INTRODUCTION TO ASSOCIATION SCHEMES J.J. Seidel Fakulteit Wiskunde en Informatica Techn. Univ. Eindhoven NL-5644 HK Eindhoven The Netherlands Abstract The present paper gives an introduction to the theory of association schemes, fol- lowing Bose-Mesner (1959), Biggs (1974), Delsarte (1973), Bannai-Ito (1984) and Brouwer-Cohen-Neumaier (1989). Apart from definitions and many examples, also several proofs and some problems are included. The paragraphs have the following titles: 1. Introduction 5. Representations 2. Distance regular graphs 6. Root lattices 3. Minimal idempotents 7. Generalizations 4. -modules 8. References A 1. Introduction x An ordinary graph on n vertices (symmetric relation Γ on an n-set Ω) is described by its symmetric n n adjacency matrix A. We paint the edges of the complete × graph on n vertices in s colours: J I = A1 + A2 + ::: + As ; − and require that the vector space = A0 = I;A1;A2; :::; As R A h i 1 is a symmetric algebra w.r.t. matrix multiplication, that is, s A A = A A = ak A ; i; j = 0; 1; :::; s : i j j i X ij k k=0 We call this algebra the Bose-Mesner algebra of the s-association scheme (Ω, id,Γ1; Γ2; f :::; Γs ), where colour i corresponds to relation (graph) Γi and adjacency matrix g k 0 Ai. The intersection numbers aij and the valencies vi = aii have the following interpretation: w Γk w' ◦@ ◦ Γi w ]vi Γi @@ Γj ◦ k ]aij These notions go back to Bose and Mesner (1959). Example 1. A strongly regular graph is a 2-association scheme, where A1 and A2 denote the adjacency matrices of the graph and its complement. In the next example we use the distance @(u; v) of the vertices u and v of a graph, and the relations Γi, defined by u; v Γi iff @(u; v) = i, for i = 0; 1; :::; d = f g 2 diameter. Example 2. The hexagon ◦6 2◦T T T T 1 4 ◦T ◦ T T T 5 3 ◦ ◦ gives rise to a 3-association scheme, since the distance i matrices Ai read: 0 J I J I 0 0 I A1 = " − # ;A2 = " − # ;A3 = " # : J I 0 0 J I I 0 − − 2 Problem. Prove that the distance relations in the cube graph form a 3-association scheme. Determine the valencies and the intersection numbers. Example 3 Hamming scheme H(v; F2). v Consider Ω := (F2) with Hamming distance @H (x; y), that is, the number of coordinates in which x and y Ω differ. Denote by Γi the relation 2 x; y Γi iff @H (x; y) = i : f g 2 Then we have a v-association scheme with v v k k v k n = 2 ; vi = ; a = − : i ij 1 (i j + k) 1 (i + j k) 2 − 2 − Example 4 Johnson scheme J(v; k). Take Ω the set of all k-subsets of a v-set, and w; w0 Γi iff w w0 = k i. f g 2 j \ j − Then v k v k n = ; vi = − : k i i In an association scheme (Ω; Γi ) we will be interested in special subsets X Ω, f g ⊂ for instance: { blue cliques X : only blue edges in X, { blue cocliques X : no blue edges in X, { code X at min. distance δ : no Γ1; Γ2; :::; Γδ 1 in X, − { few-distance sets X in Rd, etc., etc. The problem then will be to find bounds for the cardinality X of the special sub- j j sets X Ω, and to investigate the case of equality. ⊂ 2. Distance-regular graphs x In a graph Γ = (Ω;E) of diameter d we define: distance @(u; v) = length of shortest path between u; v Ω ; 2 Γi(u) := x Ω: @(x; u) = i ; Γi(u) =: ki : f 2 g j j 3 Definition. A graph Γ is distance regular if for all u Ω, for i = 0; 1; 2; :::; d , 2 each v Γi(u) has ci neighbours in Γi 1(u), 2 − has bi neighbours in Γi+1(u), has ai neighbours in Γi(u). λ a2 ai 1 ai ai+1 ad u 1 b1 c2 b2 ci 1 − bi 1 ci bi ci+1 bi+1 cd − − :::::: k k2 ki 1 ki ki+1 kd ◦ − Γ1(u)Γ2(u)Γi 1(u)Γi(u)Γi+1(u)Γd(u) − Then ai + bi + ci = k ; ki+1ci+1 = kibi ; b0 = k ; c1 = 1 ; a1 = λ : So the independent parameters are k = b0; b1; b2; :::; bd 1 ; 1 = c1; c2; :::; cd : f − g It is convenient to arrange the parameters into the (d + 1) (d + 1) tridiagonal × matrix T : 0 k 2 3 c1 a1 b1 6 7 6 c a b 7 6 2 2 2 7 6 .. 7 6 c3 a3 . 7 6 7 T := 6 .. 7 : 6 c4 . 7 6 7 6 .. 7 6 . 7 6 7 6 ... 7 6 bd 1 7 6 − 7 4 cd ad 5 The definition of distance-regularity translates in terms of the n n distance i × matrices Ai, which are defined by Ai(x; y) = 1 if @(x; y) = i ; = 0 otherwise: (So A1 = A; A0 = I:) Theorem. Γ is distance regular iff, for 1 i d 1, ≤ ≤ − AAi = bi 1Ai 1 + aiAi + ci+1Ai+1 : − − 4 Proof. (AAi)(x; y) = # z Ω: @(x; z) = 1;@(y; z) = i : f 2 g There are such z only if @(x; y) = i 1; i; i + 1 , and their numbers are bi 1, ai, − − ci+1, respectively. Corollary. In a distance regular graph the distance i matrices Ai are polynomials pi of degree i in the adjacency matrix A, for i = 0; 1; :::; d. Proof. By recursive application of the theorem. Corollary. For a distance regular graph of diameter d, the distance i relations constitute a d-association scheme. 2 d Proof. Conversely to Ai = pi(A), deg pi = i, the powers I; A; A ; :::; A are linear combinations of A0;A1; :::; Ad. This implies that A0 = I;A1 = A; A2; :::; Ad R is a h i Bose-Mesner algebra. Example. The distance 1 relation in the Hamming scheme H(d; F2) defines a distance regular d graph. The vertices are the vectors of F2, two vertices being adjacent whenever they differ in one coordinate. Hence d k = d ; ci = i ; bi = d i ; ki = : − i Problem. Find the parameters bi and ci for the distance regular graph formed by the d-subsets of an n-set, n 2d, adjacency whenever two d-subsets differ in one element. ≥ The tridiagonal matrix T , of size d + 1, is useful for eigenvalues. Lemma. The eigenvalues of A are those of T (not counting multiplicities). Proof. Let λ be an eigenvalue of A. Then Ai = pi(A) has the eigenvalue pi(λ). 5 The theorem implies λpi(λ) = bi 1pi 1(λ) + aipi(λ) + ci+1pi+1(λ) : − − But this reads t T p(λ) = λp(λ) ; for p(λ) := p0(λ); p1(λ); :::; pd(λ) : and λ is an eigenvalue of T t, hence of T . There are d + 1 distinct eigenvalues of A, hence of T . Although T and T t have the same eigenvalues, they do not have the same eigenvec- tors. We shall denote by u(#) the eigenvector of T corresponding to the eigenvalue #: t T p(λ) = λp(λ); T u(#) = #u(#); u0 = p0 = 1 ; hence ciui 1 + aiui + biui+1 = #ui ; i = 1; :::; d 1 : − − Lemma. u(#); p(λ) = 0 ; for # = λ : 6 Proof. t #u(#); p(λ) = T u(#); p(λ) = u(#);T p(λ) = λu(#); p(λ) : Theorem. Let the adjacency matrix A of a distance regular graph have the eigenvalue # of multiplicity f. Let the tridiagonal T have eigenvector u(#). Then f L := (I + u A + u A + ::: + u A ) n 1 1 2 2 d d is an idempotent matrix of rank f. Proof. If λ is any other eigenvalue of A, then the corresponding eigenvalue of L equals f d f u (#)p (λ) = u(#); p(λ) = δ : n X i i n ϑ,λ i=0 Indeed, the lemma gives 0 for λ = #. For λ = # the corresponding eigenvalue of L, 6 which also has multiplicity f, equals 1, since trace L = f. 6 Remark. The theory in this section goes back to Biggs (1974). By the present theorem a distance regular graph may be viewed as a set of vectors at equal length in Rf , at cosines ui. For certain classes of DRG this paves the way to characterization, by use of root lattices, cf. BCN (1989) and 6. x 3. Minimal idempotents x We return to the general case of an association scheme with Bose-Mesner algebra = A0 = I;A1;A2; :::; As R : A h i The commuting Ai are simultaneously diagonalizable, hence there exists a basis of minimal orthogonal idempotents: 1 = E0 = J; E1; :::; Es : A D n ER Example. s = 2, spec A = (k1; rf ; sg). 1 k s E1 = A sI − J , of rank f , r s − − n − 1 k r E2 = rI A + − J , of rank g . r s − n − The algebra is closed with respect to matrix multiplication. It is also closed with A respect to Schur (= entry-wise) multiplication with idempotents A0;A1; :::; As. We have: 7 Matrix multiplication , Schur multiplication · ◦ EiEj = δijEi , Ai Aj = δijAi s ◦ s k k AiAj = a Ak , Ei Ej = b Ek X ij ◦ X ij k=0 k=0 intersection numbers ak N , Krein parameters bk 0 ij 2 ij ≥ Transition between the two bases of : A s 1 s A = p E , E = q A k X ik i i n X ki k i=1 k=0 1 AkEi = pikEi , Ei Ak = qkiAk ◦ n valency vk = pok , multiplicity fi = qoi ∆v := diag(vk) , ∆f := diag(fi) P = [pik], the character table , Q from PQ = nI = QP : Theorem.