Search for systems of linked symmetric 2 (36 , 15 , 6) designs - PowerPoint PPT Presentation

Search for systems of linked symmetric 2 − (36 , 15 , 6) designs Matan Ziv-Av Ben-Gurion University of the Negev AGT16, Plzeˇ n, October 2016.

Outline Symmetric 2-designs 1 Systems of linked symmetric designs 2 Parameters (36 , 15 , 6) 3 Concluding remarks 4 1

Axioms A symmetric 2-design with parameters ( v , k , λ ) (also called a square 2-design) is an incidence structure ( B , P , I ) such that |P| = v ; 1 |B| = v ; 2 Each block is incident with exactly k points. 3 Each point is incident with exactly k blocks. 4 Each pair of blocks is incident with exactly λ points. 5 Each pair of points is incident with exactly λ blocks. 6 Some necessary conditions on the parameters are: λ ( v − 1) = k ( k − 1). Bruck-Ryser-Chowla: if v is even, then k − λ is a square; if v is odd then the equation x 2 − ( k − λ ) y 2 − ( − 1) 2 λ z 2 = 0 has non zero integer solutions. v − 1 2

Related designs The dual of a symmetric 2-design is a symmetric 2-design with the same parameters. The complement of a symmetric 2-( v , k , λ ) design (taking the complement of the incidence relation) is a symmetric 2-( v , v − k , v − 2 k + λ ) design. 3

Projective planes When λ = 1 we get the axioms of a projective plane. v = ( k + 1) 2 + ( k + 1) + 1. Exists when k − 1 is a prime power. Does not exist for k − 1 = 6 and for k − 1 = 10. 4

Biplanes When λ = 2 we get a biplane. v = 1 + k ( k − 1) . 2 The order of a biplane is k − 2. For k = 2 , 3 , 4 , 5, there exists a unique biplane. For k = 6 ( v = 16) there are 3 biplanes. For k = 9 there are 4 biplanes. For k = 11, there are 5 biplanes. For k = 13, there are 2 known biplanes. 5

Biplane of order 2 k = 4, v = 1 + k ( k − 1) = 7. 2 Parameters: (7 , 4 , 2). Complement design (7 , 3 , 1). Fano plane: B = {{ 1 , 2 , 4 } , { 2 , 3 , 5 } , { 3 , 4 , 6 } , { 4 , 5 , 7 } , { 1 , 5 , 6 } , { 2 , 6 , 7 } , { 1 , 3 , 7 }} Thus the unique (up to isomorphism) biplane of order 2 is B = {{ 3 , 5 , 6 , 7 } , { 1 , 4 , 6 , 7 } , { 1 , 2 , 5 , 7 } , { 1 , 2 , 3 , 6 } , { 2 , 3 , 4 , 7 } , { 1 , 3 , 4 , 5 } , { 2 , 4 , 5 , 6 }} 6

Incidence matrix With each symmetric 2-design we associate its incidence matrix. This is a v × v (0,1)-matrix with rows denoted by points and columns by blocks. The ( i , j ) entry is 1 if point i is incident to block j and 0 otherwise. The following holds for A , an incidence matrix of a symmetric 2-design: AA T = kI + λ ( J − I ) 7

Polarities A polarity of a symmetric design ( P , B , I ) is a bijection σ : B → P , such that x I σ ( y ) if and only if y I σ ( x ). It is easy to see that a polarity can be interpreted as a permutation of the columns of the incidence matrix such that the permuted matrix is symmetric. In this case we can rewrite the matrix equation: A 2 = kI + λ ( J − I ) This is the matrix equation of a ( v , k , λ, λ )-SRG. 8

Biplane of order 2 0 1 1 1 0 1 0   0 0 1 1 1 0 1 1 0 0 1 1 1 0 B = {{ 3 , 5 , 6 , 7 } , { 1 , 4 , 6 , 7 } , { 1 , 2 , 5 , 7 } , A =  0 1 0 0 1 1 1   1 0 1 0 0 1 1  { 1 , 2 , 3 , 6 } , { 2 , 3 , 4 , 7 } , { 1 , 3 , 4 , 5 } , { 2 , 4 , 5 , 6 }} 1 1 0 1 0 0 1 1 1 1 0 1 0 0 σ = (2 , 6 , 5)(4 , 7), 0 0 1 0 1 1 1   0 1 1 1 0 0 1 A ′ = 1 1 0 0 1 0 1  0 1 0 1 1 1 0   1 0 1 1 1 0 0  1 0 0 1 0 1 1 1 1 1 0 0 1 0 Also: σ = (1 , 2)(3 , 5 , 7 , 4 , 6), 1 0 0 1 0 1 1   0 0 1 0 1 1 1 0 1 1 1 0 0 1 A ′′ =  1 0 1 1 1 0 0   0 1 0 1 1 1 0  1 1 0 0 1 0 1 1 1 1 0 0 1 0 Those are not the matrices of a simple graph. 9

Absolute points If σ is a polarity of a design D = ( P , B , I ), then p ∈ P is an absolute point of σ if p ∈ σ ( p ). In the incidence matrix: if the i-th element of the diagonal is 1. Back to the biplane of order 2: 1 0 0 1 0 1 1   0 0 1 0 1 1 1 A ′′ = 0 1 1 1 0 0 1  1 0 1 1 1 0 0   0 1 0 1 1 1 0  1 1 0 0 1 0 1 1 1 1 0 0 1 0 This polarity has 4 absolute points. 10

SRG ( v , k , λ, λ ) If D = ( P , B , I ) is a design and σ its polarity with no absolute points, then its incidence matrix A is the adjacency matrix of a SRG with parameters ( v , k , λ, λ ). Conversely, if Γ = ( V , E ) is a ( v , k , λ, λ ), then ( V , { Γ( v ) | v ∈ V } , ∈ ) is a symmetric 2 − ( v , k , λ ) design. In matrices language: the adjacency matrix of Γ is an incidence matrix of a design. Isomorphism is not preserved when going from a design to a graph. Two non-isomorphic ( v , k , λ, λ ) SRGs may correspond to the same design. There are three (16 , 6 , 2) designs (biplanes of order 4), but only one of them admits a polarity with no absolute points. There are two (16 , 6 , 2 , 2) SRGs (Shrikhande graph and L 2 (4)), so they both generate the same biplane. 11

SRG ( v , k − 1 , λ − 2 , λ ) If a ( v , k , λ ) design has a polarity with v absolute points, then A − I is an adjacency matrix of a graph. Recall A 2 = kI + λ ( J − I ), so ( A − I ) 2 = ( k − 1) I + ( λ − 2) A + λ ( J − I − A ) . That is, A − I is the adjacency matrix of a ( v , k − 1 , λ − 2 , λ ) SRG. Conversely, if Γ = ( V , E ) is a ( v , k − 1 , λ − 2 , λ ), then ( V , { v ∪ Γ( v ) | v ∈ V } , ∈ ) is a symmetric 2 − ( v , k , λ ) design. In matrices language: A (Γ) + I is an incidence matrix of a design. Again, two non-isomorphic graphs may correspond to the same design. The smallest example has parameters (35 , 16 , 6 , 8). 12

Symmetric 2-designs as association schemes Let ( P , B , I ) be a symmetric ( v , k , λ ) 2-design. Let us take Ω = P ∪ B . R 0 = ∆ Ω . Let R 1 be the relation on Ω containing pairs of distinct elements of the same type. R 2 = I is the incidence relation. R 3 is the remaining pairs. Then ( R 0 , R 1 , R 2 , R 3 ) is a symmetric association scheme. Let us denote the corresponding adjacency matrices by I , A 1 , A 2 , A 3 . A 2 1 = ( v − 1) I + ( v − 2) A 1 , A 1 A 2 = ( k − 1) A 2 + kA 3 , A 1 A 3 = ( v − k ) A 2 + ( v − k − 1) A 3 , A 2 2 = kI + λ A 1 , A 2 A 3 = ( k − λ ) A 1 , A 2 3 = ( v − k ) I + ( v − 2 k − λ ) A 1 . 13

Symmetric 2-designs as association schemes Conversely, an imprimitive symmetric association scheme of rank 4 and order 2 v having one equivalence relation with two equivalence classes of the same size corresponds to a symmetric 2-design. Indeed if we take take P to be one equivalence class and B to be the other, I to be R 2 , then: there are v points and v blocks. Each block is incident to p 0 22 points, and vice versa; Every two blocks are incident to exactly p 1 22 common points, and vice versa. 14

Generalization We want to generalize symmetric designs. We look for rank 4 symmetric association schemes with R 1 an equivalence relation with f equivalence classes of the same size. We need an extra requirement: For any two equivalence classes A , B , restriction of the association scheme to A ∪ B results in a rank 4 association scheme with the same parameters. 15

Systems of linked symmetric designs (Ω 1 , . . . , Ω f , I ) is a homogeneous system of linked symmetric designs with parameters ( v , k , λ ) if: Each pair (Ω i , Ω j , I ) is a symmetric design with parameters ( v , k , λ ). For every distinct i , j , l and for any a ∈ Ω i , b ∈ Ω j , the number of c ∈ Ω l incident to both a and b depends only on whether a and b are incident. For f = 2 we get a single symmetric design. The multiplication table becomes: A 2 1 = ( v − 1) I + ( v − 2) A 1 , A 1 A 2 = ( k − 1) A 2 + kA 3 , A 1 A 3 = ( v − k ) A 2 + ( v − k − 1) A 3 , A 2 2 = ( f − 1) kI + ( f − 1) λ A 1 + ( f − 2) x 2 A 2 + ( f − 2) x 1 A 3 , A 2 A 3 = ( f − 1)( k − λ ) A 1 + ( f − 2)( k − x 2 ) A 2 + ( f − 2)( k − x 1 ) A 3 , A 2 3 = ( f − 1)( v − k ) I +( f − 1)( v − 2 k − λ ) A 1 +( f − 2)( v − 2 k + x 2 ) A 2 +( f − 2)( v − 2 k + x 1 ) A 3 . Here x 1 is the number of c when a , b are incident and x 2 is the number of c when they are not incident. 16

History of systems of linked symmetric designs Systems of linked symmetric designs were defined by Cameron (1972) for the investigation of groups with more than two pairwise non-equivalent representations as doubly transitive permutation groups. Cameron and Seidel (1973) constructed an infinite series of systems of linked designs with parameters (2 2 t +2 , 2 2 t +1 − 2 t , 2 2 t − 2 t ) ( f = 2 2 t +1 − 1) Noda (1974) added a few constraints on the parameters of such systems. Among them: k − λ = u 2 , x 1 = k ( k + u ) , x 2 = x 1 − u . v Mathon (1981) used the above results and using the method of admissible sets constructed all linked symmetric designs with parameters (16 , 6 , 2). Davis, Martin and Polhill (2013) described a new series of systems of linked designs with the same parameters. The authors suggest that there is evidence that the systems in this series are not isomorphic, but prove this only for the case t = 1. 17

Non power of 4 - smallest feasible parameters All known systems of linked designs have order power of 4: The infinite series by Cameron, Seidel and by Davis, Martin, Polhill. The 19 systems with f = 3 , 4 , 5 , 6 , 7 and parameters (16 , 6 , 2) discovered by Mathon. The smallest feasible sets of parameters for which the existence of a system of linked designs is unknown is: (36 , 15 , 6) (45 , 12 , 3). 18