Association schemes with multiple Q -polynomial structures Jianmin - PowerPoint PPT Presentation

Association schemes with multiple Q -polynomial structures Jianmin Ma Hebei Normal University Shanghai Jiaotong University 9/27/2014

� � Association Scheme Let R 0 , R 1 , . . . , R d be binary relations on set X : (a) R 0 = { ( x , x ) | x ∈ X } (b) R 0 ∪ R 1 ∪ · · · ∪ R d = X × X , R i ∩ R j = ∅ (c) R T = R j for some j (0 ≤ j ≤ d ) i (d) a △ counting for p k ij z j i � y x k ◮ Commutative AS ⇔ p i jk = p i kj . ◮ Symmetric AS ⇔ R i = R T for all i . i

Examples: Polygon, Hamming ◮ Polygons: ◮ Hamming: H(d,q) on Q d with alphabets Q of order q . ( x , y ) ∈ R i if x and y differ in i positions ◮ Why Hamming? Error-correcting Codes ◮ Hamming codes ◮ Golay codes ◮ Reed-Muller codes, . . .

Johnson J ( d , n ) Johnson: J ( d , q ) on d -subsets of Q . ( S , T ) ∈ R i if | S ∩ T | = d − i Why Johnson? Designs on subsets

Examples: finite groups ◮ (Transitive group) Association schemes are a combinatorial generalization of transitive permutation groups: if the group G is transitive on Ω, the orbits of G on Ω 2 form an association scheme. J(3,7): Group S 7 acts transitively on 3-subsets Ω of [1 .. 7]: R 1 = { ( A , B ) | | A ∩ B | = 2 } . Graph (Ω , R 1 ) has diameter 3: A i ∼ B ⇔ | A ∩ B | = 3 − i . ◮ (Group case) Let G be a finite group with conjugacy classes C 0 = { 1 } , C 1 , . . . , C d ( x , y ) ∈ R i if yx − 1 ∈ C i .

Definition of association scheme Symmetric association scheme: X = ( X , { R 0 , R 1 , . . . , R d } ). Adjacency matrices: A 0 , . . . , A d . A T A 0 + · · · + A d = J , A 0 = I , i = A i , d � p k A i A j = ij A k . k =0 Bose-Mesner algebra: A = C [ A 0 , . . . , A d ]. Note: A i ◦ A j = δ ij A i → A has a second basis: E 0 , . . . , E d , primitive idempotents. E 0 = | X | − 1 J , E T = E i , E 0 + · · · + E d = I , i k � E i ◦ E j = | X | − 1 q k ij E k . k =0 Eigenvalues/eigenmatrices: ( A 0 , A 1 , . . . , A d ) = ( E 0 , E 1 , . . . , E d ) P ( E 0 , E 1 , . . . , E d ) = | X | − 1 ( A 0 , A 1 , . . . , A d ) Q

Definition of association scheme Symmetric association scheme: X = ( X , { R 0 , R 1 , . . . , R d } ). Adjacency matrices: A 0 , . . . , A d . A T A 0 + · · · + A d = J , A 0 = I , i = A i , d � p k A i A j = ij A k . k =0 Bose-Mesner algebra: A = C [ A 0 , . . . , A d ]. Note: A i ◦ A j = δ ij A i → A has a second basis: E 0 , . . . , E d , primitive idempotents. E 0 = | X | − 1 J , E T = E i , E 0 + · · · + E d = I , i k � E i ◦ E j = | X | − 1 q k ij E k . k =0 Eigenvalues/eigenmatrices: ( A 0 , A 1 , . . . , A d ) = ( E 0 , E 1 , . . . , E d ) P ( E 0 , E 1 , . . . , E d ) = | X | − 1 ( A 0 , A 1 , . . . , A d ) Q

For a 3-cube: V = { 000 } ∪ { 001 , 010 , 100 } ∪ { 011 , 110 , 101 } ∩ { 111 } d ( x , y ) = Hammming distance A i = distance i matrix. � � Row 1 of A 1 = 0 1 1 1 0 0 0 0 A 2 = 3 I + 0 · A + 2 A 2 + 0 · A 3 ⇒ A 2 ∈ � I , A , A 2 � , A 2 ∈ � I , A , A 2 � , A 2 / ∈ � I , A � , AA 2 = 0 · I + 2 A + 0 · A 2 + 3 A 3 ⇒ A 3 ∈ � I , A , A 2 , A 3 � 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 � 1. Each A j is a polynomial in A of deg j 2. ( V , R 1 ) is a distance-regular graph

Polynomial structure, distance-regular graph Example J(3,7): Ω= 3-subsets Ω of [1 .. 7]: R 1 = { ( A , B ) | | A ∩ B | = 2 } . Graph Γ = (Ω , R 1 ) has diameter 3: A h ∼ B ⇔ | A ∩ B | = 3 − h . Take O 7 := (Ω , R 3 ), the Odd graph of diameter 3. Definition If there are polynomials v i of degree i such that A i = v i ( A 1 ), then A 0 , A 1 , . . . A d is called P-polynomial structure: i.e., A 1 A ℓ = c ℓ − 1 A ℓ − 1 + a ℓ A ℓ + b ℓ +1 A ℓ +1 . In this case, ( X , A 1 ) is a distance-regular graph. J(3,7) has two P -polynomial structures: (Ω , R 1 ) , (Ω , R 3 ).

Polynomial structure, distance-regular graph Example J(3,7): Ω= 3-subsets Ω of [1 .. 7]: R 1 = { ( A , B ) | | A ∩ B | = 2 } . Graph Γ = (Ω , R 1 ) has diameter 3: A h ∼ B ⇔ | A ∩ B | = 3 − h . Take O 7 := (Ω , R 3 ), the Odd graph of diameter 3. Definition If there are polynomials v i of degree i such that A i = v i ( A 1 ), then A 0 , A 1 , . . . A d is called P-polynomial structure: i.e., A 1 A ℓ = c ℓ − 1 A ℓ − 1 + a ℓ A ℓ + b ℓ +1 A ℓ +1 . In this case, ( X , A 1 ) is a distance-regular graph. J(3,7) has two P -polynomial structures: (Ω , R 1 ) , (Ω , R 3 ).

Q-polynomial structure Definition Suppose scheme X has primitive idempotents: E 0 , . . . , E d . If there are polynomials v ∗ i of degree i such that E i = v ∗ i ( E 1 ) with entry-wise product, then E 0 , E 1 , . . . E d is called Q-polynomial structure: i.e., E 1 ◦ E j = | X | − 1 ( c ∗ j − 1 E j − 1 + a ∗ j E j + b ∗ j +1 E j +1 ) . No combinatorial interpretation for Q -polynomial structure! There are association schemes with more than one such structures.

Q-polynomial structure Definition Suppose scheme X has primitive idempotents: E 0 , . . . , E d . If there are polynomials v ∗ i of degree i such that E i = v ∗ i ( E 1 ) with entry-wise product, then E 0 , E 1 , . . . E d is called Q-polynomial structure: i.e., E 1 ◦ E j = | X | − 1 ( c ∗ j − 1 E j − 1 + a ∗ j E j + b ∗ j +1 E j +1 ) . No combinatorial interpretation for Q -polynomial structure! There are association schemes with more than one such structures.

Main object Problem: Study association schemes with multiple Q - or P -polynomial structures. How many P -polynomial structure can a scheme have? ◮ If a scheme is from an ordinary n -gon, it has φ ( n ) P -polynomial structures, where φ ( n ) = |{ k | ( k , n ) = 1 , 1 ≤ k ≤ n }| . ◮ What about otherwise?

Multiple P -polynomial structures Ei. Bannai Et. Bannai Theorem (Bannai-Bannai, 1980) If A 0 , A 1 , . . . , A d is a P-polynomial structure for an association scheme X which is not an ordinary polygon, then any second such structure must be one of the following: (I) A 0 , A 2 , A 4 , A 6 , . . . , A 5 , A 3 , A 1 ; (II) A 0 , A d , A 1 , A d − 1 , A 1 , A 2 , A d − 2 , A 3 , A d − 3 , . . . ; (III) A 0 , A d , A 2 , A d − 2 , A 4 , A d − 4 , . . . , A d − 3 , A 3 , A d − 1 , A 1 ; (IV) A 0 , A d − 1 , A 2 , A d − 3 , A 4 , A d − 5 , . . . , A 3 , A d − 2 , A 1 , A d ; ( X , R ) admits at most two P-polynomial structures.

Multiple P- or Q- structures for DRG In the book of Bannai and Ito (1984), they proved for ( P and Q )-polynomial schemes X with Q -structure E 0 , E 1 , . . . , E d , and intersection numbers a i , b i , c i . If X is not a polygons, diameter d ≥ 34 and has another Q -structure E ′ 0 , E ′ 1 , . . . , E ′ d , then ◮ the possible form of E ′ 0 , E ′ 1 , . . . , E ′ d were determined; ◮ the possible form of a i , b i , c i in terms of Leonard’s theorem. ◮ X can have at most two P -polynomial structures. ◮ All eigenvalues are rational integers. They remarked similar assertions hold if X has instead another P -polynomial structure.

P-Polynomial structures In their 1980 paper, Ei. Bannai and Et. Bannai asked Q1. whether similar result holds for AS with multiple Q -polynomial structures? Q2. determine all association schemes with two P -structures; Q3. whether type III ( A 0 , A d , A 2 , . . . ) can hold for larger d ≥ 4? ◮ Hiroshi Suzuki answered Q3 in 1993: d ≤ 4 (published in 1996). ◮ He answer Q1 in 1996, published in 1998. See below. H. Suzuki

P-Polynomial structures In their 1980 paper, Ei. Bannai and Et. Bannai asked Q1. whether similar result holds for AS with multiple Q -polynomial structures? Q2. determine all association schemes with two P -structures; Q3. whether type III ( A 0 , A d , A 2 , . . . ) can hold for larger d ≥ 4? ◮ Hiroshi Suzuki answered Q3 in 1993: d ≤ 4 (published in 1996). ◮ He answer Q1 in 1996, published in 1998. See below. H. Suzuki

Dickie and Terwilliger’s attack on 2Q DRGs Theorem (Dickie 1995) Let Γ be a distance regular graph with diameter d ≥ 5 and valency k ≥ 3 . Then Γ has two Q-polynomial structures if and only if Γ is one of the following: (i) the cube H ( d , 2) with d even; (ii) the half cube 1 2 H (2 d + 1 , 2) ; (iii) the folded cube ˜ H (2 d + 1 , 2) ; (iv) the dual polar graph on [ 2 A 2 d − 1 ( q )] , where q ≥ 2 is a prime power. The Academy Award for “Scientific and Technical Achievement” For the primary design (Perry Kivolowitz) and for the development (Garth Dickie) of the algorithms, for the shape-driven warping and morphing subsystem of the Elastic Reality Special Effects System.

Dickie and Terwilliger’s attack on 2Q DRGs Theorem (Dickie 1995) Let Γ be a distance regular graph with diameter d ≥ 5 and valency k ≥ 3 . Then Γ has two Q-polynomial structures if and only if Γ is one of the following: (i) the cube H ( d , 2) with d even; (ii) the half cube 1 2 H (2 d + 1 , 2) ; (iii) the folded cube ˜ H (2 d + 1 , 2) ; (iv) the dual polar graph on [ 2 A 2 d − 1 ( q )] , where q ≥ 2 is a prime power. The Academy Award for “Scientific and Technical Achievement” For the primary design (Perry Kivolowitz) and for the development (Garth Dickie) of the algorithms, for the shape-driven warping and morphing subsystem of the Elastic Reality Special Effects System.