Non-commutative association schemes of rank 6 M. Muzychuk (joint - PowerPoint PPT Presentation

Non-commutative association schemes of rank 6 M. Muzychuk (joint work with A. Herman and B. Xu), Netanya Academic College, Israel , October, 2016, Pilsen, Czech Republick

Non-commutative association schemes of rank 6 1. Y. Asaba and A. Hanaki, A construction of integral standard generalized table algebras from parameters of projective geometries, Israel J. Math. , 194 , (2013), 395-408. 2. A. Hanaki and P.-H. Zieschang, on imprimitive noncommutative association schemes of order 6, Comm. Algebra , 42 (3), (2014), 1151-1199. 3. M. Yoshikawa, On noncommutative integral standard table algebras in dimension 6, Comm. Algebra , 42 (2014), 2046-2060. 4. B. Drabkin and C. French, On a class of noncommutative imprimitive association schemes of rank 6, Comm. Algebra , 43 (9), (2015), 4008-4041. 5. C. French and P.-H. Zieschang, On the normal structure of noncommutative association schemes of rank 6, Comm. Algebra , 44 (3), 2016, 1143-1170.

Notation

Notation If R , S ⊆ X 2 are binary relations on a finite set X , then 1 R ( x ) := { y ∈ X | ( x , y ) ∈ R } ; 2 R t := { ( x , y ) ∈ X 2 | ( y , x ) ∈ R } 3 RS is the relational product of R and S

Notation If R , S ⊆ X 2 are binary relations on a finite set X , then 1 R ( x ) := { y ∈ X | ( x , y ) ∈ R } ; 2 R t := { ( x , y ) ∈ X 2 | ( y , x ) ∈ R } 3 RS is the relational product of R and S If F is a field, then 1 M X ( F ) is the matrix algebra; 2 I X is the identity matrix; 3 J X is all one matrix; ⊤ is is matrix transposition; 4 5 Y is the characteristic vector of Y ⊆ X .

Association schemes

Association schemes Definition A pair X = ( X , R = { R 0 , ..., R d } ) is called an association scheme iff

Association schemes Definition A pair X = ( X , R = { R 0 , ..., R d } ) is called an association scheme iff 1 R is a partition of X 2 and R 0 = { ( x , x ) | x ∈ X } ;

Association schemes Definition A pair X = ( X , R = { R 0 , ..., R d } ) is called an association scheme iff 1 R is a partition of X 2 and R 0 = { ( x , x ) | x ∈ X } ; 2 ∀ i ∈{ 0 ,..., d } ∃ i ′ ∈{ 0 ,..., d } s.t. R t i = R i ′ ;

Association schemes Definition A pair X = ( X , R = { R 0 , ..., R d } ) is called an association scheme iff 1 R is a partition of X 2 and R 0 = { ( x , x ) | x ∈ X } ; 2 ∀ i ∈{ 0 ,..., d } ∃ i ′ ∈{ 0 ,..., d } s.t. R t i = R i ′ ; 3 for any triple i , j , k ∈ { 0 , ..., d } and any pair ( x , y ) ∈ R k the intersection number p k ij := | R i ( x ) ∩ R j ′ ( y ) | depends only on i , j , k .

Association schemes Definition A pair X = ( X , R = { R 0 , ..., R d } ) is called an association scheme iff 1 R is a partition of X 2 and R 0 = { ( x , x ) | x ∈ X } ; 2 ∀ i ∈{ 0 ,..., d } ∃ i ′ ∈{ 0 ,..., d } s.t. R t i = R i ′ ; 3 for any triple i , j , k ∈ { 0 , ..., d } and any pair ( x , y ) ∈ R k the intersection number p k ij := | R i ( x ) ∩ R j ′ ( y ) | depends only on i , j , k . 1 ( X , R i ) - basic (di)graphs of X ; 2 | X | - the order of X ; 3 |R| - the rank of X ; 4 v i := p 0 ii ′ - the valency of ( X , R i ).

Adjacency (BM-) algebra of a scheme Theorem Let A i be the adjacency matrix of the basic graph ( X , R i ). Then the linear span A F := � A 0 , ..., A d � is a subalgebra of the matrix algebra M X ( F ). Moreover I X , J X ∈ A F , A ⊤ F = A F and A i A j = � d k =0 p k ij A k . A F is called the adjacency / Bose-Mesner algebra of X . The basis A 0 , ..., A d is called the standard basis of A F .

Adjacency (BM-) algebra of a scheme Theorem Let A i be the adjacency matrix of the basic graph ( X , R i ). Then the linear span A F := � A 0 , ..., A d � is a subalgebra of the matrix algebra M X ( F ). Moreover I X , J X ∈ A F , A ⊤ F = A F and A i A j = � d k =0 p k ij A k . A F is called the adjacency / Bose-Mesner algebra of X . The basis A 0 , ..., A d is called the standard basis of A F . A scheme is called commutative if it’s BM-algebra is commutative.

Adjacency (BM-) algebra of a scheme Theorem Let A i be the adjacency matrix of the basic graph ( X , R i ). Then the linear span A F := � A 0 , ..., A d � is a subalgebra of the matrix algebra M X ( F ). Moreover I X , J X ∈ A F , A ⊤ F = A F and A i A j = � d k =0 p k ij A k . A F is called the adjacency / Bose-Mesner algebra of X . The basis A 0 , ..., A d is called the standard basis of A F . A scheme is called commutative if it’s BM-algebra is commutative. A scheme is called symmetric (antisymmetric) if all it’s non-reflexive relations are symmetric (antisymmetric. resp.).

Adjacency (BM-) algebra of a scheme Theorem Let A i be the adjacency matrix of the basic graph ( X , R i ). Then the linear span A F := � A 0 , ..., A d � is a subalgebra of the matrix algebra M X ( F ). Moreover I X , J X ∈ A F , A ⊤ F = A F and A i A j = � d k =0 p k ij A k . A F is called the adjacency / Bose-Mesner algebra of X . The basis A 0 , ..., A d is called the standard basis of A F . A scheme is called commutative if it’s BM-algebra is commutative. A scheme is called symmetric (antisymmetric) if all it’s non-reflexive relations are symmetric (antisymmetric. resp.). Proposition A symmetric scheme is commutative.

Adjacency (BM-) algebra of a scheme Theorem Let A i be the adjacency matrix of the basic graph ( X , R i ). Then the linear span A F := � A 0 , ..., A d � is a subalgebra of the matrix algebra M X ( F ). Moreover I X , J X ∈ A F , A ⊤ F = A F and A i A j = � d k =0 p k ij A k . A F is called the adjacency / Bose-Mesner algebra of X . The basis A 0 , ..., A d is called the standard basis of A F . A scheme is called commutative if it’s BM-algebra is commutative. A scheme is called symmetric (antisymmetric) if all it’s non-reflexive relations are symmetric (antisymmetric. resp.). Proposition A symmetric scheme is commutative. Conjecture (Evdokimov - Ponomarenko) Primitive antisymmetric scheme is commutative.

Imprimitive association schemes Definition The scheme is imprimitive if there exists a non-reflexive basic graph which is not strongly connected.

Imprimitive association schemes Definition The scheme is imprimitive if there exists a non-reflexive basic graph which is not strongly connected. Proposition Let X = ( X , R = { R i } d i =0 ) be an association scheme and A F = � A 0 , ..., A d � its BM-algebra, char( F ) = 0. The following conditions are equivalent (a) X is imprimitive; (b) ∃ I ⊂ { 0 , ..., d } s.t. | I | > 1 and � i ∈ I R i is an equivalence relation on X ; (c) ∃ I ⊂ { 0 , ..., d } s.t. I ′ = I and � A i � i ∈ I is a subalgebra of A F , char( F ) = 0. The subset { R i } i ∈ I is called a closed subset of R .

A concrete example  0 1 2 2 1 3 4 5 5 4  1 0 1 2 2 4 3 4 5 5     2 1 0 1 2 5 4 3 4 5     2 2 1 0 1 5 5 4 3 4     1 2 2 1 0 4 5 5 4 3   A ( X ) =   3 5 4 4 5 0 2 1 1 2     5 3 5 4 4 2 0 2 1 1     4 5 3 5 4 1 2 0 2 1     4 4 5 3 5 1 1 2 0 2   5 4 4 5 3 2 1 1 2 0

Main sources of AS

Main sources of AS 1 group theory;

Main sources of AS 1 group theory; 2 merging of classes;

Main sources of AS 1 group theory; 2 merging of classes; 3 finite geometry and design theory;

Main sources of AS 1 group theory; 2 merging of classes; 3 finite geometry and design theory; 4 the others.

Schemes coming from groups Let G = Hg 0 H ∪ ... ∪ Hg d H be a double coset decomposition of a finite group G ( g 0 = e ) w.r. to a proper subgroup H . On the set of left cosets G / H = { gH | g ∈ G } define relations R i , i = 0 , ..., d via ⇒ Hg − 1 ( g 1 H , g 2 H ) ∈ R i ⇐ 1 g 2 H = Hg i H .

Schemes coming from groups Let G = Hg 0 H ∪ ... ∪ Hg d H be a double coset decomposition of a finite group G ( g 0 = e ) w.r. to a proper subgroup H . On the set of left cosets G / H = { gH | g ∈ G } define relations R i , i = 0 , ..., d via ⇒ Hg − 1 ( g 1 H , g 2 H ) ∈ R i ⇐ 1 g 2 H = Hg i H . Proposition The set of relations R i , i = 0 , ..., d form an association scheme on the set G / H . Association schemes of this type are called Schurian. The BM-algebra of this scheme is known as the Hecke algebra of the pair ( G , H ).

Schemes coming from groups Let G = Hg 0 H ∪ ... ∪ Hg d H be a double coset decomposition of a finite group G ( g 0 = e ) w.r. to a proper subgroup H . On the set of left cosets G / H = { gH | g ∈ G } define relations R i , i = 0 , ..., d via ⇒ Hg − 1 ( g 1 H , g 2 H ) ∈ R i ⇐ 1 g 2 H = Hg i H . Proposition The set of relations R i , i = 0 , ..., d form an association scheme on the set G / H . Association schemes of this type are called Schurian. The BM-algebra of this scheme is known as the Hecke algebra of the pair ( G , H ). Example If H = { e } , then the relations R i are permutations of G which form a regular permutation group on G isomorphic to G . All basic relations of this scheme are thin (have valency one). The BM-algebra of this scheme is isomorphic to F [ G ].

Class merging (fusion and fission schemes) Definition i =0 ) and X ′ = ( X , R ′ = { R ′ i } d ′ Let X = ( X , R = { R i } d i =0 ) be two association schemes with the same point set X . We say that X ′ is a fusion of X (or X is a fission of X ′ ) iff each R ′ i is a union of some R j .