Introduction to Association Schemes

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1 Introduction to Association Schemes Akihiro Munemasa Tohoku University June 5 6, 24 Algebraic Combinatorics Summer School, Sendai Assumed results (i) Vandermonde determinant: a a m =. a m a m m i<j m (a j a i ) if a,..., a m are distinct. (ii) If A,..., A n Mat n (R) are pairwise commutative symmetric matrices, then there exists an orthogonal matrix T Mat n (R) such that T A i T is diagonal for all i. (iii) If E Mat n (R) is positive semidefinite symmetric matrix, then there exists an n rank(e) matrix F such that E = F F. J stands for a matrix all of whose entries are. For a positive integer m, denote by [m] the set {, 2,..., m}. Unless otherwise noted, X will denote a finite set with X = n. 2 Graphs and their adjacency matrices Let Γ = (X, E) be a undirected simple graph. The adjacency matrix A of Γ is defined as { if {x, y} E, (A) xy = otherwise. Then (A 2 ) xy = z X(A) xz (A) zy = # path of length 2 from x to y { deg x if x = y, = # common neighbors otherwise

2 (x, y) = distance between x and y. The i-th distance matrix is defined as For the 3-cube, (A i ) xy = δ i, (x,y). A 2 = 3I + A + 2A 2 + A 3 = A 2 I, A, A 2, A 2 I, A, A 2, A 2 / I, A, AA 2 = I + 2A + A 2 + 3A 3 = A 3 I, A, A 2, A 3, By induction A n I, A, A 2, A 3 for all n N. Thus A 3 I, A, A 2, A 3, A 3 / I, A, A 2, AA 3 = A 2 = A 4 I, A, A 2, A 3 I, A, A 2, A 3. R[A] = I, A, A 2, A 3. () 2 where R[A] is the subalgebra of the full matrix algebra Mat X (R) generated by A. Observe that R[A] is an algebra with respect to matrix multiplication, while I, A, A 2, A 3 is in general just a vector subspace. However, by considering the entrywise product, I, A, A 2, A 3 is also an algebra. This implies that () is closed under both multiplications. 3 Coherent algebras lem: Let F be a field. An algebra A over F is a commutative ring with which is also an F-vector space, satisfying come compatibility axioms. An example is a subring of Mat n (F) which is also closed under scalar multiplications. A simpler example is the vector space F m with entrywise multiplication. Its identity as a ring is (,..., ) F m. Suppose that A F m is an algebra over F. Using the Vandermonde determinant, we find F m A a, i j, a i a j = A = F m. The same proof shows F m A a, i [m], e j A. (2) j [m] a j =a i Lemma. Let A F m be a subalgebra. Define i j a A, a i = a j, and denote by I,... I d its equivalence classes. Then A = i I e i,..., i I d e i. Proof. We may assume I, and it suffices to show i I e i A. For b A, set I(b) = {i [m] b i = b }. Then by (2), e i A. i I(b) 2

3 For any k [m] \ I, there exists b (k) A such that k / I(b (k) ). Then I(b (k) ) = I, so e i = k [m]\i i I k [m]\i i I(b (k) ) e i A. Definition 2. A coherent algebra A is a subalgebra of Mat n (C) containing J, closed under the entrywise product and transposition. Definition 3. A coherent configuration is a pair (X, {R i } d i= ) where {R i} d i= is a partition of X X such that (i) {(x, x) x X} is a union of some R i s, (ii) For each i {,,..., d}, t R i = R i for some i {,,..., d}, where t R i = {(x, y) X X (y, x) R i }. (iii) For h, i, j {,,..., d}, the number of z X such that (x, z) R i and (z, y) R j is constant whenever (x, y) R h. This constant is denoted by p h ij. These constants are called intersection numbers. The adjacency matrices of the relations of a coherent configuration form a basis of a coherent algebra, and every coherent algebra arises in this way. partition of X X J A (i) I A (ii) closed under transposition (iii) closed under multiplication, A i A j = d p h ija h. Definition 4. If a coherent configuration (X, {R i } d i= ) satisfies the additional property (iv) R = {(x, x) x X} then it is called an association scheme. If an association scheme satisfies the additional property (v) p h ij = ph ji for all h, i, j {,,..., d}, then it is called commutative. If it satisfies the additional property (vi) t R i = R i for all i {,,..., d} then it is called symmetric. A symmetric association scheme is commutative. For simplicity, we consider symmetric association schemes in what follows. 4 Primitive idempotents Let (X, {R i } d i= ) be a symmetric association scheme. The coherent algebra A spanned by its adjacency matrices is called the Bose Mesner algebra. Since the adjacency matrices are pairwise commutative symmetric matrices, there exists an orthogonal matrix T such that 3

4 T AT is a subalgebra of diagonal matrices, that is, T AT R n. By Lemma, T AT has a basis of the form ,..., etc Call these matrices D i. Then D i D j = δ ij D i. Define E i = T D i T A. Then E i E j = δ ij E i. Since A = A, A,..., A d has dimension d +, we have dim T AT = d +, so there are d + E i s. These E i s are called the primitive idempotents. Since J A, writing J = d i= c ie i, squaring both sides gives c i {, }. Comparing rank shows n J = E i for some i, so we may assume n J = E without loss of generality. We have To be complete, we need A i A j = δ ij A i, d A i A j = p h ija h, E i E j = δ ij E i. E i E j = n d qije h h. The coefficients q h ij are called Krein parameters. It is known that qh ij. The nonsingular matrix P defined by (A, A,..., A d ) = (E, E,..., E d )P is called the first eigenmatrix, and Q = np is called the second eigenmatrix. Example 5. A j = E j = n d P ij E i, i= d Q ij A i. i= A = I, A =, A 2 =, 4

5 E = 4, E = 2, E 2 = 4. Moreover, A 2 = 2A + 2A 2, A A 2 = A, A 2 2 = A, E E = 4 (2E + 2E 2 ), E E 2 = 4 E, E 2 E 2 = 4 E. 5 Orthogonality relations Write 2 P = Q =. 2 k j = P j that is, A j J = k j J, m j = rank E j = tr E j = d n tr Q ij A i = Q j. i= Then Q hj k h = Q hj n tr A2 h = d n tr Q ij A h A i i= = tr A h E j d = tr P ih E i E j i= = tr P jh E j = P jh m j. This implies the orthogonality relations: δ ij n = (P Q) ij = δ ij n = (QP ) ij = d P ih Q hj = m i d Q ih P hj = k i d Q hi Q hj k h, d P hi P hj m h 5

6 6 Examples (i) Complete graphs. A = I, J I. (ii) Polygons. A = C + C, where C =... { I, C + C, C 2 + (C ) 2,..., C m + (C ) m A = I, C + C, C 2 + (C ) 2,..., C m (2m + )-gon, 2m-gon (iii) Let G be a finite group of order n. Define (A g ) xy = δ x,gy. Then A g A h = A gh. A = A g g G is commutative if and only if G is abelian, A is symmetric if and only if g 2 = for all g G. If H is a subgroup of Aut G, with orbits S = {}, S, S d, then A = g S i A g i d is also a coherent algebra, defining an association scheme. (iv) Let G be a transitive permutation group. Let {R i } d i= be the G-orbits on X X. Then (X, {R i } d i= ) is an association scheme. (v) Let F be a finite set with q 2 elements, and set X = F d. Then X is a metric space with respect to the Hamming distance d H (x, y) = {i i [d], x i y i }. Then (X, {R i } d i= ) is a symmetric association scheme called the Hamming scheme H(d, q), where R i = {(x, y) X X d H (x, y) = i}. (vi) Let Ω be a finite set with v elements, and let X be the collection of all d-element subsets of Ω. Define R i = {(x, y) X X x y = d i}, (i =,,..., d). Then (X, {R i } d i= ) is a symmetric association scheme called the Johnson scheme J(v, d). (vii) Let Γ be a connected regular graph of diameter 2. If there are integers λ, µ such that any adjacent (resp. non-adjacent) pair of vertices have λ (resp. µ) common neighbors, then Γ is called a strongly regular graph. Let A be the adjacency matrix. Then one obtains an association scheme with Bose Mesner algebra A = I, A, J I A. (viii) Let (P, B) be a quasi-symmetric 2-design, that is, in addition to being a 2-design, we assume that two distinct blocks intersect with x or y points, where x and y are distinct integers. Then X = B naturally carries a structure of a strongly regular graph. (ix) A Steiner triple system is a 2-(v, 3, ) design. any pair of distinct blocks intersect with or points, so it is a quasi-symmetric design, and hence carries a structure of a strongly regular graph. 6