Product of P-polynomial association schemes

Transcription

1 Product of P-polynomial association schemes Ziqing Xiang Shanghai Jiao Tong University Nov 19, / 20

2 P-polynomial association scheme Definition 1 P-polynomial association scheme A = (X, {A i } 0 i < d+1 ). A i is X -by-x symmetric (0, 1) matrix. i A i = J. A i = p i (A 1 ), polynomials p i (x) = t i x i + o(x i ), t i > 0. 2 / 20

3 P-polynomial association scheme Definition 1 P-polynomial association scheme A = (X, {A i } 0 i < d+1 ). A i is X -by-x symmetric (0, 1) matrix. i A i = J. A i = p i (A 1 ), polynomials p i (x) = t i x i + o(x i ), t i > 0. Distance-regular graph G. V (G) = X. dist(x, y) = i, A i (x, y) = 1. 2 / 20

4 Layers x 0 X i-th layer X i = {x dist(x 0, x) = i}. i-th valency k i = X i. Shells form a partition of the ground set X. 3 / 20

5 Hamming scheme Binary Hamming scheme H v Ground set X = F v 2. dist(x, y) is Hamming distance between x and y. t i = 1 i!, p i(x) = 1 i! x i + o(x i ). Layer X i consists of elements with exactly i ones. 4 / 20

6 Johnson scheme Johnson scheme J v,k Ground set X = X Hv k. dist(x, y) is half of Hamming distance between x and y. t i = 1 (i!) 2, p i (x) = 1 (i!) 2 x i + o(x i ). 5 / 20

7 Design Definition 2 (X, µ) is a measure space. V is a vector space consisting of measurable functions on X. Y X is a V -design if there exists a measure ν on Y satisfying: f dµ = f dν for every f V. Some assumptions on measures are omitted. X Y 6 / 20

8 A block matrix A is a P-polynomial association scheme. A(X i, X j ) = A j i (X i, X j ) (a) A 0 (b) A 1 (c) A 2 (d) A 3 (e) A 4 (f) A 7 / 20

9 Relative design A is a P-polynomial association scheme. Pol t (X S ) is the row space of A( i t X i, X S ). Definition 3 A relative t-design on layers X S = s S X s is a subset Y of X S such that f (x)w 0 (x) = f (y)w 1 (y) x X S y Y for every f Pol t (X S ). Some assumptions on weights w 0 and w 1 are omitted. 8 / 20

10 Key property A is a P-polynomial association scheme. Let Ã(X i, X j ) = t 1 j i A(X i, X j ). Lemma 4 Ã(X i, X j )Ã(X j, X k ) = Ã(X i, X k ) Proof. Ã(X i, X j )Ã(X j, X k ) =A j i 1 (X i, X j )A k j 1 (X j, X k ) =A j i 1 (X i, X )A k j 1 (X, X k ) =A k i 1 (X i, X k ) =Ã(X i, X k ) 9 / 20

11 Figure : From 10 / 20

12 Cake Definition 5 A cake C consists of: sets X i, 0 i < d + 1, which are called layers, a block (0, 1) matrix C, a positive sequence t i, a modified block matrix C(X i, X j ) = t 1 j i C(X i, X j ), and C(X i, X j ) C(X j, X k ) = C(X i, X k ). All P-polynomial association schemes are cakes. 11 / 20

13 Product cake A, B are cakes. Definition 6 The product cake C = A B is constructed as follow. Xi C = Xi A Xi B. C(X i, X j ) = A(X i, X j ) B(X i, X j ), namely C = A B. t C i = t A i t B i : Khatri-Rao product of two block matrices. 12 / 20

14 Product cake A, B are cakes. Definition 6 The product cake C = A B is constructed as follow. Xi C = Xi A Xi B. C(X i, X j ) = A(X i, X j ) B(X i, X j ), namely C = A B. t C i = t A i t B i : Khatri-Rao product of two block matrices. Lemma 7 Johnson cake J v,k is the product of Hamming cakes H k and H v k. The product cake may not be an association scheme. 12 / 20

15 Cake design C is a cake. Pol t (X S ) is the row space of C( i t X i, X S ). Definition 8 A cake t-design on layers X S = s S X s is a subset Y of X S such that f (x)w 0 (x) = f (y)w 1 (y) x X S y Y for every f Pol t (X S ). Some assumptions on weights w 0 and w 1 are omitted. 13 / 20

16 Lower bound of sizes of cake designs One possible approach to establish the lower bound consists of three steps. Step 1: Cheesecake. Step 2: Pol i (X S )Pol j (X S ) Pol k (X S ). Step 3: C(X i, X i+1 ) has full row rank. Theorem 9 Y is a relative t-design of Johnson scheme on layers X S = s S X s, then under some assumption on elements in S, it holds Y k a i, where 2a t S i < S 14 / 20

17 Step 1: Cheesecake A matrix is almost strictly totally positive if every minor is nonnegative and the determinant of square submatrices with positive main diagonal is positive. Definition 10 A cake is a cheesecake if the Toeplitz matrix T i,j = t j i is almost strictly totally positive. 15 / 20

18 Step 1: Cheesecake A matrix is almost strictly totally positive if every minor is nonnegative and the determinant of square submatrices with positive main diagonal is positive. Definition 10 A cake is a cheesecake if the Toeplitz matrix T i,j = t j i is almost strictly totally positive. Lemma 11 Under some assumption, a cake is a cheesecake if and only if the generating function f (z) = t i z i i has only real zeros. 15 / 20

19 Step 1: Cheesecake Lemma 11 Under some assumption, a cake is a cheesecake if and only if the generating function f (z) = t i z i i has only real zeros. Corollary 12 Hamming cakes and Johnson cakes are cheesecakes. Problem 13 Classify P-polynomial association schemes which are cheesecakes. 16 / 20

20 Step 2: Pol i (X S ) A, B are cakes, and C = A B. Lemma 14 If C is a cheesecake, Pol A i (X S )Pol A j (X S ) Pol A k (X S), and Pol B i (X S )Pol B j (X S ) Pol B k (X S), then under some assumption on elements in S, Pol C i (X S )Pol C j (X S ) Pol C k+ S 1 (X S). The bound k + S 1 is sharp. 17 / 20

21 Step 2: Pol i (X S ) Lemma 15 For Hamming cake, it holds Pol i (X )Pol j (X ) Pol i+j (X ). Corollary 16 For Johnson cake, under some assumption on elements in S, it holds Pol i (X S )Pol j (X S ) Pol i+j+ S 1 (X S ). 18 / 20

22 Step 3: C(X i, X i+1 ) Lemma 17 If A(X i, X i+1 ) = X i and B(X i, X i+1 ) = X i, then for C = A B, C(X i, X i+1 ) = X i. Lemma 18 For Hamming cake H v, it holds H(X i, X i+1 ) = X i for 2i v. Corollary 19 For Johnson cake J v,k, it holds J(X i, X i+1 ) = X i for 2i min{k, v k}. 19 / 20

23 Thanks for your attention. Figure : From Ziqing Xiang ZiqingXiang@gmail.com Personal Homepage: 20 / 20