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Coherent Configurations with Two Fibers

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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 = fA; B; C; Dg: Two more points E, F are not collinear with any two elements of Q. M. Klin (BGU) Configurations with two fibers 7 / 62 fA; B; C; D; E; F g 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 = f(y; x)j(x; y) 2 F g. 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).
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