z ⊆ r x,z , w z,q · v * q,y ⊆ s z,y , u x,q · v * q,y ⊆ t x,y . The relations u x,q · w * q,z and r x,z are matchings, because x ∼ q ∼ z, and x ∼ z. By the first inclusion in (15), this implies the first of the following two equalities, the second one is proved similarly: u x,q · w * q,z = r x,z and w z,q · v * q,y = s z,y . Now the third inclusion in (15) yields (14). If x ∼ y, then the relation t x,y is a matching and we are done by the above argument.
At this point, we need an auxiliary statement. In the geometric language, the conclusion of this statement means that the lines rs and x * y have a unique common point. Then |rs ∩ x * y| = 1 for all r ∈ x * z and s ∈ z * y.
The statement below establishes two sufficient conditions for relations r and s to be linked with respect to (x, y, z).
Then formula (10)  Therefore formula (11) also holds with t = r(β, γ). Finally, t ∈ rs in view of (18). It follows that the configuration formed by x, y, z, u, v, w, r, s, t, and q is Desarguesin. Thus r and s are linked with respect to (x, y, z).
Now we arrive to the main definition in this section. Namely, the scheme X is said to be Desarguesian with respect to S k if for all x, y, z ∈ S k and all r, s ∈ S satisfying (12), the elements r and s are linked with respect to (x, y, z). When the scheme is two-valenced, the mention of S k is omitted. The following statement immediately follows from the definitions.

Lemma 4.5. Let X and X ′ be algebraically isomorphic two-valenced schemes. Then X is Desarguesian if and only if so is X ′ .
Example: schemes of affine spaces (continuation). Let X be the scheme associated with an affine space A of order q and dimension at least 3. Then from formulas (7) and (8) it follows that for any three parallel classes P , Q, and R there exists a parallel class T such that e T ∈ (e P e Q ∪ e P e R ∪ e R e Q ).
It follows that the statement (L2) of Corollary 4.4 holds for q = e T , x = e P , y = e Q , and z = e R . Therefore any relations r ∈ e P e R and s ∈ e R e Q are linked with respect to (e P , e Q , e R ). Thus, the scheme X is Desarguesian. If the space A is an affine plane, i.e., an affine space of dimension 2, then X is not Desarguesian.
From the definition of the graph X, it follows that the f -image is uniquely determined for all δ ∈ ∆ 0 ∪ ∆ 1 . However, if δ ∈ ∆ 2 , then the image depends on the choice of the point γ.
We complete the proof by verifying that the mapping f is ϕ-faithful, i.e., for all δ, ǫ ∈ Ω. Note that this is true by the definition of f whenever α ∈ {δ, ǫ}. In what follows we need the following lemma.

Proofs of corollaries and Theorem 1.4
Proof of Corollary 1.2 Set f (x) = 3x 6 + 1. By Theorem 1.1, it suffices to verify that any pseudocyclic scheme X of degree n and valency k, is saturated and Desarguesian, whenever (20) n > 3k 6 .
By [9, Theorem 3.2], we have n s = k and c(s) = k − 1 for any irreflexive basis relation s of X . It follows that for k ≥ 2, Thus, X is saturated by Theorem 3.4.
To prove that X is Desarguesian, let x, y, z ∈ S k (here S k = S # and we do not assume that x ∼ z ∼ y). By Corollary 4.4, it suffices to find q ∈ S k such that condition (17) is satisfied. Assume on the contrary that no q satisfies this condition. Then for a fixed α ∈ Ω and each q ∈ S k , there exists β q ∈ α (xx * yy * ∪ xx * zz * ∪ zz * yy * ) T other than α and such that r(α, β q ) ∈ qq * . It follows that where the set Ω α,βq is as in (9). Since |αT | ≤ n 2 x n 2 y + n 2 y n 2 z + n 2 z n 2 x = 3k 4 and |S k | = there exist a point β = α such that β = β q for at least k relations q ∈ S 2 . In view of (21), this implies that for s = r(α, β) we have Proof of Corollary 1.3. Let X = (Ω, S) be a quasi-thin scheme satisfying the condition n s * s = 2 for all s ∈ S. Without loss of generality, we may assume that |Ω| > 24.
By Theorem 1.1, it suffices to verify that X is saturated and Desarguesian. The former follows from the Claim. To prove that X is Desarguesian let x, y, z ∈ S k and r, s ∈ S be basis relations of X that form an initial configuration. We have to verify that r and s are linked with respect to (x, y, z). Note that since n z = 2, the vertex z forms a loop of the graph X. Therefore if (xx * yy * ) ∩ zz * = {1}, then we are done by Corollary 4.4 (see the statement (L1)).
Let zz * ⊆ xx * ∩ yy * . Then the intersection of S 2 with the set xx * yy * ∪ xx * zz * ∪ zz * yy * = {1, x ⊥ , y ⊥ } ∪ x ⊥ y ⊥ ∪ x ⊥ z ⊥ ∪ z ⊥ y ⊥ contains at most 8 elements, where for any s ∈ S 2 , we set s ⊥ to be the unique non-thin element of ss * . On the other hand, in view of the Claim, the graph X contains at least 9 vertices. Thus there exists q ′ ∈ S 2 such that q ′ ∈ xx * yy * ∪ xx * zz * ∪ zz * yy * .
In view of formula (22) and the assumption n ss * = 2, one can find q ∈ S 2 , for which q ′ = q ⊥ . Now the required statement follows from Corollary 4.4 (see the statement (L2)).
Proof of Theorem 1.4. Let X = (Ω, S) be a meta-thin {1, p}-scheme. Then the group formed by the thin basis relations of X (with respect to the composition) contains a subgroup (the thin residue) generated by the sets ss * , s ∈ S. Assume that this subgroup is isomorphic to an elementary abelian p-group of rank greater than two. By Theorem 1.1, it suffices to verify that X is saturated and Desarguesian.
To prove that X is saturated, let s i ∈ S p , 1 ≤ i ≤ 4. By the assumption there exist u 1 , u 2 , u 3 ∈ S such that u 1 u * 1 , u 2 u * 2 , u 3 u * 3 ≃ E p 3 . Denote by T the set {s 1 s * 1 , . . . , s 4 s * 4 }. First, assume that u j u * j / ∈ T for some j.
In view of [9, Lemma 2.3], this implies that u j is adjacent in the graph X with each of the s i . Now without loss of generality we may assume that u j u * j ∈ T for all j. Since u j u * j ∩ u k u * k = {1} for all distinct j, k, there exist i ∈ {1, 2, 3, 4} and j ∈ {1, 2, 3} such that s i s * i = u j u * j = s k s * k , k = i. Taking into account that every element in S p has a loop, we conclude that s i is adjacent in the graph X with each of the s i with i ∈ {1, 2, 3, 4}. Thus the scheme X is saturated.
To prove that the scheme X is Desarguesian, let x, y, z ∈ S k and r, s ∈ S be basis relations of X that form an initial configuration. We have to verify that r and s are linked with respect to (x, y, z). If xx * , yy * , zz * ≃ E p 3 , then this follows from Corollary 4.4 (see the statement (L1)). Assume that this condition is not satisfied, i.e., the group on the left-side is isomorphic to E p or E p 2 . Then by the assumption of the theorem, there exists q ∈ S p such that qq * ∩ xx * , yy * , zz * = {1}.
It follows that condition (17) is satisfied. Thus the required statement follows from Corollary 4.4 (see the statement (L2)).