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
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
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.
Mikhail Klin New families of non-Schurian AS 2/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
DefinitionsNon-Schurian association schemes on 18 points
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 .
Mikhail Klin New families of non-Schurian AS 3/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
DefinitionsNon-Schurian association schemes on 18 points
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
i
for all i ∈ I .
Mikhail Klin New families of non-Schurian AS 4/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
DefinitionsNon-Schurian association schemes on 18 points
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.
Mikhail Klin New families of non-Schurian AS 5/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
DefinitionsNon-Schurian association schemes on 18 points
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.
Mikhail Klin New families of non-Schurian AS 6/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
DefinitionsNon-Schurian association schemes on 18 points
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 .
Mikhail Klin New families of non-Schurian AS 7/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
DefinitionsNon-Schurian association schemes on 18 points
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.
Mikhail Klin New families of non-Schurian AS 8/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
DefinitionsNon-Schurian association schemes on 18 points
Non-Schurian association schemes on 18 points
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).
Mikhail Klin New families of non-Schurian AS 9/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
DefinitionsNon-Schurian association schemes on 18 points
Non-Schurian association schemes on 18 points
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).
Mikhail Klin New families of non-Schurian AS 10/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
DefinitionsNon-Schurian association schemes on 18 points
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.
Mikhail Klin New families of non-Schurian AS 11/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
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.
Mikhail Klin New families of non-Schurian AS 12/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
Biaffine planesThe master coherent configurationFour color graphsGroups of combinatorial automorphisms
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.
Mikhail Klin New families of non-Schurian AS 13/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
Biaffine planesThe master coherent configurationFour color graphsGroups of combinatorial automorphisms
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.
Mikhail Klin New families of non-Schurian AS 14/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
Biaffine planesThe master coherent configurationFour color graphsGroups of combinatorial automorphisms
The master coherent configuration
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).
Mikhail Klin New families of non-Schurian AS 15/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
Biaffine planesThe master coherent configurationFour color graphsGroups of combinatorial automorphisms
The master coherent configuration
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.
Mikhail Klin New families of non-Schurian AS 16/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
Biaffine planesThe master coherent configurationFour color graphsGroups of combinatorial automorphisms
The master coherent configuration
Definition
The structure M = (Ω, 2−Orb(H)) is called a (master) biaffinecoherent configuration.
Mikhail Klin New families of non-Schurian AS 17/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
Biaffine planesThe master coherent configurationFour color graphsGroups of combinatorial automorphisms
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.
Mikhail Klin New families of non-Schurian AS 18/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
Biaffine planesThe master coherent configurationFour color graphsGroups of combinatorial automorphisms
Theorem 1
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.
Mikhail Klin New families of non-Schurian AS 19/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
Biaffine planesThe master coherent configurationFour color graphsGroups of combinatorial automorphisms
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.
Mikhail Klin New families of non-Schurian AS 20/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
Biaffine planesThe master coherent configurationFour color graphsGroups of combinatorial automorphisms
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.
Mikhail Klin New families of non-Schurian AS 21/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
Biaffine planesThe master coherent configurationFour color graphsGroups of combinatorial automorphisms
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.
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PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
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.
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PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
The Pappus graphMcKay-Miller-Siran graphsWenger graphs
Links to other combinatorial structures
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.
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PreliminariesBiaffine coherent configurations
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The Pappus graphMcKay-Miller-Siran graphsWenger graphs
Links to other combinatorial structures
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 ′
Cj .
H5 is the well-known Hoffman-Singleton graph.
These graphs were also considered by Hafner and Siagiova.
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PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
The Pappus graphMcKay-Miller-Siran graphsWenger graphs
Links to other combinatorial structures
4 0
3 1
2 0
43
4 0
3 1
2 0
43
4 0
3 1
2 0
43
4 0
3 1
2 0
43
4 0
3 1
2 0
43
P0 P1 P2 P3 P4
Q0 Q1 Q2 Q3 Q4
2 22
2 2
1 1 1 11
Adjacencies are between i in Pj and i ⊕ jk in Qk for all0 ≤ i , j , k ≤ 4. (Robertson)
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PreliminariesBiaffine coherent configurations
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The Pappus graphMcKay-Miller-Siran graphsWenger graphs
Links to other combinatorial structures
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.
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PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
The Pappus graphMcKay-Miller-Siran graphsWenger graphs
Links to other combinatorial structures
n = 1
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.
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PreliminariesBiaffine coherent configurations
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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.
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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
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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
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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
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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
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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
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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
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Observation 7
Assume p = 4k + 1, 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
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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
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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
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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
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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
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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
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Observation 9
Assume p = 4k + 3, then there exists non-Schurian rank 5 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
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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
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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
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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
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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
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Example
Example 10
For p = 13 there exists a non-Schurian rank 5 merging withvalencies 1, 12, 52, 117, 156.
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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.
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Computer tools
COCO
COCO II (still in development)
GAP
GRAPE together with nauty
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PreliminariesBiaffine coherent configurations
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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.
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Thank you
Thank you for your attention.
Mikhail Klin New families of non-Schurian AS 51/ 52
PreliminariesBiaffine coherent configurations
Links to other combinatorial structuresResearch in progress
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What?
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
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