On the isomorphism of certain Q -polynomial association schemes - PowerPoint PPT Presentation

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... On the isomorphism of certain Q -polynomial association schemes Giusy Monzillo (joint work with Alessandro Siciliano) 18th June 2019

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... Association Schemes

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... Association Schemes X = finite set, | X | ≥ 2 d = positive integer R = { R 0 , ..., R d } , R i ⊆ X × X

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... Association Schemes X = finite set, | X | ≥ 2 d = positive integer R = { R 0 , ..., R d } , R i ⊆ X × X Definition ( X , R ) is a d − class association scheme if : A1. R is a partition of X × X with R 0 = { ( x , x ) | x ∈ X } ; A2. R − 1 = { ( y , x ) | ( x , y ) ∈ R i } = R i , i = 0 , ..., d ; i A3. for each ( x , y ) ∈ R k , p ( k ) i , j = |{ z ∈ X | ( x , z ) ∈ R i , ( z , y ) ∈ R j }| = p ( k ) j , i does not depend on ( x , y ).

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... Definition Two schemes ( X , { R i } 0 ≤ i ≤ d ) and ( X ′ , { R ′ i } 0 ≤ i ≤ d ) are isomorphic if there exists a bijection ϕ from X to X ′ and a permutation σ of { 1 , . . . , d } such that ⇒ ( ϕ ( x ) , ϕ ( y )) ∈ R ′ ( x , y ) ∈ R i ⇐ σ ( i ) .

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... The Bose − Mesner Algebra

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... The Bose − Mesner Algebra R ( X , X ) = the set of all the | X | -matrices over R Definition A i ∈ R ( X , X ) with � 1 if ( x , y ) ∈ R i A i ( x , y ) = 0 otherwise is called the adjacency matrix of R i .

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... Theorem (Bose-Mesner, 1952) Let ( X , R ) be an association scheme with d classes. Then A = � A 0 , ..., A d � R is a commutative subalgebra in R ( X , X ) such that: i. dim A = d + 1; ii. D = D T , for each D ∈ A . A is the so-called Bose-Mesner algebra of ( X , R ).

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... Corollary i. A admits d + 1 common maximal eigen-spaces V 0 , ..., V d , where V 0 = � 1 � , such that R | X | = V 0 ⊥ ... ⊥ V d .

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... Corollary i. A admits d + 1 common maximal eigen-spaces V 0 , ..., V d , where V 0 = � 1 � , such that R | X | = V 0 ⊥ ... ⊥ V d . ii. A admits a unique basis of minimal idempotent matrices { E 0 , ..., E d } .

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... The Eigenmatrices Definition The matrices P and Q such that ( A 0 A 1 . . . A d ) = ( E 0 E 1 . . . E d ) P and ( E 0 E 1 . . . E d ) = | X | − 1 ( A 0 A 1 . . . A d ) Q are the first and the second eigenmatrix of ( X , R ), respectively.

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... Definition A scheme is P-polynomial if, after a reordering of the relations, there are polynomials p i of degree i such that A i = p i ( A 1 ).

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... Definition A scheme is P-polynomial if, after a reordering of the relations, there are polynomials p i of degree i such that A i = p i ( A 1 ). A scheme is Q-polynomial if, after a reordering of the eigenspaces, there are polynomials q i of degree i such that E i = q i ( E 1 ), where multiplication is done entrywise.

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... The Hollmann-Xiang association scheme

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... The Hollmann-Xiang association scheme Let C be a non-degenerate conic in PG (2 , q 2 ): C = {� (1 , t , t 2 ) � : t ∈ F q 2 } ∪ {� (0 , 0 , 1) �} A line ℓ of PG (2 , q 2 ) is called a passant if | ℓ ∩ C| = 0.

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... The Hollmann-Xiang association scheme Let C be a non-degenerate conic in PG (2 , q 2 ): C = {� (1 , t , t 2 ) � : t ∈ F q 2 } ∪ {� (0 , 0 , 1) �} A line ℓ of PG (2 , q 2 ) is called a passant if | ℓ ∩ C| = 0. Let C be the extension of C in PG (2 , q 4 ). An elliptic line of C is the extension ℓ of a passant ℓ of C .

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... The Hollmann-Xiang association scheme Let C be a non-degenerate conic in PG (2 , q 2 ): C = {� (1 , t , t 2 ) � : t ∈ F q 2 } ∪ {� (0 , 0 , 1) �} A line ℓ of PG (2 , q 2 ) is called a passant if | ℓ ∩ C| = 0. Let C be the extension of C in PG (2 , q 4 ). An elliptic line of C is the extension ℓ of a passant ℓ of C . Then ℓ ∩ C = {� (1 , t , t 2 ) � , � (1 , t q 2 , t 2 q 2 ) �} , for some t ∈ F q 4 \ F q 2 .

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... E = the set of all the elliptic lines of C X = the set of all pairs t = { t , t q 2 } with t in F q 4 \ F q 2

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... E = the set of all the elliptic lines of C X = the set of all pairs t = { t , t q 2 } with t in F q 4 \ F q 2 The identification F q 4 ∪ {∞} ← → C ξ : � (1 , t , t 2 ) � t ← → ∞ ← → � (0 , 0 , 1) � induces the bijection X ← → E t = { t , t q 2 } ← → ℓ t , where ℓ t = ℓ with ℓ ∩ C = {� (1 , t , t 2 ) � , � (1 , t q 2 , t 2 q 2 ) �} .

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... q even For any two distinct pairs s = { s , s q 2 } , t = { t , t q 2 } ∈ X , let ρ ( s , t ) = ( s + t )( s q 2 + t q 2 ) ( s + t q 2 )( s q 2 + t ) ∈ F q 2 \ { 0 , 1 } Note that ρ ( s , t ) is the cross-ratio of ( s , s q 2 , t , t q 2 ).

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... q even For any two distinct pairs s = { s , s q 2 } , t = { t , t q 2 } ∈ X , let ρ ( s , t ) = ( s + t )( s q 2 + t q 2 ) ( s + t q 2 )( s q 2 + t ) ∈ F q 2 \ { 0 , 1 } Note that ρ ( s , t ) is the cross-ratio of ( s , s q 2 , t , t q 2 ). From the properties of the cross-ratio it is possible to define the cross-ratio of { s , t } as the pair { ρ ( s , t ) , ρ ( s , t ) − 1 } .

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... Theorem (Hollmann-Xiang, 2006) Under the identification ξ , the action of PGL (2 , q 2 ) on E × E gives rise to an association scheme on X with q 2 / 2 − 1 classes R { λ,λ − 1 } , λ ∈ F q 2 \ { 0 , 1 } , where ⇒ { ρ ( s , t ) , ρ ( s , t ) − 1 } = { λ, λ − 1 } . ( s , t ) ∈ R { λ,λ − 1 } ⇐

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... The fusion scheme

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... The fusion scheme T 0 ( q r ) = the set of all the elements of F q r with absolute trace zero T 0 = T 0 ( q 2 ); S ∗ 0 = T 0 ( q ) \ { 0 } ; S 1 = F q \ S 0 .

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... The fusion scheme T 0 ( q r ) = the set of all the elements of F q r with absolute trace zero T 0 = T 0 ( q 2 ); S ∗ 0 = T 0 ( q ) \ { 0 } ; S 1 = F q \ S 0 . For any two distinct pairs s , t ∈ X , define 1 ρ ( s , t ) = � ρ ( s , t ) + ρ ( s , t ) − 1 Since � � 2 � � 1 1 ρ ( s , t ) = + , � ρ ( s , t ) + 1 ρ ( s , t ) + 1 then Im � ρ ⊂ T 0 .

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... Theorem (Hollmann-Xiang, 2006) The following relations are defined on X : ρ ( s , t ) ∈ S ∗ R 1 : ( s , t ) ∈ R 1 if and only � 0 ; R 2 : ( s , t ) ∈ R 2 if and only � ρ ( s , t ) ∈ S 1 ; R 3 : ( s , t ) ∈ R 3 if and only � ρ ( s , t ) ∈ T 0 \ F q .

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... Theorem (Hollmann-Xiang, 2006) The following relations are defined on X : ρ ( s , t ) ∈ S ∗ R 1 : ( s , t ) ∈ R 1 if and only � 0 ; R 2 : ( s , t ) ∈ R 2 if and only � ρ ( s , t ) ∈ S 1 ; R 3 : ( s , t ) ∈ R 3 if and only � ρ ( s , t ) ∈ T 0 \ F q . Then ( X , { R i } 3 i =0 ) is a 3-class association scheme which is a fusion of the previous scheme.

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... The Penttila-Williford association schemes

Preliminaries The construction of the relative hemisystem of Penttila-Williford Betting... The Penttila-Williford association schemes Assume q even, and let H (3 , q 2 ) be the unitary polar space of rank 2 of PG (3 , q 2 ); W (3 , q ) be a symplectic polar space of rank 2 embedded in H (3 , q 2 ); Q − (3 , q ) be an orthogonal polar space of rank 1 embedded in W (3 , q ) .