New families of non-Schurian association schemes · (p + 7)=2, respectively. Mikhail Klin New...

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PreliminariesBiaffine coherent configurations

Links to other combinatorial structuresResearch in progress

New families of non-Schurian association schemesrelated to Heisenberg groups and biaffine planes 1

Mikhail Klin(joint work with Stefan Gyurki)

Ben-Gurion University of the Negev, Beer Sheva, Israel

Villanova, June 2014

1This research was supported at Matej Bel University (Slovakia) by theEuropean Social Fund, ITMS code: 26110230082.

Mikhail Klin New families of non-Schurian AS 1/ 52

DefinitionsNon-Schurian association schemes on 18 points

Preliminaries

Color graph

Under a color graph Γ we will mean an ordered pair (V ,R), whereV is a set of vertices and R a partition of V × V into binaryrelations. The elements of R will be called as colors, and thenumber of colors is the rank of Γ.

Preliminaries

Coherent configuration

A coherent configuration is a color graph N = (Ω,R),R = Ri | i ∈ I, such that the following axioms are satisfied:

(i) The diagonal relation ∆Ω = (x , x) | x ∈ Ω is a union ofrelations ∪i∈I ′Ri , for a suitable subset I ′ ⊆ I .

(ii) For each i ∈ I there exists i ′ ∈ I such that RTi = Ri ′ , where

RTi = (y , x) | (x , y) ∈ Ri is the relation transposed to Ri .

(iii) For any i , j , k ∈ I , the number cki ,j of elements z ∈ Ω such

that (x , z) ∈ Ri and (z , y) ∈ Rj is a constant depending onlyon i , j , k, and independent on the choice of (x , y) ∈ Rk .

Preliminaries

The numbers cki ,j are called intersection numbers, or sometimes

structure constants of N .

An association scheme N = (Ω,R) is a homogeneous coherentconfiguration, i.e. where the diagonal relation ∆Ω does belong toR.

A coherent configuration N is called commutative, if for alli , j , k ∈ I we have ck

ij = ckji ; and it is called symmetric if Ri = RT

for all i ∈ I .

Preliminaries

The orbits of a group G on the set Ω× Ω are called 2-orbits, ororbitals.

If 2−Orb(G ,Ω) is the set of 2-orbits of a permutation group(G ,Ω), then (Ω, 2−Orb(Ω)) is a coherent configuration. Thosecoherent configurations which can be obtained in this manner arecalled Schurian, otherwise non-Schurian. Thus, Schurianassociation schemes are coming from transitive permutationgroups.

Preliminaries

The (combinatorial) group of automorphisms Aut(N ) consists of

permutations φ : Ω→ Ω which preserve the relations, i.e. Rφi = Ri

for all Ri ∈ R.

The color automorphisms preserve relations setwise, i.e. forφ : Ω→ Ω we have φ ∈ CAut(N ) if and only if for all i ∈ I there

exists j ∈ I such that Rφi = Rj .

An algebraic automorphism is a bijection φ : R → R whichsatisfies ck

ij = ckφ

iφjφ.

Preliminaries

Let K be a subgroup of the group of algebraic automorphisms of acoherent configuration. Let R/K denote the set of orbits of K onR. For each O ∈ R/K define O+ to be the union of all relationsfrom O.

Then the set of relations O+ |O ∈ R/K forms a coherentconfiguration on Ω. We will call it as algebraic merging of R withrespect to K .

Non-Schurian association schemes on 18 points

This research has been started in year 2012 during the visit ofStefan Gyurki at BGU.

A careful analysis of known association schemes on 18 points,which are available from the homepage of Hanaki and Miyamoto,was our starting point.

The main interest was to understand two non-Schurian associationschemes.

0 1 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 72 0 1 4 4 4 5 5 5 3 3 3 6 6 6 7 7 71 2 0 5 5 5 3 3 3 4 4 4 6 6 6 7 7 73 5 4 0 6 7 2 6 7 1 6 7 3 4 5 3 4 53 5 4 7 0 6 7 2 6 7 1 6 4 5 3 5 3 43 5 4 6 7 0 6 7 2 6 7 1 5 3 4 4 5 35 4 3 1 6 7 0 6 7 2 6 7 5 3 4 5 3 45 4 3 7 1 6 7 0 6 7 2 6 3 4 5 4 5 35 4 3 6 7 1 6 7 0 6 7 2 4 5 3 3 4 54 3 5 2 6 7 1 6 7 0 6 7 4 5 3 4 5 34 3 5 7 2 6 7 1 6 7 0 6 5 3 4 3 4 54 3 5 6 7 2 6 7 1 6 7 0 3 4 5 5 3 47 7 7 3 5 4 4 3 5 5 4 3 0 2 1 6 6 67 7 7 5 4 3 3 5 4 4 3 5 1 0 2 6 6 67 7 7 4 3 5 5 4 3 3 5 4 2 1 0 6 6 66 6 6 3 4 5 4 5 3 5 3 4 7 7 7 0 2 16 6 6 5 3 4 3 4 5 4 5 3 7 7 7 1 0 26 6 6 4 5 3 5 3 4 3 4 5 7 7 7 2 1 0

The color matrix of the non-Schurian association scheme on 18points of rank 8 (nr. 62 in the catalogue).

0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 5 5 51 0 1 5 5 5 3 3 3 4 4 4 2 2 2 5 5 51 1 0 5 5 5 3 3 3 4 4 4 5 5 5 2 2 22 5 5 0 3 4 2 5 5 2 5 5 1 3 4 1 3 42 5 5 4 0 3 5 2 5 5 2 5 4 1 3 4 1 32 5 5 3 4 0 5 5 2 5 5 2 3 4 1 3 4 14 4 4 2 5 5 0 1 1 3 3 3 5 2 5 5 5 24 4 4 5 2 5 1 0 1 3 3 3 5 5 2 2 5 54 4 4 5 5 2 1 1 0 3 3 3 2 5 5 5 2 53 3 3 2 5 5 4 4 4 0 1 1 5 5 2 5 2 53 3 3 5 2 5 4 4 4 1 0 1 2 5 5 5 5 23 3 3 5 5 2 4 4 4 1 1 0 5 2 5 2 5 55 2 5 1 3 4 5 5 2 5 2 5 0 3 4 1 3 45 2 5 4 1 3 2 5 5 5 5 2 4 0 3 4 1 35 2 5 3 4 1 5 2 5 2 5 5 3 4 0 3 4 15 5 2 1 3 4 5 2 5 5 5 2 1 3 4 0 3 45 5 2 4 1 3 5 5 2 2 5 5 4 1 3 4 0 35 5 2 3 4 1 2 5 5 5 2 5 3 4 1 3 4 0

The color matrix of the non-Schurian association scheme on 18points of rank 6 (nr. 41 in the catalogue).

What was done?

Finally, we realized that, for each prime p, we can:

work with an intransitive permutation group H of order p3,acting on two orbits of length p2,

construct a corresponding coherent configuration M of rank6p − 2 with two fibers,

detect in M four association schemes.

Biaffine planesThe master coherent configurationFour color graphsGroups of combinatorial automorphisms

Biaffine planes

The biaffine plane Bp consists of two copies of Zp × Zp:points P and “non-vertical” lines L.

Points: P = [x , y ].

Lines: ` = (k, q), y = k · x + q.

Incidence: P = [x , y ] is incident to ` = (k , q) if and only ify = k · x + q.

Biaffine planes

[0, 2]

[0, 1]

[0, 0]

[1, 2]

[1, 1]

[1, 0]

[2, 2]

[2, 1]

[2, 0]

Points : Lines `k = 0 : Lines `k = 1 : Lines `k = 2 :

Figure: The objects of the biaffine plane B3.

The master coherent configuration

Take an action of the permutation group H = (Zp)2 o Zp on theset Ω = P ∪ L.

Usually, this group is called the Heisenberg group of order p3.

H ∼= 〈t1,0, t0,1, φ〉, where

ta,b : [x , y ] 7→ [x + a, y + b], (k , q) 7→ (k , b + q − ak),

φ : [x , y ] 7→ [x , y − x ], (k, q) 7→ (k − 1, q).

The group H has 6p − 2 orbits on Ω× Ω:

(P1,P2) ∈ Ai ⇐⇒ x1 = x2 and y2 − y1 = i , where i ∈ Zp,

(P1,P2) ∈ Bi ⇐⇒ x2 − x1 = i 6= 0, where i ∈ Zp \ 0,(`1, `2) ∈ Ci ⇐⇒ k1 = k2 and q2 − q1 = i , where i ∈ Zp,

(`1, `2) ∈ Di ⇐⇒ k2 − k1 = i 6= 0, where i ∈ Zp \ 0,(P1, `1) ∈ Ei ⇐⇒ k1 · x1 + q1 − y1 = i , where i ∈ Zp,

(`1,P1) ∈ Fi ⇐⇒ y1 − k1x1 − q1 = i , where i ∈ Zp.

Definition

The structure M = (Ω, 2−Orb(H)) is called a (master) biaffinecoherent configuration.

Four color graphs

We determine four color graphs M1, M2, M3, M4 as suitablemergings of relations in the master coherent configuration M.

Their appearance was motivated by computer experiments forsmall values of p, like p = 3, 5, 7, 11.

Theorem 1

The following holds:

(a) M1,M2,M3,M4 are association schemes.

(b) Combinatorial groups of automorphisms of M1,M2,M3,M4

contain a subgroup isomorphic to H = Z2p o Zp.

Groups of combinatorial automorphisms

Theorem 2

Let Aut(M1),Aut(M2),Aut(M3) and Aut(M4) are thecombinatorial groups of automorphisms of M1,M2,M3 and M4,respectively. Then the followings hold:

(a) Aut(M1) ≤ Aut(M2) = Aut(M3) ≤ Aut(M4),

(b) |Aut(M1)| = p3,

(c) |Aut(M2)| = 2p3,

(d) |Aut(M3)| = 2p3,

(e) |Aut(M4)| = 8p3.

Corollary

Corollary 3

For each p > 3 there exist at least four non-Schurian associationschemes M1,M2,M3, and M4 with ranks 3p − 1, 2p, p + 3, and(p + 7)/2, respectively.

Algebraic groups

Theorem 4

AAut(M) ∼= (Z2p−1 o Z2)× AGL(1, p).

Proposition 5

Association schemes M1,M2,M3,M4 are algebraic mergings ofM.

Theorem 6

AAut(M1) ∼= Z2p−1.

The Pappus graphMcKay-Miller-Siran graphsWenger graphs

Links to other combinatorial structures

For p = 3 one of the basic graphs at the scheme M1 is the Pappusgraph.

McKay-Miller-Siran graphs

Let p be an odd prime and put Vp = Z2 ×Zp ×Zp as vertex set ofHp. Let ω be a primitive element.

If p = 4r + 1 then define X = 1, ω2, ω4, . . . , ωp−3,X ′ = ω, ω3, . . . , ωp−2.If p = 4r + 3 then define X = ±1,±ω2, . . . ,±ω2r,X ′ = ±ω,±ω3, . . . ,±ω2r+1.

The adjacency in the graph Hp is defined as follows:

(0, x , y) is adjacent to (0, x , y ′) if and only if y − y ′ ∈ X ,

(1, k , q) is adjacent to (1, k, q′) if and only if q − q′ ∈ X ′,

(0, x , y) is adjacent to (1, k , q) if and only if y = kx + q.

Hp = E0 ∪ F0 ∪⋃i∈X

Ai ∪⋃j∈X ′

H5 is the well-known Hoffman-Singleton graph.

These graphs were also considered by Hafner and Siagiova.

P0 P1 P2 P3 P4

Q0 Q1 Q2 Q3 Q4

1 1 1 11

Adjacencies are between i in Pj and i ⊕ jk in Qk for all0 ≤ i , j , k ≤ 4. (Robertson)

Wenger graphs

The graph Wn(q) has as vertex set two copies P and L of the(n + 1)-dimensional vector space over Fq. The adjacency betweenpoints P = [p1, . . . , pn+1] and “lines” L = (l1, . . . , ln+1) is given bythe system:

l2 + p2 = p1l1

l3 + p3 = p1l2...

ln+1 + pn+1 = p1ln.

Wenger graphs W1(p) are isomorphic to the graphs defined by U0.

Here U0 = E0 ∪ F0, this is, in fact, the set of all flags in biaffineplane Bp.

For example, W1(3) is isomorphic to the Pappus graph.

Wenger graphs belong to a richer family of so-called algebraicallydefined graphs, which are described by a system of equations.

The rank of schemes in the four families above grows linearly withincreasing p.

Recently, we discovered new families of non-Schurian mergings ofour master coherent configuration, which have constant rank.

This part of our results is still research in progress.

Observation 7

Assume p = 4k + 3, then there exists non-Schurian rank 6 mergingof M with the following intersection matrices:

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

0 1 0 0 0 0

p−1 p−2 0 0 0 00 0 p−1 0 0 00 0 0 p−1 0 00 0 0 0 0 10 0 0 0 p−1 p−2

0 0 1 0 0 00 0 p−1 0 0 00 0 (p2−3p)/4 (p2+p)/4 0 0

(p2−p)/2 (p2−p)/2 (p2−3p)/4 (p2−3p)/4 0 00 0 0 0 (p−1)/2 (p−1)/2

0 0 0 0 (p−1)2/2 (p−1)2/2

0 0 0 1 0 00 0 0 p−1 0 0

(p2−p)/2 (p2−p)/2 (p2−3p)/4 (p2−3p)/4 0 0

0 0 (p2+p)/4 (p2−3p)/4 0 00 0 0 0 (p−1)/2 (p−1)/2

0 0 0 0 (p−1)2/2 (p−1)2/2

0 0 0 0 1 00 0 0 0 0 10 0 0 0 (p−1)/2 (p−1)/20 0 0 0 (p−1)/2 (p−1)/2p 0 1 1 0 00 p p−1 p−1 0 0

0 0 0 0 0 10 0 0 0 p−1 p−20 0 0 0 (p−1)2/2 (p−1)2/2

0 0 0 0 (p−1)2/2 (p−1)2/20 p p−1 p−1 0 0

p2−p p2−2p (p−1)2 (p−1)2 0 0

Observation 7

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

0 1 0 0 0 0

p−1 p−2 0 0 0 00 0 p−1 0 0 00 0 0 p−1 0 00 0 0 0 0 10 0 0 0 p−1 p−2

0 0 1 0 0 00 0 p−1 0 0 0

(p2−p)/2 (p2−p)/2 (p2−5p)/4 (p2−p)/4 0 0

0 0 (p2−p)/4 (p2−p)/4 0 00 0 0 0 (p−1)/2 (p−1)/2

0 0 0 0 (p−1)2/2 (p−1)2/2

0 0 0 1 0 00 0 0 p−1 0 00 0 (p2−p)/4 (p2−p)/4 0 0

(p2−p)/2 (p2−p)/2 (p2−p)/4 (p2−5p)/4 0 00 0 0 0 (p−1)/2 (p−1)/2

0 0 0 0 (p−1)2/2 (p−1)2/2

0 0 0 0 1 00 0 0 0 0 10 0 0 0 (p−1)/2 (p−1)/20 0 0 0 (p−1)/2 (p−1)/2p 0 1 1 0 00 p p−1 p−1 0 0

0 0 0 0 0 10 0 0 0 p−1 p−20 0 0 0 (p−1)2/2 (p−1)2/2

0 0 0 0 (p−1)2/2 (p−1)2/20 p p−1 p−1 0 0

p2−p p2−2p (p−1)2 (p−1)2 0 0

Observation 9

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

0 1 0 0 0p−1 p−2 0 0 0

0 0 p−1 0 00 0 0 (p−3)/2 (p−1)/20 0 0 (p+1)/2 (p−1)/2

0 0 1 0 00 0 p−1 0 0

p2−p p2−p p2−2p 0 00 0 0 (p−1)2/2 (p−1)2/2

0 0 0 (p2−1)/2 (p2−1)/2

0 0 0 1 00 0 0 (p−3)/2 (p−1)/2

0 0 0 (p−1)2/2 (p−1)2/2

(p2−p)/2 (p2−3p)/4 (p−1)2/4 0 0

0 (p2+p)/4 (p2−1)/4 0 0

0 0 0 0 10 0 0 (p+1)/2 (p−1)/2

0 0 0 (p2−1)/2 (p2−1)/2

0 (p2+p)/4 (p2−1)/4 0 0

(p2+p)/2 (p2+p)/4 (p+1)2/4 0 0

Example

Example 10

For p = 13 there exists a non-Schurian rank 5 merging withvalencies 1, 12, 52, 117, 156.

Remaining tasks

to round obtained results and to get for them full theoreticaljustification;

to understand spectral properties of the appearing graphs;

to make a more deep comparison with other results;

to get more full and more beautiful justification of theproperties of the schemes Mi , 1 ≤ i ≤ 4.

Computer tools

COCO II (still in development)

GRAPE together with nauty

References

Faradzev I.A., Klin M.H.:Computer package for computationswith coherent configurations, Proc. ISSAC-91, pp. 219–223.

Hafner P.R.: Geometric realization of the graphs ofMcKay-Miller-Siran, J. Comb. Th. B, 90(2) (2004), 223–232.

Klin M.H., Muzychuk M.E., Pech C., Woldar A.J., ZieschangP-H.: Association schemes on 28 points as mergings of ahalf-homogeneous coherent configuration, Eur. J. Combin.28(7) (2007), 1994–2025.

Wild P.: Biaffine planes and divisible semiplanes, J. Geom.,25(2) (1985), 121–130.

Thank you

Thank you for your attention.

Summer School in Coherent Configurations, Permutation Groupsand Applications in Algebraic Graph Theory 2014

When? Where?

August 31 - September 5, 2014, High Tatras, Slovakia

Lecturers:

Mikhail Klin and Gareth Jones

Organized by:

Slovak Mathematical Society & Matej Bel University

New families of non-Schurian association schemes · (p + 7)=2, respectively. Mikhail Klin New...

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