Tight relative 2-designs on 2 shells in Johnson scheme Yan Zhu, - PowerPoint PPT Presentation

Tight relative 2-designs on 2 shells in Johnson scheme Yan Zhu, Eiichi Bannai and Etsuko Bannai zhuyan870311@sina.com Shanghai Jiao Tong University March 15, 2014

Outline 1. Introduction 2. Main results 3. Construction of some examples 4. Future work 2 / 27

Association scheme X = a finite set, { R 0 , R 1 , . . . , R d } = the set of relations on X (i.e. R i ⊆ X × X ), � R 0 = { ( x , x ) | x ∈ X } . � R 1 � . . . � R d = X × X , and R i � R j = ∅ if i � = j . � R 0 � t R i = R j for some j ∈ { 0 , 1 , . . . , d } ,where t R i = { ( y , x ) | ( x , y ) ∈ R i } . � |{ z ∈ X | ( x , z ) ∈ R i , ( z , y ) ∈ R j }| = p k i , j is a constant whenever ( x , y ) ∈ R k . Then X = ( X , { R i } 0 ≤ i ≤ d ) is an association scheme. Moreover, it is symmetric if t R i = R i . 3 / 27

Adjacency matrix The i-th adjacency matrix A i of X is defined by � 1 , if ( x , y ) ∈ R i ( A i ) xy = otherwise 0 , � A 0 = I . � A 0 + A 1 + . . . + A d = J . � t A i = A j � A i A j = � d i = 0 p k i , j A k = A j A i . { A 0 , A 1 , . . . , A d } form an associative commutative algebra which is called the Bose-Mesner algebra of the association scheme. 4 / 27

Matrix version � Symmetric association scheme: X = ( X , { R i } i = 0 ,..., d ) . � Adjacency matrices: A 0 , . . . , A d . � Primitive idempotents: E 0 , . . . , E d . � Bose-Mesner algebra: C [ A 0 , . . . , A d ] = C [ E 0 , . . . , E d ] d d � � 1 p k q k A i A j = i , j A k and E i ◦ E j = i , j E k . | X | i = 0 i = 0 � Eigenmatrices: ( A 0 , . . . , A d ) = ( E 0 , . . . , E d ) P (1) 1 ( E 0 , . . . , E d ) = | X | ( A 0 , . . . , A d ) Q (2) d d � � 1 i.e. , A i = P i ( j ) E j and E i = Q i ( j ) A j | X | j = 0 j = 0 5 / 27

X = ( X , { R i } 0 ≤ i ≤ d ) is called a P-polynomial scheme with respect to the ordering A 0 , A 1 , . . . , A d , if there exist some polynomials v i ( x ) of degree i such that A i = v i ( A 1 ) . X = ( X , { R i } 0 ≤ i ≤ d ) is called a Q-polynomial scheme with respect to the ordering E 0 , E 1 , . . . , E d , if there exist some polynomials v ∗ i ( x ) of degree i such that E i = v ∗ i ( E 1 ) . Definition 1.1 Let V be a set of cardinality v and let d be a positive integer with d ≤ v 2 . Let X be the set of d-element subsets of V . Define R i by ( x , y ) ∈ R i if | x ∩ y | = d − i . Then X = ( X , { R i } 0 ≤ i ≤ d ) is a symmetric association scheme of class d and is called Johnson scheme J ( v , d ) . 6 / 27

Some notations X = ( X , { R i } 0 ≤ i ≤ d ) : a symmetric association scheme. u 0 ∈ X : a fixed point arbitrarily. X i = { x ∈ X | ( u 0 , x ) ∈ R i } , then X 0 , X 1 , . . . , X d are called shells of X . F ( X ) : the vector space consists of all the real valued functions on X . L j ( X ) : the subspace of F ( X ) spanned by all the columns of E j . F ( X ) = L 0 ( X ) ⊥ L 1 ( X ) ⊥ . . . ⊥ L d ( X ) . Denote m j = dim ( L j ( X )) = rank ( E j ) , k i = |{ y ∈ X | ( u 0 , y ) ∈ R i }| . 7 / 27

Definition 1.2 [1] Let ( Y , w ) be a weighted subset of X with positive function w on Y . ( Y , w ) is called a relative t-design with respect to u 0 if the following condition holds. p � � � W r i | X r i | f ( x ) = w ( y ) f ( y ) (3) i = 1 x ∈ X ri y ∈ Y for any function f ∈ L 0 ( X ) ⊥ L 1 ( X ) ⊥ . . . ⊥ L t ( X ) , where W r i = � y ∈ Y ri w ( y ) , i = 1 , 2 , . . . , p . � Y � = ∅ } and S = X r 1 � X r 2 � . . . � X r p , we Let { r 1 , r 2 , . . . , r p } = { r | X r say Y is supported by p shells. Denote Y r i = Y � X r i , i = 1 , 2 , . . . , p . 8 / 27

Theorem 1.3 [1] Let ( Y , w ) be a relative 2e-design of a Q-polynomial scheme. Then the following inequality holds. | Y | ≥ dim ( L 0 ( S ) + L 1 ( S ) + . . . + L e ( S )) , (4) where L j ( S ) = { f | S , f ∈ L j ( X ) } , j = 0 , 1 , . . . , e . Definition 1.4 If equality holds in (4), then ( Y , w ) is called a tight relative 2e-design with respect to u 0 . Theorem 1.5 [2] Let X = ( X , { R i } 0 ≤ i ≤ d ) be a Q-polynomial scheme. Let G be the automorphism group of X . Let ( Y , w ) be a tight relative 2 e-design with respect to u 0 . Assume that the stabilizer G u 0 of u 0 acts transitively on every shell X r , 1 ≤ r ≤ d . Then the weighted function w of any tight relative 2 e-design ( Y , w ) is constant on each Y r i ( 1 ≤ i ≤ p ) . 9 / 27

Formula for some parameters i � � j �� d − j �� v − d − j � (− 1 ) t P i ( j ) = . (5) i − t i − t t t = 0 Q j ( i ) = P i ( j ) m j (6) . k i � v � v � � � d �� v − d � m j = − and k i = (7) . j j − 1 i i We consider 2-designs on 2 shells, i.e., e = 1 and p = 2 . d � 1 1 E 0 = | X | J , E 1 = Q 1 ( j ) A j . | X | j = 0 10 / 27

Theorem 2.1 Take a sequence elements from X as u 0 = { 1 , 2 , . . . , d } , ( 1 ≤ i ≤ v − d − 1 ) u i = { 1 , 2 , . . . , d − 1 , d + i + 1 } , u i = { 1 , 2 , . . . , d , d + 1 } \ { i − ( v − d ) + 1 } , ( v − d ≤ i ≤ v − 1 ) i.e., u 1 = { 1 , 2 , . . . , d − 1 , d + 2 } = { 1 , 2 , . . . , d − 1 , d + 3 } u 2 . . . u v − d − 1 = { 1 , 2 , . . . , d − 1 , v } = { 2 , 3 , . . . , d − 1 , d , d + 1 } u v − d u v − d + 1 = { 1 , 3 , . . . , d − 1 , d , d + 1 } . . . = { 1 , 2 , . . . , d − 1 , d + 1 } u v − 1 Then { φ 0 | S , φ 1 | S , . . . , φ v − 1 | S } is a basis of L 0 ( S ) + L 1 ( S ) , where � X r 2 . S = X r 1 11 / 27

Some notations φ 0 ( x ) = φ ( 0 ) u 0 ( x ) = | X | E 0 ( x , u 0 ) ≡ 1 (8) φ i ( x ) = φ ( 1 ) (9) u i ( x ) = | X | E 1 ( x , u i ) � Inner product is defined by 2 � � W r i < f , g > = f ( x ) g ( x ) . (10) | X r i | i = 1 x ∈ X ri = < φ 0 , φ 0 >, d 0 1 ≤ i ≤ v − 1 c 0 = < φ i , φ i >, for = < φ 0 , φ i >, for 1 ≤ i ≤ v − 1 c 1 , 5 c 1 , 1 = < φ i , φ j >, for 1 ≤ i � = j ≤ v − d − 1 for v − d ≤ i � = j ≤ v − 2 c 1 , 2 = < φ i , φ j >, = < φ i , φ v − 1 >, for 1 ≤ i ≤ v − d − 1 c 1 , 3 v − d ≤ i ≤ v − 2 c 1 , 4 = < φ i , φ v − 1 >, for = < φ i , φ j >, for 1 ≤ i ≤ v − d − 1 , v − d ≤ j ≤ v − 2 c 2 c 1 , 1 = c 1 , 3 and c 1 , 2 = c 1 , 4 12 / 27

idea of the proof: � X r 2 . Denote U ∗ = U \ { u 0 } and S = X r 1 Let M be a submatrix of E 1 . M is indexed by S × U ∗ whose ( x , u i ) -entry is defined by for any ( x , u i ) ∈ S × U ∗ . ( M ) x , u i = | X | E 1 ( x , u i ) 2 � � W ri ( t MM )( u j , u k ) = φ j ( x ) φ k ( x ) . | X ri | i = 1 x ∈ X ri   · · · · · · c 0 c 1 , 1 c 1 , 1 c 2 c 2 c 2 c 1 , 1 c 1 , 1 c 0 · · · c 1 , 1 c 2 c 2 · · · c 2 c 1 , 1     . . . . . . . ...   . . . . . . . . . . . . . .      · · · · · ·  c 1 , 1 c 1 , 1 c 0 c 2 c 2 c 2 c 1 , 1   t MM =   c 2 c 2 · · · c 2 c 0 c 1 , 2 · · · c 1 , 2 c 1 , 2 (11)     · · · · · · c 2 c 2 c 2 c 1 , 2 c 0 c 1 , 2 c 1 , 2     . . . . . . . ...  . . . . . . .  . . . . . . .      c 2 c 2 · · · c 2 c 1 , 2 c 1 , 2 · · · c 0 c 1 , 2    · · · · · · c 1 , 1 c 1 , 1 c 1 , 1 c 1 , 2 c 1 , 2 c 1 , 2 c 0 13 / 27

Orthonormal basis Gram-Schmidt’s method: { φ 1 , . . . , φ v − 1 , φ 0 } − → { ϕ 1 , ϕ 2 , . . . , ϕ v } . ϕ 1 = φ 1 , c 0 � � < φ 1 , φ 1 > < φ 2 , φ 1 > . . . < φ i , φ 1 > � � � � < φ 1 , φ 2 > < φ 2 , φ 2 > . . . < φ i , φ 2 > � � � � 1 . . � � √ D i − 1 D i . . ϕ i = · · · · · · . . � � � � � � < φ 1 , φ i − 1 > < φ 2 , φ i − 1 > . . . < φ i , φ i − 1 > � � � � φ 1 φ 2 . . . φ i � � (12) The Gram determinant D i is given by � � < φ 1 , φ 1 > < φ 2 , φ 1 > . . . < φ i , φ 1 > � � � � < φ 1 , φ 2 > < φ 2 , φ 2 > . . . < φ i , φ 2 > � � � � . . � � . . D i = . · · · · · · . � � � � � � < φ 1 , φ i − 1 > < φ 2 , φ i − 1 > . . . < φ i , φ i − 1 > � � � � < φ 1 , φ i > < φ 2 , φ i > . . . < φ i , φ i > � � 14 / 27

Property of orthonormal basis Let H be a matrix whose rows are indexed by Y with v columns whose � ( y , i ) -entry is defined by w ( y ) ϕ i ( y ) . Then ( t HH ) i , j = δ i , j and ( H t H ) x , y = δ x , y imply  � w ( y ) ϕ i ( y ) ϕ j ( y ) = δ i , j    y ∈ Y v � w ( y ) ϕ i ( x ) ϕ i ( y ) = δ x , y    i = 1 � x ∈ X r , x = { 1 , 2 , . . . , d − r , d + 1 , d + 2 , . . . , d + r } � 1 ϕ 2 = i ( x ) . (13) w r x ∈ Y r 15 / 27