Coherent Configurations with Two Fibers

Coherent Configurations 7: Coherent configurations with two fibers

Mikhail Klin (Ben-Gurion University)

September 1-5, 2014

M. Klin (BGU) Conﬁgurations with two ﬁbers 1 / 62 Fibers

An association scheme is a particular case of coherent configuration (briefly CC): all loops form one basic graph. In general, in a coherent configuration the full reflexive relation is split into a few parts, each one corresponds to a fiber. A fiber is a combinatorial analogue of an orbit of a permutation group.

M. Klin (BGU) Configurations with two fibers 2 / 62 Two fiber coherent configurations

In a series of papers, D. Higman attacked coherent configurations of “small type”, that is non-trivial types of coherent configurations with two fibers, each of rank at most three. Following Higman, we investigate irreducible ones, that is those that can not be decomposed as direct or wreath products of configurations of a smaller order.

M. Klin (BGU) Configurations with two fibers 3 / 62 The type of a CC W with two fibers is a a b matrix , where a is the rank of the c AS on the first fiber, c on the second fiber, b the amount of relations from first to second fiber. Thus rank(W ) = a + 2b + c. Clearly a ≥ 2, c ≥ 2, and for non-trivial cases, b ≥ 2.

M. Klin (BGU) Configurations with two fibers 4 / 62 In addition, we require that the AS on each fiber be symmetric. 2 2 The smallest non-trivial type is . 2 It is easy to understand that it is equivalent to the case of symmetric block designs.

M. Klin (BGU) Conﬁgurations with two ﬁbers 5 / 62 The next interesting “small” type is 2 2 . 3 It is equivalent to quasi-symmetric block designs. This case was carefully investigated by A.R. Calderbank, A. Neumaier, J.J.Seidel, M.S. Shrikhande et al.

M. Klin (BGU) Conﬁgurations with two ﬁbers 6 / 62 Example 1 Hyperovals in the projective plane Π of order 4. Start from any quadrangle Q in Π, Q = {A, B, C, D}: Two more points E, F are not collinear with any two elements of Q.

M. Klin (BGU) Conﬁgurations with two ﬁbers 7 / 62 {A, B, C, D, E, F } is an example of a (hyper-) oval. 21·20·16·9 There are 4! = 2520 quadrangles. 6 Each oval contains 2 = 15 quadrangles. 2520 There are 15 = 168 ovals.

M. Klin (BGU) Conﬁgurations with two ﬁbers 8 / 62 More about ovals: The simple group PSL(3, 4) of order 20160 acts transitively on points of Π and has 3 orbits of equal size 56 on the set of ovals. Pick one such orbit X . Two ovals from X are adjacent if they are disjoint.

M. Klin (BGU) Conﬁgurations with two ﬁbers 9 / 62 Now points of Π are points of the design D, ovals from X are blocks of D. We get a biplane D with 21 points and 56 blocks. On blocks we have a primitive triangle-free SRG(56, 10, 0, 2), the Gewirtz graph.. Altogether we have a quasisymmetric BIBD.

M. Klin (BGU) Conﬁgurations with two ﬁbers 10 / 62 Strongly regular designs

We consider the third type (due to 3 2 Higman), that is ( 3 ), which was called by him strongly regular designs (SRD). These are rank 10 coherent conﬁgurations with two ﬁbers. They will be subject of main attention in this lecture.

M. Klin (BGU) Configurations with two fibers 11 / 62 Each fiber in an SRD induces an SRG (that is, an AS with two classes, or a rank 3 AS). Extra special requirements imply axiomatics of partial geometries. Very beautiful and quite rare structures.

M. Klin (BGU) Conﬁgurations with two ﬁbers 12 / 62 Example 2 (Partial geometry pg(8, 9, 4)) Start with the Fano plane F , also known as PG(2, 2). 3

5 6 7 1 4 2 Aut(F ) =∼ PSL(3, 2) is a simple group of order 168.

M. Klin (BGU) Conﬁgurations with two ﬁbers 13 / 62 Add one more point 8 to the point set of F . Consider the action of the alternating group A8 on all images of such a structure (under S8). There are two orbits of equal length 8! 2·168 = 120. Pick one such orbit V .

M. Klin (BGU) Configurations with two fibers 14 / 62 Define a graph Γ with vertex set V . Two copies of F are adjacent ⇐⇒ they do not share a common line. Get an SRG Γ with parameters

(120, 63, 30, 36).

M. Klin (BGU) Conﬁgurations with two ﬁbers 15 / 62 The graph Γ contains 2025 cliques of size 8.

Among the orbits of A8 on these cliques there are orbits of lengths 120 and 15. Pick these orbits and get a set L of size 135 consisting of 8-element subsets of V .

M. Klin (BGU) Conﬁgurations with two ﬁbers 16 / 62 The structure (V , L) provides an example of a partial geometry pg(8, 9, 4). Details will be communicated during the exercises. This model is due to Klin-Reichard (1997).

M. Klin (BGU) Conﬁgurations with two ﬁbers 17 / 62 Extra comments

∼ 6 In fact, Aut(Γ) = (Z) .A8. There are altogether four partial geometries with the same point graph Γ, having groups 1 of order 28!, 1440, 1344 (twice). The graph Γ was discovered by Brouwer-A.V. Ivanov-Klin (1989).

M. Klin (BGU) Conﬁgurations with two ﬁbers 18 / 62 Altogether (up to isomorphism) there are eight known pg(8, 9, 4). Contributions by A.M. Cohen, R. Mathon and A.P. Street. Full list of all pg(8, 9, 4) is not yet known.

M. Klin (BGU) Conﬁgurations with two ﬁbers 19 / 62 An SRD is called proper if it does not come from a partial geometry. An SRD is called primitive if both point and line graphs are primitive SRG’s.

M. Klin (BGU) Conﬁgurations with two ﬁbers 20 / 62 Structure of SRDs

Each ﬁber in SRD induces a strongly regular graph (that is, a rank 3 scheme). Higman required that both SRGs are primitive. This is a strong requirement: there are just three parameter sets on up to 1000 vertices (for certain proper cases, those that are not coming from partial geometries). We omit requirements of primitivity.

M. Klin (BGU) Conﬁgurations with two ﬁbers 21 / 62 Results

The list of feasible parameter sets increases drastically.

Let n1, n2 be the sizes of the two ﬁbers, n = max(n1, n2). Then we get 22 parameter sets for n ≤ 20, 318 for n ≤ 100 and 869 for n ≤ 200. Using computer, smallest parameter sets were investigated, all SRDs were classiﬁed (a few feasible parameter sets were killed).

M. Klin (BGU) Conﬁgurations with two ﬁbers 22 / 62 Recall that a graph Γ is called coherent if it is a basic graph of its WL-closure W (Γ). Special interest goes to coherent non-Schurian graphs. We combine this requirement with consideration of an incidence graph of an SRD.

M. Klin (BGU) Configurations with two fibers 23 / 62 Thus we consider the incidence structure (P, B, F ), where P is a set of points, B a set of blocks, and F a set of flags, here F ⊆ P × B, and P and B are disjoint non-empty sets. The dual incidence structure is (B, P, F T ), where F T = {(y, x)|(x, y) ∈ F }.

M. Klin (BGU) Configurations with two fibers 24 / 62 A strongly regular design (SRD) is a finite incidence structure with n1 points and n2 blocks which satisfies the following conditions (together with the corresponding conditions, formulated for their duals):

1 Each block is incident to S1 points. 2 Two distinct blocks are incident with either a1 or b1 points, a1 > b1, and both cases occur.

M. Klin (BGU) Configurations with two fibers 25 / 62 This allows us to define the block graph Γ2 to be the graph whose vertices are blocks, two distinct blocks being adjacent if they are incident with a1 common points. 3 The number of blocks incident with a point x and adjacent to a block y is N2 or P2, according as x and y are incident or not.

M. Klin (BGU) Configurations with two fibers 26 / 62 There are also the dual conditions 1’, 2’, 3’: we understand the same conditions 1, 2, 3, just formulated for the dual structure. In the constants applying to the dual we change the index from 1 to 2 or vice versa. Thus, by definition the SRD’s form a self-dual class of designs.

M. Klin (BGU) Conﬁgurations with two ﬁbers 27 / 62 An SRD Γ is called algebraically (or formally) self-dual (or symmetric) if all parameters of Γ coincide with the corresponding parameters of its dual ΓT . A symmetric SRD (SSRD) Γ is (combinatorially) self-dual if it allows a duality, that is, if Γ and ΓT are isomorphic incidence structures.

M. Klin (BGU) Conﬁgurations with two ﬁbers 28 / 62 Higman lists 15 numerical equations for the parameter sets of an SRD. Unfortunately there is a (typographical?) mistake in the formulation of Condition 15. This is why in our consideration we were omitting this Condition 15.

M. Klin (BGU) Configurations with two fibers 29 / 62 The first job was to generate the list of feasible parameters of SRD’s with n up to 200, n = max(n1, n2). The list contains for each of the two designs Γ and ΓT the parameters (ni , si , ai , bi , Ni , Pi ), i = 1, 2, as well as the parameters (k, λ, µ, r, s, f , g) for both accompanying SRG’s (point and block graphs).

M. Klin (BGU) Configurations with two fibers 30 / 62 3 2 Note that a given CC of type can 3 be generated by four different SRD’s, some of which may be isomorphic (for known SRD’s, this information is included to the list).

M. Klin (BGU) Conﬁgurations with two ﬁbers 31 / 62 We have 22 parameter sets for n ≤ 20, 318 for n ≤ 100, and 869 for n ≤ 200. Besides the completeness of the list of parameters, our new input to it refers only to proper SRD’s.

M. Klin (BGU) Configurations with two fibers 32 / 62 The next stage was the constructive enumeration of all SRD’s (up to isomorphism) with a given parameter set. At this moment full results have been obtained for the initial 25 parameter sets (n ≤ 24). The first hole in the constructive enumeration corresponds to set # 26 with n1 = n2 = 24.

M. Klin (BGU) Conﬁgurations with two ﬁbers 33 / 62 The computer returns non-existence of SSRD’s for parameter sets 5, 7, 15, 16, 22, 23, 31 (n = 10, 12, 16, 16, 20, 21, 26) thus pushing us to formulate some plausible conjectures about the non-existence of SSRD’s in some general situations.

M. Klin (BGU) Configurations with two fibers 34 / 62 The first case with a pleasant surprise is provided by parameter set #6 (n1 = 8, n2 = 12) where there exists a unique SRD that is non-Schurian. In addition we mention results corresponding to parameter sets #8, 12, 13, 18, 21, 24, 28, 33. In each such case there exists at least one non-Schurian example.

M. Klin (BGU) Configurations with two fibers 35 / 62 The smallest non-Schurian SRD with 8 points and 12 lines This is the first example (# 6 in our list) of a non-Schurian SRD. It has the following parameters: n1 = 8, S1 = 4, a1 = 2, b1 = 0, N1 = 2, P1 = 2, n2 = 12, S2 = 6, a2 = 3, b2 = 2, N2 = 5, P2 = 5, (n1, k1, l1, µ1, λ1) = (8, 4, 3, 4, 0) and (n2, k2, l2, µ2, λ2) = (12, 10, 1, 10, 8).

M. Klin (BGU) Configurations with two fibers 36 / 62 Up to isomorphisms there is just one CC with two fibres of size n1 = 8 and n2 = 12. This CC is non-Schurian, the SRD and the one with complement incidence structure are isomorphic. The automorphism group G of this CC, denoted now by S, has order 192, acts transitively on the fibres and, as intransitive permutation group of degree 20, has rank 11.

M. Klin (BGU) Conﬁgurations with two ﬁbers 37 / 62 Recall that the WL-closure of an SRD has always rank 10. Thus this inequality, observed by us on a computational level, allows to claim that the SRD # 6 is non-Schurian.

M. Klin (BGU) Conﬁgurations with two ﬁbers 38 / 62 The point graph of S is isomorphic to the complete bipartite graph Γ1 = K4,4, while the block graph Γ2 has the form 6 ◦ K2. Each subset of the vertex set V corresponding to a block induces a quadrangle (or 4-cycle) in Γ1.

M. Klin (BGU) Conﬁgurations with two ﬁbers 39 / 62 Thus it is convenient to consider the block set B of S as the set of 12 induced quadrangles in Γ1. It is listed below as follows: (0, 1, 2, 3) (0, 1, 4, 6) (0, 1, 5, 7) (0, 2, 4, 7) (0, 2, 5, 6) (0, 3, 6, 7) (1, 2, 4, 5) (1, 3, 4, 7) (1, 3, 5, 6) (2, 3, 4, 6) (2, 3, 5, 7) (4, 5, 6, 7)

M. Klin (BGU) Conﬁgurations with two ﬁbers 40 / 62 Example 3 (A mnemonic model) Consider the Latin square L, Cayley table of e a b c the group E4, that is a e c b L = b c e a c b a e Identify rows and columns of L with points of M. 4 Prove that L has 2 · 2 = 12 Latin subsquares of order 2. Identify these subsquares as lines in M. Incidence is natural.

M. Klin (BGU) Configurations with two fibers 41 / 62 It is a little bit more tricky to observe the group of order 192. For this purpose let us consider a suitable amorphic association scheme which is defined on the cells of L. The automorphism group of this scheme has 2 order |E4| · |Aut(E4)| · 2 = 16 · 6 · 2 = 192. This group is isomorphic to our group G.

M. Klin (BGU) Configurations with two fibers 42 / 62 In fact we were able to elaborate four additional models of this strongly regular design. These models refer to such diverse structures as the affine plane of order 4, a reaction graph on cubes, extra manipulations inside of K4,4, and the consideration of the famous Shrikhande graph.

M. Klin (BGU) Conﬁgurations with two ﬁbers 43 / 62 Using these models we get diverse names for G (with the aid of GAP):

∼ G = [E16 : S3].Z2 ∼ = E8 : S4 ∼ = E4 :(S3 o S2) ∼ = [E16 : Z3]: E4.

M. Klin (BGU) Configurations with two fibers 44 / 62 The Reye configuration and its non-Schurian mates

We are now working with the next interesting case provided by the parameter set #8. Here n1 = n2, and the structure is formally self-dual, thus it is enough to list the parameters of the initial SRD, which are n = 12, S = 6, a = 3, b = 2, N = 4, P = 4, (n, k, l, λ, µ) = (12, 8, 3, 4, 8).

M. Klin (BGU) Conﬁgurations with two ﬁbers 45 / 62 Both the point and the block graphs are of the form 3 ◦ K4. By computer search we found that there exist exactly three solutions up to isomorphism.

M. Klin (BGU) Conﬁgurations with two ﬁbers 46 / 62 Notation Schurity self-dual # SRD’s R1 yes yes 2 R2 no no 4 R3 no yes 2 CC AS Notation |Aut| rank Aut |Aut| rank Aut pos R1 576 10 (S2 o S4) 1152 5 S2 o S4 R2 24 52 A4 × S2 24 52 A4 × S2 R3 32 58 (D4 × S2): S2 64 29 ((C4 × C4): S2): S2

M. Klin (BGU) Configurations with two fibers 47 / 62 Reye configuration

A brief sketch of the classical structure, usually called Reye conﬁguration, which corresponds to our solution R1. The structure was discovered by T. Reye. It appeared in terms of projective geometry and included 12 points, 16 lines, and 12 planes.

M. Klin (BGU) Configurations with two fibers 48 / 62 The pair (points, lines) indeed forms a configuration of type (124, 163) in the sense which was coined by Reye himself (1876) - a partial linear space in modern terms. The pair (lines, planes) is also a configuration (with dual parameters), while the pair (points, planes) is not literally a configuration but a general incidence structure.

M. Klin (BGU) Configurations with two fibers 49 / 62 The projective model of the Reye configuration is described in terms of the cube Q3. Points are the eight vertices of the cube, its centre, and three points at infinity corresponding to the directions of the edges of Q3. The 16 lines, each of size 3, consist of the 12 edges of Q3 and the four space diagonals with natural incidence.

M. Klin (BGU) Conﬁgurations with two ﬁbers 50 / 62 The 12 planes are the six faces of Q3 and the six planes passing through diagonally opposite pairs of edges (again with natural incidence). Typically, symmetries of this structure are considered strictly in the projective framework and thus are explained in terms of the geometrical symmetries of Q3.

M. Klin (BGU) Configurations with two fibers 51 / 62 For us the pair (points, planes) appears as “abstract” incidence structure. This is why finally its automorphism group turns out to be much larger and has order 576 (if we are thinking of the CC).

M. Klin (BGU) Conﬁgurations with two ﬁbers 52 / 62 Extra feature of R1

It turns out that the incidence and non-incidence graphs of R1 are not isomorphic. (Recall that both graphs have the same parameter sets.)

M. Klin (BGU) Configurations with two fibers 53 / 62 For the justification it is enough to prove that the incidence graph of R1 and of the complement R1 to R1 are not isomorphic.

M. Klin (BGU) Conﬁgurations with two ﬁbers 54 / 62 Consider any line of the Reye geometry. It is clear that the three points and the three lines incident to it form an induced subgraph of the incidence graph (see Figure 1) that is isomorphic to the graph K3,3. Altogether there are 16 such subgraphs; one example is given by the vertices 0, 1, 2, 16, 21, 23.

M. Klin (BGU) Conﬁgurations with two ﬁbers 55 / 62 It turns out that the non-incidence graph does not have any subgraph isomorphic to K3,3.

M. Klin (BGU) Conﬁgurations with two ﬁbers 56 / 62 0 • 11• •1 21• 10• 20• •22 •2 19• •23 9• 18• •12 •3

17• •13 • • 16• 14• 8 15• 4 • • 7 • 5 6

Figure: The incidence graph of the Reye conﬁguration

M. Klin (BGU) Conﬁgurations with two ﬁbers 57 / 62 •0 •6 18• 12• 11• •1 17• •19 5• 23• •13 •7

• • 10• 16 20 •2 22• 14• 4• •8 15• 21• •9 •3

Figure: The non-incidence graph of the Reye conﬁguration

M. Klin (BGU) Configurations with two fibers 58 / 62 Pseudo-Reye configurations

Two other SRD’s, denoted by R2 and R3 are non-Schurian. They are obtained as so-called algebraic mates of R1 (the concept to be discussed in further lectures). This is why we call them pseudo-Reye SRD’s.

M. Klin (BGU) Conﬁgurations with two ﬁbers 59 / 62 There is an ongoing challenge to describe these objects without essential dependence on the use of computers.

M. Klin (BGU) Conﬁgurations with two ﬁbers 60 / 62 References

1 M. Klin, S. Reichard: Construction of Small Strongly Regular Designs. Trudy Inst. Mat. i Mekh. UrO RAN, 2013, Volume 19, Number 3, Pages 164–178. www.mathnet.ru/eng/timm974

M. Klin (BGU) Conﬁgurations with two ﬁbers 61 / 62 Thank you!

M. Klin (BGU) Conﬁgurations with two ﬁbers 62 / 62