Primitivity of homogeneous m -schemes

Chapter VI: Constructing new P -schemes from old ones

6.6 Primitivity of homogeneous m -schemes

Let U be a nonempty subset of T of cardinality at most m. We claim i_U maps each block ofC_G⁰ 0

U to a block ofC_GU. The rest of the proof focuses on this claim.

Combining it with the two diagrams above, we can derive the various properties ofC⁰(compatibility, regularity, invariance, antisymmetry and strong antisymmetry) from the corresponding properties ofC in a straightforward manner.

LetB be a block ofC_G⁰

U and B⁰ be the block ofC_GU containingi_U(B). Assume to the contrary thati_U(B)6=B⁰. ChooseG_Ug⁻¹, G_Ug⁰⁻¹ ∈G_U\G, represented by g⁻¹, g⁰⁻¹ ∈Grespectively, such thatGUg⁻¹ ∈iU(B)andGUg⁰⁻¹ ∈B⁰−iU(B). We may assumeg ∈G⁰ and hence^gz ∈^gT =T for allz ∈T. Also noteB =i⁻¹_U (B⁰) by construction. So fromG_Ug⁰⁻¹ ∈B⁰ −i_U(B)we knowG_Ug⁰⁻¹ 6∈i_U(G⁰_U\G⁰). Assume there exists z ∈ U such that ^g0z 6∈ T. As G_Ug⁻¹ and G_Ug⁰⁻¹ are in the same block B⁰ ofC_GU, by compatibility of C we knowπ_GU,Gz(G_Ug⁻¹) = G_zg⁻¹ andπ_GU,Gz(G_Ug⁰⁻¹) = G_zg⁰⁻¹ are in the same block of C_Gz. On the other hand, we haveG_zg⁻¹ =λ_z(^gz)∈λ_z(Gz∩T)andG_zg⁰⁻¹ =λ_z(^g0z)6∈λ_z(Gz∩T)since

gz ∈Gz∩T,^g0z 6∈ T andλz :Gz →Gz\Gis a bijection. But this contradicts the assumption thatλz(Gz∩T)is a disjoint union of blocks ofCGz.

Now assume ^g0z ∈ T for all z ∈ U. Suppose U = {x₁, . . . , x_k}, where x_i are distinct and ordered in an arbitrary way. Letx= (x₁, . . . , x_k)∈T^(k). Then^g0xis in Gx∩T^(k)and hence inG⁰xby the second condition in Definition 6.7. So^g0x=^g00x for someg⁰⁰ ∈G⁰. Theng⁰⁻¹g⁰⁰ ∈G_x =G_U. SoG_Ug⁰⁻¹ =G_Ug⁰⁰⁻¹ =i_U(G⁰_Ug⁰⁰⁻¹), contradicting the fact G_Ug⁰⁻¹ 6∈ i_U(G⁰_U\G⁰)above. This proves the claim thati_U maps each block ofC_G⁰0

U to a block ofC_GU.

The reader familiar with primitivity of association schemes (see, e.g., [CGS78]) may recognize that whenm ≥3, Definition 6.8 simply definesΠ = {P₁, . . . , P_m} to be primitive iff P(Π⁰) is primitive, where Π⁰ denotes the homogeneous 3- scheme{P₁, P₂, P₃}andP(Π⁰)is the corresponding association scheme (see Defi- nition 2.16).

Remark. Our definition of primitivity coincides with the notion of primitivity at level 2 introduced in the full version of [IKS09]. The same paper also generalizes the notion of primitivity to higher levels. We will not discuss their generalization in this thesis, but refer the interested reader to [IKS09] for further details.

Restricting to a connected component. We note that restricting a homogeneous m-scheme to a connected component yields another homogeneousm-scheme:

Lemma 6.16. LetΠ = {P₁, . . . , P_m}be a homogeneous m-scheme on a finite set S wherem ≥ 3. For eachB ∈ P₂ and a connected componentT ⊆ S ofG_B, the m-collectionΠk_T (see Definition 6.6) is a homogeneousm-scheme onT. Moreover, ifΠis antisymmetric (resp. strongly antisymmetric), then so isΠk_T. And ifΠhas no matching, then neither doesΠkT.

Proof. Let T ⊆ S be as in the lemma. It is well known that there exist blocks B1, . . . , Bk ∈ P2 such that the union of these blocks and1S = {(x, x) : x ∈ S}

yields an equivalence relation∼onS, andT is one of its equivalence classes (see, e.g., [CGS78]).

Fork ∈ [m], define the equivalence relation∼_k onS^(k) such that(x₁, . . . , x_k)∼_k (y₁. . . , y_k)iffx_i ∼y_ifor alli∈[k]. These equivalence relations are respected by the mapsπ_i^kandc^k_g. The various properties ofΠk_T then follow from the corresponding properties ofΠin a straightforward manner.

Primitivity of homogeneous orbitm-schemes. The next lemma states that primi- tivity of homogeneous orbitm-schemes is equivalent to primitivity of the associated permutation group.

Lemma 6.17. A homogeneous orbit m-scheme on a finite set S associated with K ⊆Sym(S)is primitive iffK is a primitive permutation group onS.

Proof. LetΠ ={P₁, . . . , P_m}be a homogeneous orbitm-scheme associated with a groupK ⊆ Sym(S). ThenK acts transitively onS. The graphsGBforB ∈ P2

are known as the non-diagonal (undirected)orbital graphs. The lemma then follows from Definition 6.8 and the well known fact that a transitive permutation group is primitive iff every non-diagonal orbital graph is connected [Hig67].

In general, we can obtain a primitive orbit m-scheme from a possibly imprimitive one by restricting to a minimal set that is a connected component:

Lemma 6.18. Let Π = {P₁, . . . , P_m} be a homogeneous orbit m-scheme on S associated with K ⊆ Sym(S), where |S| > 1. Let T be a minimal subset of S such that T is a connected component of GB for some B ∈ P2. Let K⁰ be the image of the permutation representationK_{T} →Sym(T). ThenΠk_T is a primitive homogeneous orbitm-scheme onT, and is the orbitm-scheme associated withK⁰. Proof. As already noted, for B ∈ P₂ and any connected component T⁰ of G_B, there exist blocks B₁, . . . , B_k ∈ P₂ such that the union of these blocks and 1_S = {(x, x) : x ∈ S}yields an equivalence relation on S where T⁰ is an equivalence class [CGS78]. Primitivity ofΠthen follows from minimality ofT.

ChooseB ∈P₂ such thatT is a connected component ofG_B. Note that forg ∈K and(u, v)∈ S⁽²⁾, the edge(u, v)is inG_B iff(^gu,^gv)is inG_B. So forg ∈K, the set^gT is a connected component ofGB. It follows thatT is a set of imprimitivity ofK, i.e.,^gT ∩T =∅or^gT =T for allg ∈K.

Considerk ∈[m]andx, y ∈T^(k)in the same block ofP_k|_T(k) ∈Πk_T. There exists g ∈ K sendingxto y. AsT is a set of imprimitivity ofK, we have ^gT = T and henceg ∈K⁰. SoΠk_T is the orbitm-scheme onT associated withK⁰.

Antisymmetric homogeneous orbit m-schemes for m ≥ 3. As an application, we prove that for m ≥ 3, an antisymmetric homogeneous orbit m-scheme Π on a finite set S where |S| > 1always has a matching. In particular, it is not strongly antisymmetric by Lemma 2.10. The same claim form ≥4was proved in [IKS09].

Note that strongly antisymmetric homogeneous orbitm-schemes on setsS where

|S|>1do exist form = 1andm= 2(see Section 2.5).

We need the following result from finite group theory.

Lemma 6.19. LetG be a primitive solvable permutation group on a finite set S.

The set S can be identified with a finite dimensional vector space V over a finite fieldF such thatGacts on it as a subgroup of thegeneral affine group

AGL(V) = {φg,u :g ∈GL(V), u∈V},

whereφ_g,u sendsx∈V to^gx+u. Moreover, the groupGcontains the translation φ_e,u :x7→x+ufor allu∈V.

See [Sup76, Section I.4] for its proof. We have

Theorem 6.6. LetΠ = {P₁, . . . , P_m}be an antisymmetric homogeneous orbitm- scheme on a finite setSassociated with a group K ⊆Sym(S), wherem ≥3and

|S|>1. ThenΠhas a matching.

Proof. We may assumem= 3. Assume to the contrary thatΠhas no matching. Let T be a minimal subset ofS such thatT is a connected component ofG_B for some B ∈P₂. LetK⁰ be the image of the permutation representationK{T} →Sym(T). By Lemma 6.16 and Lemma 6.18, the m-scheme Πk_T is the orbit m-scheme on T associated withK⁰ which is antisymmetric, homogeneous, primitive and has no matching. By replacingΠwithΠk_T, SwithT, andK withK⁰, we may assumeΠ is primitive. ThenK is a primitive permutation group onS by Lemma 6.17. Also note that|K|is odd by Lemma 2.16. It follows by theOdd Order Theorem[FT63]

thatK is solvable. We conclude thatK is a primitive solvable permutation group onS of odd order.

By Lemma 6.19, we can identifySwith a finite dimensional vector spaceV over a finite fieldF, andK with a subgroup ofAGL(V)acting onV that contains all the translationsφ_e,u, u∈V. Moreover, we havechar(F)6= 2since|K|is odd.

Choosev ∈ V − {0}. Let x = (0, v,2v) ∈ S⁽³⁾, y = π³₃(x) = (0, v) ∈ S⁽²⁾ and z = π₁³(x) = (v,2v) ∈ S⁽²⁾. LetB = Kx ∈ P₃, B⁰ = π₃³(B) = Ky ∈ P₂ and B⁰⁰ =π₁³(B) = Kz ∈P₂. We claim thatBtogether with the mapsπ₃³|_B :B →B⁰, π₁³|_B :B →B⁰⁰ is a matching ofΠ, which contradicts the assumption. To see this, note that the translationφ_e,v :x7→x+vis inKand sendsytoz. SoB⁰ =B⁰⁰. We also need to prove|B|=|B⁰|. By the orbit-stabilizer stabilizer theorem, it suffices to showKx =Ky, which holds since2v lies on the affine line spanned by0andv, andK acts affine linearly onV. The claim follows.

Remark. The first half of our proof basically follows [IKS09] which reduces to the case thatK is primitive solvable. In [IKS09], the proof is completed by a result of Seress [Ser96] that bounds the minimal base size of primitive solvable permutation groups of odd order. This result allows them to prove the theorem for m ≥ 4. We substitute it with the more elementary fact in Lemma 6.19, and use the above argument to prove the theorem form≥3.