Crested products R. A. Bailey r.a.bailey@qmul.ac.uk From - PowerPoint PPT Presentation

Crested products R. A. Bailey r.a.bailey@qmul.ac.uk From Higman-Sims to Urysohn: a random walk through groups, graphs, designs, and spaces August 2007

A story of collaboration

Time-line ◮ Pre-Cambrian: ◮ association schemes; ◮ transitive permutation groups; ◮ direct products (crossing); ◮ wreath products (nesting); ◮ partitions; ◮ orthogonal block structures.

Association schemes An association scheme of rank r on a finite set Ω is a colouring of the elements of Ω × Ω by r colours such that

Association schemes An association scheme of rank r on a finite set Ω is a colouring of the elements of Ω × Ω by r colours such that (i) one colour is exactly the main diagonal; (ii) each colour is symmetric about the main diagonal; (iii) if ( α , β ) is yellow then there are exactly p yellow red , blue points γ such that ( α , γ ) is red and ( γ , β ) is blue (for all values of yellow, red and blue).

Association schemes An association scheme of rank r on a finite set Ω is a colouring of the elements of Ω × Ω by r colours such that (i) one colour is exactly the main diagonal; (ii) each colour is symmetric about the main diagonal; (iii) if ( α , β ) is yellow then there are exactly p yellow red , blue points γ such that ( α , γ ) is red and ( γ , β ) is blue (for all values of yellow, red and blue). The set of pairs given colour i is called the i -th associate class.

Adjacency matrices The adjacency matrix A i for colour i is the Ω × Ω matrix with � 1 if ( α , β ) has colour i A i ( α , β ) = 0 otherwise.

Adjacency matrices The adjacency matrix A i for colour i is the Ω × Ω matrix with � 1 if ( α , β ) has colour i A i ( α , β ) = 0 otherwise. Colour 0 is the diagonal, so (i) A 0 = I (identity matrix); (ii) every A i is symmetric; (iii) A i A j = ∑ p k ij A k ; k (iv) ∑ A i = J (all-1s matrix). i

Permutation groups If G is a transitive permutation group on Ω , it induces a permutation group on Ω × Ω . Give ( α , β ) the same colour as ( γ , δ ) iff there is some g in G with ( α g , β g ) = ( γ , δ ) . The colour classes are the orbitals of G .

Permutation groups If G is a transitive permutation group on Ω , it induces a permutation group on Ω × Ω . Give ( α , β ) the same colour as ( γ , δ ) iff there is some g in G with ( α g , β g ) = ( γ , δ ) . The colour classes are the orbitals of G . association scheme permutation group (i) A 0 = I ⇐ ⇒ transitivity (ii) every A i is symmetric ⇐ ⇒ the orbitals are self-paired (iii) A i A j = ∑ p k ij A k always satisfied k (iv) ∑ A i = J always satisfied i

Permutation groups If G is a transitive permutation group on Ω , it induces a permutation group on Ω × Ω . Give ( α , β ) the same colour as ( γ , δ ) iff there is some g in G with ( α g , β g ) = ( γ , δ ) . The colour classes are the orbitals of G . association scheme permutation group (i) A 0 = I ⇐ ⇒ transitivity (ii) every A i is symmetric ⇐ ⇒ the orbitals are self-paired (iii) A i A j = ∑ p k ij A k always satisfied k (iv) ∑ A i = J always satisfied i Some of the theory extends if (ii) is weakened to ‘if A i is an adjacency matrix then so is A ⊤ i ’, which is true for permutation groups.

The Bose–Mesner algebra and the character table (i) A 0 = I ; (ii) every A i is symmetric; (iii) A i A j = ∑ p k ij A k ; k (iv) ∑ A i = J (all-1s matrix). i The set of all real linear combinations of the A i forms a commutative algebra A , the Bose–Mesner algebra of the association scheme.

The Bose–Mesner algebra and the character table (i) A 0 = I ; (ii) every A i is symmetric; (iii) A i A j = ∑ p k ij A k ; k (iv) ∑ A i = J (all-1s matrix). i The set of all real linear combinations of the A i forms a commutative algebra A , the Bose–Mesner algebra of the association scheme. It contains the projectors S 0 , S 1 , . . . , S r onto its mutual eigenspaces.

The Bose–Mesner algebra and the character table (i) A 0 = I ; (ii) every A i is symmetric; (iii) A i A j = ∑ p k ij A k ; k (iv) ∑ A i = J (all-1s matrix). i The set of all real linear combinations of the A i forms a commutative algebra A , the Bose–Mesner algebra of the association scheme. It contains the projectors S 0 , S 1 , . . . , S r onto its mutual eigenspaces. The character table gives each A i as a linear combination of S 0 , . . . , S r .

The Bose–Mesner algebra and the character table (i) A 0 = I ; (ii) every A i is symmetric; (iii) A i A j = ∑ p k ij A k ; k (iv) ∑ A i = J (all-1s matrix). i The set of all real linear combinations of the A i forms a commutative algebra A , the Bose–Mesner algebra of the association scheme. It contains the projectors S 0 , S 1 , . . . , S r onto its mutual eigenspaces. The character table gives each A i as a linear combination of S 0 , . . . , S r . Its inverse expresses each S j as a linear combination of A 0 , . . . , A r .

Direct product (crossing) association set adjacency index Bose–Mesner scheme matrices set algebra Ω 1 i ∈ K 1 Q 1 A i A 1 Q 2 Ω 2 j ∈ K 2 A 2 B j

Direct product (crossing) association set adjacency index Bose–Mesner scheme matrices set algebra Ω 1 i ∈ K 1 Q 1 A i A 1 Q 2 Ω 2 j ∈ K 2 A 2 B j Ω 2     ∗    Ω 1 colour in Q 1  ∗      � �� � colour in Q 2 The underlying set of Q 1 × Q 2 is Ω 1 × Ω 2 . The adjacency matrices of Q 1 × Q 2 are A i ⊗ B j for i in K 1 and j in K 2 . A = A 1 ⊗ A 2

Direct product of permutation groups Ω 2        permute rows by an Ω 1 element of G 1       � �� � permute columns by an element of G 2 If G 1 is transitive on Ω 1 with self-paired orbitals and association scheme Q 1 , and G 2 is transitive on Ω 2 with self-paired orbitals and association scheme Q 2 , then G 1 × G 2 is transitive on Ω 1 × Ω 2 with self-paired orbitals and association scheme Q 1 × Q 2 .

Wreath product (nesting) The underlying set of Q 1 / Q 2 is Ω 1 × Ω 2 . Ω 2  The adjacency matrices of    ∗  Q 1 / Q 2 are   colour in Q 1 Ω 1 if colour � = 0  ∗  A i ⊗ J for i in K 1 \{ 0 }    

Wreath product (nesting) The underlying set of Q 1 / Q 2 is Ω 1 × Ω 2 . Ω 2  The adjacency matrices of    ∗  Q 1 / Q 2 are   colour in Q 1 Ω 1 if colour � = 0  ∗  A i ⊗ J for i in K 1 \{ 0 }     and I ⊗ B j for j in K 2 . Ω 2 ∗ ∗ Ω 1 � �� � colour in Q 2

Wreath product (nesting) The underlying set of Q 1 / Q 2 is Ω 1 × Ω 2 . Ω 2  The adjacency matrices of    ∗  Q 1 / Q 2 are   colour in Q 1 Ω 1 if colour � = 0  ∗  A i ⊗ J for i in K 1 \{ 0 }     and I ⊗ B j for j in K 2 . Ω 2 So ∗ ∗ A = A 1 ⊗� J � + � I �⊗ A 2 Ω 1 � �� � colour in Q 2

Wreath product (nesting) The underlying set of Q 1 / Q 2 is Ω 1 × Ω 2 . Ω 2  The adjacency matrices of    ∗  Q 1 / Q 2 are   colour in Q 1 Ω 1 if colour � = 0  ∗  A i ⊗ J for i in K 1 \{ 0 }     and I ⊗ B j for j in K 2 . Ω 2 So ∗ ∗ A = A 1 ⊗� J � + � I �⊗ A 2 Ω 1 NB A 1 � I � = A 1 and � �� � colour in Q 2 � J � A 2 = � J �

Wreath product of permutation groups Ω 2        permute rows by an Ω 1 element of G 1       � �� � permute the cells in each row by its own element of G 2 If G 1 is transitive on Ω 1 with self-paired orbitals and association scheme Q 1 , and G 2 is transitive on Ω 2 with self-paired orbitals and association scheme Q 2 , then G 2 ≀ G 1 is transitive on Ω 1 × Ω 2 with self-paired orbitals and association scheme Q 1 / Q 2 .

Time-line ◮ Pre-Cambrian: association schemes; transitive permutation groups; direct products (crossing); wreath products (nesting); partitions; orthogonal block structures

Time-line ◮ Pre-Cambrian: association schemes; transitive permutation groups; direct products (crossing); wreath products (nesting); partitions; orthogonal block structures ◮ March 1999: 45th German Biometric Colloquium, Dortmund RAB idea!—use a partition in the top scheme

Inherent partitions A partition F of Ω is inherent in the association scheme Q on Ω if there is a subset L of the colours such that α and β are in the same part of F ⇐ ⇒ the colour of ( α , β ) is in L

Inherent partitions A partition F of Ω is inherent in the association scheme Q on Ω if there is a subset L of the colours such that α and β are in the same part of F ⇐ ⇒ the colour of ( α , β ) is in L The relation matrix R F for partition F is the Ω × Ω matrix with � 1 if α and β are in the same part of F R F ( α , β ) = 0 otherwise.

Inherent partitions A partition F of Ω is inherent in the association scheme Q on Ω if there is a subset L of the colours such that α and β are in the same part of F ⇐ ⇒ the colour of ( α , β ) is in L The relation matrix R F for partition F is the Ω × Ω matrix with � 1 if α and β are in the same part of F R F ( α , β ) = 0 otherwise. If F is inherent then R F = ∑ A i . i ∈ L