Q -polynomial Association Schemes Jason Williford University of - - PowerPoint PPT Presentation

Q-polynomial Association Schemes

Jason Williford University of Wyoming Modern Trends in Algebraic Graph Theory June 4th, 2014

Jason Williford University of Wyoming Q-polynomial Association Schemes

Association Schemes

A d-class symmetric association scheme is a collection of relations R0, ..., Rd on a set X which satisfy: The relations R0, . . . Rd partition X × X R0 is the identity relation on X Ri = RT

i

there are constants pk

ij such that for all x k

∼y we have that there are exactly pk

ij points z such that x i

∼z and z

j

∼y.

k i j k pij x y Jason Williford University of Wyoming Q-polynomial Association Schemes

Schurian Schemes

Let H be a group which is generously transitive; i.e. a group acting transitively on a set X with the property that for all x, y ∈ X there is a h ∈ H with xh = y and yh = x. Then the orbitals (the

• rbits of H on X × X) form a symmetric association scheme.

We will call these schemes Schurian.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Imprimitivity

We can view each relation Ri as a graph with vertex set X. An association scheme is called primitive if the graphs R1, . . . , Rd are all connected, and is called imprimitive otherwise. Imprimitive schemes have associated subschemes and quotient schemes.

Jason Williford University of Wyoming Q-polynomial Association Schemes

The Bose-Mesner Algebra

Let A0, . . . Ad be the adjacency matrices of the relations R0, . . . , Rd of an association scheme. These matrices satisfy: AiAj =

k pk ijAk

Let A denote the span of the matrices A0, . . . Ad over the real numbers. Since the matrices are all symmetric, the matrix algebra A is commutative, and is called the Bose-Mesner algebra of the association scheme.

Jason Williford University of Wyoming Q-polynomial Association Schemes

The Bose-Mesner Algebra

Let A0, . . . Ad be the adjacency matrices of the relations R0, . . . , Rd of an association scheme. These matrices satisfy: AiAj =

k pk ijAk

Let A denote the span of the matrices A0, . . . Ad over the real numbers. Since the matrices are all symmetric, the matrix algebra A is commutative, and is called the Bose-Mesner algebra of the association scheme.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Primitive Idempotents

The Bose-Mesner algebra A of an association scheme is a commutative algebra of real symmetric matrices, and so can be simultaneously diagonalized by a real orthogonal matrix. This gives a decomposition of R|X| into d + 1 orthogonal common eigenspaces V0, V1, . . . Vd. Since all of the graphs are regular and J is in A, one of these eigenspaces is the one-dimensional eigenspace V0 =< (1, 1, . . . , 1) >.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Primitive Idempotents

The Bose-Mesner algebra A of an association scheme is a commutative algebra of real symmetric matrices, and so can be simultaneously diagonalized by a real orthogonal matrix. This gives a decomposition of R|X| into d + 1 orthogonal common eigenspaces V0, V1, . . . Vd. Since all of the graphs are regular and J is in A, one of these eigenspaces is the one-dimensional eigenspace V0 =< (1, 1, . . . , 1) >.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Primitive Idempotents

The primitive idempotents E0, E1, . . . , Ed which project onto these eigenspaces can also be seen to be in A, and form another basis of A. Since the idempotents project onto orthogonal eigenspaces, we have EiEj = 0 if i = j and EiEj = Ei if i = j.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Primitive Idempotents

The primitive idempotents E0, E1, . . . , Ed which project onto these eigenspaces can also be seen to be in A, and form another basis of A. Since the idempotents project onto orthogonal eigenspaces, we have EiEj = 0 if i = j and EiEj = Ei if i = j.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Entrywise Multiplication

The Bose-Mesner Algebra of an association scheme is also closed under entrywise multiplication, since for all i, j we have Ai ◦ Aj = 0 if i = j, and Ai ◦ Aj = Ai if i = j. We define the Krein parameters qk

ij to satisfy :

Ei ◦ Ej =

1 |X|

d

k=0 qk ijEk

The Krein parameters do not have to be integral or even rational, but they must be non-negative. The parameters q0

ii are equal to

the multiplicities of the eigenvalues of the scheme, and so must be positive integers.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Entrywise Multiplication

The Bose-Mesner Algebra of an association scheme is also closed under entrywise multiplication, since for all i, j we have Ai ◦ Aj = 0 if i = j, and Ai ◦ Aj = Ai if i = j. We define the Krein parameters qk

ij to satisfy :

Ei ◦ Ej =

1 |X|

d

k=0 qk ijEk

The Krein parameters do not have to be integral or even rational, but they must be non-negative. The parameters q0

ii are equal to

the multiplicities of the eigenvalues of the scheme, and so must be positive integers.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Entrywise Multiplication

The Bose-Mesner Algebra of an association scheme is also closed under entrywise multiplication, since for all i, j we have Ai ◦ Aj = 0 if i = j, and Ai ◦ Aj = Ai if i = j. We define the Krein parameters qk

ij to satisfy :

Ei ◦ Ej =

1 |X|

d

k=0 qk ijEk

The Krein parameters do not have to be integral or even rational, but they must be non-negative. The parameters q0

ii are equal to

the multiplicities of the eigenvalues of the scheme, and so must be positive integers.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Formal Duality

The matrices Ai satisfy: A0 = I d

i=0 Ai = J

AiAj =

k pk ijAk

Ai ◦ Aj = δijAi The matrices Ei satisfy: E0 =

1 |X|J

d

i=0 Ei = I

Ei ◦ Ej =

1 |X|

• k qk

ijEk

EiEj = δijEi

Jason Williford University of Wyoming Q-polynomial Association Schemes

Formal Duality

The matrices Ai satisfy: A0 = I d

i=0 Ai = J

AiAj =

k pk ijAk

Ai ◦ Aj = δijAi The matrices Ei satisfy: E0 =

1 |X|J

d

i=0 Ei = I

Ei ◦ Ej =

1 |X|

• k qk

ijEk

EiEj = δijEi

Jason Williford University of Wyoming Q-polynomial Association Schemes

Bipartite double

We can construct new association schemes from old using Kronecker products. The bipartite double of a scheme {A0, . . . , Ad} has matrices: Ai Ai

• and

Ai Ai

• for 0 ≤ i ≤ d.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Polynomial Association Schemes

An association scheme is called P-polynomial (metric) provided that, after suitably reordering the Ai, there are polynomials pk(x)

• f degree k for 0 ≤ k ≤ d such that Ak = pk(A1). This is

equivalent to saying R1 is a distance-regular graph of diameter d. An association scheme is called Q-polynomial (cometric) provided that, after suitably reordering the Ei, there are polynomials qk(x)

• f degree k for 0 ≤ k ≤ d such that Ek = qk(E1), where

multiplication is done entrywise. Conjecture (Bannai, Ito) For sufficiently large d, a primitive scheme is P-polynomial if and only if it is Q-polynomial.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Polynomial Association Schemes

An association scheme is called P-polynomial (metric) provided that, after suitably reordering the Ai, there are polynomials pk(x)

• f degree k for 0 ≤ k ≤ d such that Ak = pk(A1). This is

equivalent to saying R1 is a distance-regular graph of diameter d. An association scheme is called Q-polynomial (cometric) provided that, after suitably reordering the Ei, there are polynomials qk(x)

• f degree k for 0 ≤ k ≤ d such that Ek = qk(E1), where

multiplication is done entrywise. Conjecture (Bannai, Ito) For sufficiently large d, a primitive scheme is P-polynomial if and only if it is Q-polynomial.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Polynomial Association Schemes

An association scheme is called P-polynomial (metric) provided that, after suitably reordering the Ai, there are polynomials pk(x)

• f degree k for 0 ≤ k ≤ d such that Ak = pk(A1). This is

equivalent to saying R1 is a distance-regular graph of diameter d. An association scheme is called Q-polynomial (cometric) provided that, after suitably reordering the Ei, there are polynomials qk(x)

• f degree k for 0 ≤ k ≤ d such that Ek = qk(E1), where

multiplication is done entrywise. Conjecture (Bannai, Ito) For sufficiently large d, a primitive scheme is P-polynomial if and only if it is Q-polynomial.

Jason Williford University of Wyoming Q-polynomial Association Schemes

T-designs in Q-polynomial Schemes

Let Y be a subset of a Q-polynomial scheme, let χ be the characteristic vector of Y , and let T ⊂ {1, . . . d}. The set Y is called a T-design provided that Eiχ = 0 for all i ∈ T. If T = {1, . . . , t}, we call Y a t-design. In the Johnson Scheme J(v, k), Y is a t-design if and only if it is a t − (v, k, λ) design for some λ. Delsarte’s “conjecture”: T-designs will often have interesting properties.

Jason Williford University of Wyoming Q-polynomial Association Schemes

T-designs in Q-polynomial Schemes

Let Y be a subset of a Q-polynomial scheme, let χ be the characteristic vector of Y , and let T ⊂ {1, . . . d}. The set Y is called a T-design provided that Eiχ = 0 for all i ∈ T. If T = {1, . . . , t}, we call Y a t-design. In the Johnson Scheme J(v, k), Y is a t-design if and only if it is a t − (v, k, λ) design for some λ. Delsarte’s “conjecture”: T-designs will often have interesting properties.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Subsets in Q-polynomial Schemes

Theorem (Delsarte 1973) Let Y be a subset of a Q-polynomial scheme. Define the degree s of Y to be the number of nontrivial relations

• ccurring in Y , and suppose Eiχ = 0 for all 1 ≤ i ≤ t, where χ is

the characteristic vector of Y . If t ≥ 2s − 2 then Y is a Q-polynomial subscheme. Theorem (Brouwer, Godsil, Koolen, Martin 2003) Let Y be a subset of a Q-polynomial scheme. Define the degree s of Y to be the number

• f nontrivial relations occurring in Y , and suppose

w∗ = max{i : Eiχ = 0} where χ is the characteristic vector of Y . Then w∗ ≥ d − s. If equality holds, then Y is a Q-polynomial subscheme.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Subsets in Q-polynomial Schemes

Theorem (Delsarte 1973) Let Y be a subset of a Q-polynomial scheme. Define the degree s of Y to be the number of nontrivial relations

• ccurring in Y , and suppose Eiχ = 0 for all 1 ≤ i ≤ t, where χ is

the characteristic vector of Y . If t ≥ 2s − 2 then Y is a Q-polynomial subscheme. Theorem (Brouwer, Godsil, Koolen, Martin 2003) Let Y be a subset of a Q-polynomial scheme. Define the degree s of Y to be the number

• f nontrivial relations occurring in Y , and suppose

w∗ = max{i : Eiχ = 0} where χ is the characteristic vector of Y . Then w∗ ≥ d − s. If equality holds, then Y is a Q-polynomial subscheme.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Splitting Fields of Q-polynomial Schemes

The splitting field of an association scheme is the field generated by the eigenvalues of the scheme. Theorem (Suzuki ’98) A Q-polynomial scheme can have at most 2 Q-polynomial orderings. A corollary of this is that the splitting field of a Q-polynomial scheme is at most a degree 2 extension of the rationals. Theorem (Martin, W ’09) For each integer m > 2 there are finitely many Q-polynomial schemes with rk(E1) = m.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Splitting Fields of Q-polynomial Schemes

The splitting field of an association scheme is the field generated by the eigenvalues of the scheme. Theorem (Suzuki ’98) A Q-polynomial scheme can have at most 2 Q-polynomial orderings. A corollary of this is that the splitting field of a Q-polynomial scheme is at most a degree 2 extension of the rationals. Theorem (Martin, W ’09) For each integer m > 2 there are finitely many Q-polynomial schemes with rk(E1) = m.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Splitting Fields of Q-polynomial Schemes

The splitting field of an association scheme is the field generated by the eigenvalues of the scheme. Theorem (Suzuki ’98) A Q-polynomial scheme can have at most 2 Q-polynomial orderings. A corollary of this is that the splitting field of a Q-polynomial scheme is at most a degree 2 extension of the rationals. Theorem (Martin, W ’09) For each integer m > 2 there are finitely many Q-polynomial schemes with rk(E1) = m.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Parameters of Distance-Regular Graphs

A graph Γ of diameter d is distance-regular if and only if there are constants b0, . . . bd−1, c1, . . . , cd such that given any two vertices x and y of distance i, we have that the number of z which are distance i − 1 from x and adjacent to y is ci and the number of z which are distance i + 1 from x and adjacent to y is bi.

i

(x)

i-1 (x) c b x y

i

i (x) i+1(x) i-1(x)

i

The constants relate to the pk

ij of the resulting association scheme

by: bi = pi

1,i+1, ci = pi 1,i−1, ai = pi 1,i.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Parameters of Distance-Regular Graphs

A graph Γ of diameter d is distance-regular if and only if there are constants b0, . . . bd−1, c1, . . . , cd such that given any two vertices x and y of distance i, we have that the number of z which are distance i − 1 from x and adjacent to y is ci and the number of z which are distance i + 1 from x and adjacent to y is bi.

i

(x)

i-1 (x) c b x y

i

i (x) i+1(x) i-1(x)

i

The constants relate to the pk

ij of the resulting association scheme

by: bi = pi

1,i+1, ci = pi 1,i−1, ai = pi 1,i.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Imprimitivite Distance-Regular Graphs

An imprimitive distance-regular graph Γ of diameter d which is not a cycle falls into one of three types: Γ is bipartite, meaning the vertices can be partitioned into two sets so that all edges have an endpoint in each set. This is equivalent to having pk

ij = 0 whenever i + j + k is odd. The

distance two graph of Γ is called the halved graph, and is also distance-regular. Γ is antipodal, meaning that the graph Γd is a union of complete graphs (distance d in the graph Γ is an equivalence relation). This is equivalent to having bj = cd−j for all j except possibly j = ⌊ d

2 ⌋. The antipodal quotient is also

distance-regular, and called the folded graph. Γ is both antipodal and bipartite.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Imprimitivite Distance-Regular Graphs

An imprimitive distance-regular graph Γ of diameter d which is not a cycle falls into one of three types: Γ is bipartite, meaning the vertices can be partitioned into two sets so that all edges have an endpoint in each set. This is equivalent to having pk

ij = 0 whenever i + j + k is odd. The

distance two graph of Γ is called the halved graph, and is also distance-regular. Γ is antipodal, meaning that the graph Γd is a union of complete graphs (distance d in the graph Γ is an equivalence relation). This is equivalent to having bj = cd−j for all j except possibly j = ⌊ d

2 ⌋. The antipodal quotient is also

distance-regular, and called the folded graph. Γ is both antipodal and bipartite.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Imprimitivite Distance-Regular Graphs

An imprimitive distance-regular graph Γ of diameter d which is not a cycle falls into one of three types: Γ is bipartite, meaning the vertices can be partitioned into two sets so that all edges have an endpoint in each set. This is equivalent to having pk

ij = 0 whenever i + j + k is odd. The

distance two graph of Γ is called the halved graph, and is also distance-regular. Γ is antipodal, meaning that the graph Γd is a union of complete graphs (distance d in the graph Γ is an equivalence relation). This is equivalent to having bj = cd−j for all j except possibly j = ⌊ d

2 ⌋. The antipodal quotient is also

distance-regular, and called the folded graph. Γ is both antipodal and bipartite.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Q-polynomial Association Schemes

An association scheme is called Q-polynomial provided that, after suitably reordering the Ei, there are polynomials qk(x) of degree k for 0 ≤ k ≤ d such that Ek = qk(E1), where multiplication is done entrywise. We define constants b∗

i and c∗ i by: b∗ i = qi 1,i+1, c∗ i = qi 1,i−1,

a∗

i = qi 1,i.

These are analogous to the bi, ci and ai of a distance-regular graph.

Jason Williford University of Wyoming Q-polynomial Association Schemes

The Imprimitive case

Theorem (Suzuki ’98) An imprimitive Q-polynomial association scheme which is not a cycle is one of the following: Q-bipartite, where qk

ij = 0 when i + j + k is odd.

Q-antipodal, where b∗

j = c∗ d−j for all j except possibly

j = ⌊ d

2 ⌋.

Both Q-bipartite and Q-antipodal. One of two hypothetical families of schemes with d = 4 or 6. Theorem (Cerzo, Suzuki ’09) The hypothetical family of exceptions for d = 4 do not exist. Theorem (Tanaka, Tanaka ’11) The hypothetical family of exceptions for d = 6 do not exist.

Jason Williford University of Wyoming Q-polynomial Association Schemes

The Imprimitive case

Theorem (Suzuki ’98) An imprimitive Q-polynomial association scheme which is not a cycle is one of the following: Q-bipartite, where qk

ij = 0 when i + j + k is odd.

Q-antipodal, where b∗

j = c∗ d−j for all j except possibly

j = ⌊ d

2 ⌋.

Both Q-bipartite and Q-antipodal. One of two hypothetical families of schemes with d = 4 or 6. Theorem (Cerzo, Suzuki ’09) The hypothetical family of exceptions for d = 4 do not exist. Theorem (Tanaka, Tanaka ’11) The hypothetical family of exceptions for d = 6 do not exist.

Jason Williford University of Wyoming Q-polynomial Association Schemes

The Imprimitive case

Theorem (Suzuki ’98) An imprimitive Q-polynomial association scheme which is not a cycle is one of the following: Q-bipartite, where qk

ij = 0 when i + j + k is odd.

Q-antipodal, where b∗

j = c∗ d−j for all j except possibly

j = ⌊ d

2 ⌋.

Both Q-bipartite and Q-antipodal. One of two hypothetical families of schemes with d = 4 or 6. Theorem (Cerzo, Suzuki ’09) The hypothetical family of exceptions for d = 4 do not exist. Theorem (Tanaka, Tanaka ’11) The hypothetical family of exceptions for d = 6 do not exist.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Q-antipodal schemes

Theorem (Martin, Muzychuk, W ’07) A Q-antipodal scheme is “dismantlable”: it can be partitioned into w > 1 Q-antipodal classes of equal size, with ⌊ d

2 ⌋ nontrivial relations occurring

between vertices in the same class, and the rest occurring between vertices in different classes, where any collection of the Q-antipodal classes induces a Q-polynomial subscheme. Furthermore w ≤ rk(E1) if d is odd, and w ≤ rk(E2) if d is even.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Q-antipodal schemes

Theorem (van Dam, Martin, Muzychuk ’13) A Q-polynomial scheme is Q-antipodal if and only if it is dismantable.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Extended Q-bipartite double

Theorem An imprimitive Q-polynomial scheme is Q-bipartite if and only if there is a relation which is a perfect matching. Theorem (Martin, Muzychuk, W ’07) If a Q-polynomial scheme satisfies b∗

j + c∗ j+1 = m1 + 1 for all j < d then there is a Q-bipartite scheme

with d + 1 classes which can be built from the original scheme. It is called the extended Q-bipartite double of the original scheme, and its quotient is a fusion of this scheme. The resulting scheme is a fusion of the bipartite double: A0 A0

• ,
• Ai

Ad+1−i Ad+1−i Ai

• , and

A0 A0

• where A0, . . . , Ad is the natural ordering of the adjacency matrices.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Extended Q-bipartite double

Theorem An imprimitive Q-polynomial scheme is Q-bipartite if and only if there is a relation which is a perfect matching. Theorem (Martin, Muzychuk, W ’07) If a Q-polynomial scheme satisfies b∗

j + c∗ j+1 = m1 + 1 for all j < d then there is a Q-bipartite scheme

with d + 1 classes which can be built from the original scheme. It is called the extended Q-bipartite double of the original scheme, and its quotient is a fusion of this scheme. The resulting scheme is a fusion of the bipartite double: A0 A0

• ,
• Ai

Ad+1−i Ad+1−i Ai

• , and

A0 A0

• where A0, . . . , Ad is the natural ordering of the adjacency matrices.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Extended Q-bipartite double

Theorem An imprimitive Q-polynomial scheme is Q-bipartite if and only if there is a relation which is a perfect matching. Theorem (Martin, Muzychuk, W ’07) If a Q-polynomial scheme satisfies b∗

j + c∗ j+1 = m1 + 1 for all j < d then there is a Q-bipartite scheme

with d + 1 classes which can be built from the original scheme. It is called the extended Q-bipartite double of the original scheme, and its quotient is a fusion of this scheme. The resulting scheme is a fusion of the bipartite double: A0 A0

• ,
• Ai

Ad+1−i Ad+1−i Ai

• , and

A0 A0

• where A0, . . . , Ad is the natural ordering of the adjacency matrices.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Schemes which are Q-antipodal and Q-bipartite

Theorem (Martin, Le Compte, Owens ’10) An association scheme with 4 classes which is Q-antipodal and Q-bipartite must be a scheme generated by a set of mutually unbiased bases of a real vector

• space. Furthermore, any set of k > 1 mutually unbiased bases

gives rise to such a scheme, and this scheme is not P-polynomial for k > 2. Theorem (Bannai, Ito ’84) The bipartite double of an almost dual bipartite scheme (Q-polynomial, a∗

1 = . . . a∗ d−1 = 0 but a∗ d = 0) is

Q-bipartite and Q-antipodal. In particular, the bipartite doubles of the hermitian dual polar space graph is q-polynomial for all d.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Schemes which are Q-antipodal and Q-bipartite

Theorem (Martin, Le Compte, Owens ’10) An association scheme with 4 classes which is Q-antipodal and Q-bipartite must be a scheme generated by a set of mutually unbiased bases of a real vector

• space. Furthermore, any set of k > 1 mutually unbiased bases

gives rise to such a scheme, and this scheme is not P-polynomial for k > 2. Theorem (Bannai, Ito ’84) The bipartite double of an almost dual bipartite scheme (Q-polynomial, a∗

1 = . . . a∗ d−1 = 0 but a∗ d = 0) is

Q-bipartite and Q-antipodal. In particular, the bipartite doubles of the hermitian dual polar space graph is q-polynomial for all d.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Examples of schemes that are P and Q-polynomial

distance-regular graphs with classical parameters (Hamming, Johnson, bilinear, quadratic forms, Grassmann, dual polar space, etc.) partition graphs Double covers of complete graphs (Taylor Graphs) incidence graphs if symmetric designs certain families of regular near polygons strongly regular graphs, complete graphs

Jason Williford University of Wyoming Q-polynomial Association Schemes

Infinite families of imprimitive strictly Q

linked systems of symmetric designs (Q-antipodal, d = 3) * duals of Kasami code graphs (primitive, d = 3) relative hemisystems of generalized quadrangles (primitive, d = 3) * certain strongly regular decompositions of strongly regular graphs, hemisystems (Q-antipodal, d = 4) * doubly subtended quadrangles (Q-bipartite, d = 4) * duals of extended Kasami code graphs (Q-bipartite, d = 4) real mutually unbiased bases in even dimension (Q-antipodal, Q-bipartite, d = 4) * Higman’s examples from triality (Q-antipodal, d = 5) bipartite doubles of Hermitian dual polar space graphs (Q-antipodal, Q-bipartite, d odd) double cover of symplectic dual polar space graphs ( Q-bipartite, d odd) *

Jason Williford University of Wyoming Q-polynomial Association Schemes

Infinite families of imprimitive strictly Q

linked systems of symmetric designs (Q-antipodal, d = 3) * duals of Kasami code graphs (primitive, d = 3) relative hemisystems of generalized quadrangles (primitive, d = 3) * certain strongly regular decompositions of strongly regular graphs, hemisystems (Q-antipodal, d = 4) * doubly subtended quadrangles (Q-bipartite, d = 4) * duals of extended Kasami code graphs (Q-bipartite, d = 4) real mutually unbiased bases in even dimension (Q-antipodal, Q-bipartite, d = 4) * Higman’s examples from triality (Q-antipodal, d = 5) bipartite doubles of Hermitian dual polar space graphs (Q-antipodal, Q-bipartite, d odd) double cover of symplectic dual polar space graphs ( Q-bipartite, d odd) *

Jason Williford University of Wyoming Q-polynomial Association Schemes

Infinite families of imprimitive strictly Q

linked systems of symmetric designs (Q-antipodal, d = 3) * duals of Kasami code graphs (primitive, d = 3) relative hemisystems of generalized quadrangles (primitive, d = 3) * certain strongly regular decompositions of strongly regular graphs, hemisystems (Q-antipodal, d = 4) * doubly subtended quadrangles (Q-bipartite, d = 4) * duals of extended Kasami code graphs (Q-bipartite, d = 4) real mutually unbiased bases in even dimension (Q-antipodal, Q-bipartite, d = 4) * Higman’s examples from triality (Q-antipodal, d = 5) bipartite doubles of Hermitian dual polar space graphs (Q-antipodal, Q-bipartite, d odd) double cover of symplectic dual polar space graphs ( Q-bipartite, d odd) *

Jason Williford University of Wyoming Q-polynomial Association Schemes

Infinite families of imprimitive strictly Q

linked systems of symmetric designs (Q-antipodal, d = 3) * duals of Kasami code graphs (primitive, d = 3) relative hemisystems of generalized quadrangles (primitive, d = 3) * certain strongly regular decompositions of strongly regular graphs, hemisystems (Q-antipodal, d = 4) * doubly subtended quadrangles (Q-bipartite, d = 4) * duals of extended Kasami code graphs (Q-bipartite, d = 4) real mutually unbiased bases in even dimension (Q-antipodal, Q-bipartite, d = 4) * Higman’s examples from triality (Q-antipodal, d = 5) bipartite doubles of Hermitian dual polar space graphs (Q-antipodal, Q-bipartite, d odd) double cover of symplectic dual polar space graphs ( Q-bipartite, d odd) *

Jason Williford University of Wyoming Q-polynomial Association Schemes

Sporadic primitive strictly Q-polynomial schemes

The shortest vectors of the Leech lattice form a 6-class Q-bipartite scheme. The quotient of this scheme is a primitive 3-class Q-polynomial scheme. The subgraph of the Leech lattice scheme formed by taking all vectors which have angles from a fixed vector satisfying cos(θ) = 1

4 is a primitive 5-class Q-polynomial scheme.

The subgraph of the previous scheme formed by taking all vectors which have angles from a fixed vector satisfying cos(θ) = −1

3 is a primitive 4-class Q-polynomial scheme.

The subgraph of the previous 5-class scheme formed by taking all vectors which have angles from a fixed vector satisfying cos(θ) = 7

15 is a primitive 3-class Q-polynomial scheme.

The block scheme of the 4 − (11, 5, 1) design is a primitive 3-class Q-polynomial scheme. The block scheme of the 4 − (47, 11, 8) design arising from a quadratic residue code is a primitive 3-class scheme.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Sporadic strictly Q-polynomial schemes

• voids in O+(8, 2), O(7, 3)

fusion of scheme on pairs of disjoint planes in O+(6, 2) schemes from Golay codes shortest vectors of E6, E7, E8, Leech lattice, Martinet lattice bipartite doubles of SRG’s with q1

11 = 0 (strongly regular

subconstituents)

Jason Williford University of Wyoming Q-polynomial Association Schemes

Linked systems of symmetric designs

A linked system of symmetric designs can be thought of as a multipartite graph on the vertex set V0 ∪ · · · ∪ Vl, l ≥ 1 such that: The induced subgraph on Vi ∪ Vj for i = j is the incidence graph of a 2 − (v, k, λ) symmetric design. There are constants σ, τ such that for all distinct i, j, k, x ∈ Γi, y ∈ Γj we have |Γ(x) ∩ Γ(y) ∩ Vk| = σ or τ depending

• n whether x and y are adjacent or not adjacent, respectively.

For l = 1 the second condition is vacuous and we simply have the incidence graph of a symmetric design. Note that a necessary condition for l ≥ 2 is that the symmetric design has two-intersection sets of cardinality k with intersection sizes σ, τ.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Linked systems of symmetric designs

A linked system of symmetric designs can be thought of as a multipartite graph on the vertex set V0 ∪ · · · ∪ Vl, l ≥ 1 such that: The induced subgraph on Vi ∪ Vj for i = j is the incidence graph of a 2 − (v, k, λ) symmetric design. There are constants σ, τ such that for all distinct i, j, k, x ∈ Γi, y ∈ Γj we have |Γ(x) ∩ Γ(y) ∩ Vk| = σ or τ depending

• n whether x and y are adjacent or not adjacent, respectively.

For l = 1 the second condition is vacuous and we simply have the incidence graph of a symmetric design. Note that a necessary condition for l ≥ 2 is that the symmetric design has two-intersection sets of cardinality k with intersection sizes σ, τ.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Linked systems of symmetric designs

Theorem (van Dam ’99) Every Q-antipodal 3-class Q-polynomial association scheme arises from a linked system of symmetric designs. Theorem (Cameron ’72) There is a system of linked (22t+2, 22t+1 − 2t, 22t − 2t) designs with l = 22t+1 − 1. More examples found by Mathon, and Davis, Martin Polhill, with the same design parameters. Question Which symmetric 2-designs can be extended to linked systems with l ≥ 2?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Linked systems of symmetric designs

Theorem (van Dam ’99) Every Q-antipodal 3-class Q-polynomial association scheme arises from a linked system of symmetric designs. Theorem (Cameron ’72) There is a system of linked (22t+2, 22t+1 − 2t, 22t − 2t) designs with l = 22t+1 − 1. More examples found by Mathon, and Davis, Martin Polhill, with the same design parameters. Question Which symmetric 2-designs can be extended to linked systems with l ≥ 2?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Linked systems of symmetric designs

Theorem (van Dam ’99) Every Q-antipodal 3-class Q-polynomial association scheme arises from a linked system of symmetric designs. Theorem (Cameron ’72) There is a system of linked (22t+2, 22t+1 − 2t, 22t − 2t) designs with l = 22t+1 − 1. More examples found by Mathon, and Davis, Martin Polhill, with the same design parameters. Question Which symmetric 2-designs can be extended to linked systems with l ≥ 2?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Linked systems of symmetric designs

Theorem (van Dam ’99) Every Q-antipodal 3-class Q-polynomial association scheme arises from a linked system of symmetric designs. Theorem (Cameron ’72) There is a system of linked (22t+2, 22t+1 − 2t, 22t − 2t) designs with l = 22t+1 − 1. More examples found by Mathon, and Davis, Martin Polhill, with the same design parameters. Question Which symmetric 2-designs can be extended to linked systems with l ≥ 2?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Linked systems of symmetric designs

Some open cases: 2 − (36, 15, 6), σ, τ = 8, 5 2 − (45, 12, 3), σ, τ = 1, 4 Results for known infinite families of symmetric designs? Do they even contain a 2-intersection set of the right type?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Generalized Quadrangles

A generalized quadrangle of order (s, t) is a point-line incidence structure satisfying: Each point is on t + 1 lines, and each line contains s + 1 points. Any pair of points lie together in at most one line. If P is a point not on the line l, then there is a unique line l′ such that P ∈ l′ and |l′ ∩ l| = 1 . A GQ of order (t, s) can be constructed from a quadrangle of order (s, t) by taking the points and lines of the original quadrangle as the lines and points, respectively, of the new quadrangle, with incidence reversed. This is called the dual of the GQ.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Generalized Quadrangles

A generalized quadrangle of order (s, t) is a point-line incidence structure satisfying: Each point is on t + 1 lines, and each line contains s + 1 points. Any pair of points lie together in at most one line. If P is a point not on the line l, then there is a unique line l′ such that P ∈ l′ and |l′ ∩ l| = 1 . A GQ of order (t, s) can be constructed from a quadrangle of order (s, t) by taking the points and lines of the original quadrangle as the lines and points, respectively, of the new quadrangle, with incidence reversed. This is called the dual of the GQ.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Examples of generalized quadrangles

Below are a few of the known families of GQ: W (q), the set of points of PG(3, q) together with all lines which are totally isotropic with respect to a symplectic polarity, s = t = q Q(4, q), consisting of the points and totally isotropic lines of a parabolic quadric in PG(4, q), s = t = q Q−(5, q), consisting of the points and totally isotropic lines of a elliptic quadric in PG(5, q), s = q, t = q2 H(3, q2), consisting of the points and totally isotropic lines of a hermitian variety in PG(3, q2), s = q2, t = q The quadrangles Q(4, q) and W (q) are duals of one another, as are Q−(5, q) and H(3, q2).

Jason Williford University of Wyoming Q-polynomial Association Schemes

Hemisystems of GQ

Let S be a generalized quadrangle of order (q, q2). A hemisystem

• f S is a partition of the points into two sets of equal size such

that each line has half of its points in each set. A hemisystem was constructed in the quadrangle (3, 9) by Segre in 1965, but no others were known until 2005. Theorem (Cossidente, Penttila ’05) Hemisystems exist in all classical quadrangles of order (q, q2) for odd prime powers q.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Hemisystems of GQ

Let S be a generalized quadrangle of order (q, q2). A hemisystem

• f S is a partition of the points into two sets of equal size such

that each line has half of its points in each set. A hemisystem was constructed in the quadrangle (3, 9) by Segre in 1965, but no others were known until 2005. Theorem (Cossidente, Penttila ’05) Hemisystems exist in all classical quadrangles of order (q, q2) for odd prime powers q.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Schemes from hemisystems

Theorem (Cameron, Goethals, Seidel ’78) Hemisystems give a strongly regular subgraph of the collinearity graph of the GQ. Haemers and Higman further defined the notion of a strongly regular decomposition of a strongly regular graph, and that hemisystems give an example. Theorem (van Dam, Martin, Muzychuk ’13) Nonexceptional strongly regular decompositions of strongly regular graphs into srg’s with the same parameters give Q-antipodal schemes.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Schemes from hemisystems

Theorem (Cameron, Goethals, Seidel ’78) Hemisystems give a strongly regular subgraph of the collinearity graph of the GQ. Haemers and Higman further defined the notion of a strongly regular decomposition of a strongly regular graph, and that hemisystems give an example. Theorem (van Dam, Martin, Muzychuk ’13) Nonexceptional strongly regular decompositions of strongly regular graphs into srg’s with the same parameters give Q-antipodal schemes.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Schemes from hemisystems

Theorem (Cameron, Goethals, Seidel ’78) Hemisystems give a strongly regular subgraph of the collinearity graph of the GQ. Haemers and Higman further defined the notion of a strongly regular decomposition of a strongly regular graph, and that hemisystems give an example. Theorem (van Dam, Martin, Muzychuk ’13) Nonexceptional strongly regular decompositions of strongly regular graphs into srg’s with the same parameters give Q-antipodal schemes.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Doubly subtended quadrangles

A set O of points of a GQ is called an ovoid if every line of the GQ meets the set O in one point. Let S be a generalized quadrangle of order (q, q2), containing a subquadrangle S′ of order (q, q). For any X ∈ S\S′ we have that the set θX of all points in S′ collinear with X is an ovoid of S′. We say that θX is subtended by the point X. We will call any two points which subtend the same ovoid

• antipodes. If an ovoid is subtended by two points we call it doubly

subtended, and we say that S′ is doubly subtended in S if all of the subtended ovoids θX are doubly subtended. If two points are not antipodes, the ovoids they subtend meet in 1

• r q + 1 points.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Doubly subtended quadrangles

A set O of points of a GQ is called an ovoid if every line of the GQ meets the set O in one point. Let S be a generalized quadrangle of order (q, q2), containing a subquadrangle S′ of order (q, q). For any X ∈ S\S′ we have that the set θX of all points in S′ collinear with X is an ovoid of S′. We say that θX is subtended by the point X. We will call any two points which subtend the same ovoid

• antipodes. If an ovoid is subtended by two points we call it doubly

subtended, and we say that S′ is doubly subtended in S if all of the subtended ovoids θX are doubly subtended. If two points are not antipodes, the ovoids they subtend meet in 1

• r q + 1 points.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Doubly subtended quadrangles

A set O of points of a GQ is called an ovoid if every line of the GQ meets the set O in one point. Let S be a generalized quadrangle of order (q, q2), containing a subquadrangle S′ of order (q, q). For any X ∈ S\S′ we have that the set θX of all points in S′ collinear with X is an ovoid of S′. We say that θX is subtended by the point X. We will call any two points which subtend the same ovoid

• antipodes. If an ovoid is subtended by two points we call it doubly

subtended, and we say that S′ is doubly subtended in S if all of the subtended ovoids θX are doubly subtended. If two points are not antipodes, the ovoids they subtend meet in 1

• r q + 1 points.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Doubly subtended quadrangles

A set O of points of a GQ is called an ovoid if every line of the GQ meets the set O in one point. Let S be a generalized quadrangle of order (q, q2), containing a subquadrangle S′ of order (q, q). For any X ∈ S\S′ we have that the set θX of all points in S′ collinear with X is an ovoid of S′. We say that θX is subtended by the point X. We will call any two points which subtend the same ovoid

• antipodes. If an ovoid is subtended by two points we call it doubly

subtended, and we say that S′ is doubly subtended in S if all of the subtended ovoids θX are doubly subtended. If two points are not antipodes, the ovoids they subtend meet in 1

• r q + 1 points.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Doubly subtended quadrangles

A set O of points of a GQ is called an ovoid if every line of the GQ meets the set O in one point. Let S be a generalized quadrangle of order (q, q2), containing a subquadrangle S′ of order (q, q). For any X ∈ S\S′ we have that the set θX of all points in S′ collinear with X is an ovoid of S′. We say that θX is subtended by the point X. We will call any two points which subtend the same ovoid

• antipodes. If an ovoid is subtended by two points we call it doubly

subtended, and we say that S′ is doubly subtended in S if all of the subtended ovoids θX are doubly subtended. If two points are not antipodes, the ovoids they subtend meet in 1

• r q + 1 points.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Doubly subtended quadrangles

Theorem (Matt Brown, 1998) Let S be a quadrangle of order (q, q2) which doubly subtends a subquadrangle S′ of order (q, q). Then the set

• f subtended ovoids of S′ and the rosettes of S′ (sets of ovoids

subtended by the points of a line of S\S′) forms a semipartital geometry.

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Q-bipartite schemes

(Penttila, Williford ’11) An association scheme can be constructed

• n the points of S\S′ as follows:

R1 : All pairs (X, Y ) where X and Y are not collinear and |θX ∩ θY | = 1. R2 : All pairs (X, Y ) where X and Y are not collinear and |θX ∩ θY | = q + 1. R3 : All pairs (X, Y ) where X and Y are collinear (here |θX ∩ θY | = 1). R4 : All pairs of antipodes. This is a 4-class Q-bipartite scheme which is not Q-antipodal. The quotient scheme is the SRG of elliptic quadrics of Q(4, q).

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Q-bipartite schemes

(Penttila, Williford ’11) An association scheme can be constructed

• n the points of S\S′ as follows:

R1 : All pairs (X, Y ) where X and Y are not collinear and |θX ∩ θY | = 1. R2 : All pairs (X, Y ) where X and Y are not collinear and |θX ∩ θY | = q + 1. R3 : All pairs (X, Y ) where X and Y are collinear (here |θX ∩ θY | = 1). R4 : All pairs of antipodes. This is a 4-class Q-bipartite scheme which is not Q-antipodal. The quotient scheme is the SRG of elliptic quadrics of Q(4, q).

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Q-bipartite schemes

(Penttila, Williford ’11) An association scheme can be constructed

• n the points of S\S′ as follows:

R1 : All pairs (X, Y ) where X and Y are not collinear and |θX ∩ θY | = 1. R2 : All pairs (X, Y ) where X and Y are not collinear and |θX ∩ θY | = q + 1. R3 : All pairs (X, Y ) where X and Y are collinear (here |θX ∩ θY | = 1). R4 : All pairs of antipodes. This is a 4-class Q-bipartite scheme which is not Q-antipodal. The quotient scheme is the SRG of elliptic quadrics of Q(4, q).

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Q-bipartite schemes

(Penttila, Williford ’11) An association scheme can be constructed

• n the points of S\S′ as follows:

R1 : All pairs (X, Y ) where X and Y are not collinear and |θX ∩ θY | = 1. R2 : All pairs (X, Y ) where X and Y are not collinear and |θX ∩ θY | = q + 1. R3 : All pairs (X, Y ) where X and Y are collinear (here |θX ∩ θY | = 1). R4 : All pairs of antipodes. This is a 4-class Q-bipartite scheme which is not Q-antipodal. The quotient scheme is the SRG of elliptic quadrics of Q(4, q).

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Q-bipartite schemes

(Penttila, Williford ’11) An association scheme can be constructed

• n the points of S\S′ as follows:

R1 : All pairs (X, Y ) where X and Y are not collinear and |θX ∩ θY | = 1. R2 : All pairs (X, Y ) where X and Y are not collinear and |θX ∩ θY | = q + 1. R3 : All pairs (X, Y ) where X and Y are collinear (here |θX ∩ θY | = 1). R4 : All pairs of antipodes. This is a 4-class Q-bipartite scheme which is not Q-antipodal. The quotient scheme is the SRG of elliptic quadrics of Q(4, q).

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Q-bipartite schemes

(Penttila, Williford ’11) An association scheme can be constructed

• n the points of S\S′ as follows:

R1 : All pairs (X, Y ) where X and Y are not collinear and |θX ∩ θY | = 1. R2 : All pairs (X, Y ) where X and Y are not collinear and |θX ∩ θY | = q + 1. R3 : All pairs (X, Y ) where X and Y are collinear (here |θX ∩ θY | = 1). R4 : All pairs of antipodes. This is a 4-class Q-bipartite scheme which is not Q-antipodal. The quotient scheme is the SRG of elliptic quadrics of Q(4, q).

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Q-bipartite schemes

Examples : The GQ Q(4, q) consisting of the points and lines of a parabolic quadric in PG(4, q) is doubly subtended in Q−(5, q) (the points and lines of an elliptic quadric in PG(5, q)). Certain flock GQ constructed by Kantor also doubly subtend a quadrangle isomorphic to Q(4, q) for q odd.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Relative Hemisystems

Let S be a GQ of order (q, q2) that doubly subtends a subquadrangle S′ of order (q, q). A relative hemisystem of S is a partition of the points of S\S′ into two sets such that every line meets each set in q/2 points. In this case, q must be even. Dualizing: Let S be a quadrangle of order (q2, q) containing a quadrangle S′ of order (q, q). A set H of half of the lines of S which are disjoint from S′ is called a relative hemisystem of S provided that for any point X of S exactly half of the lines through X which are disjoint from S′ are in H.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Relative Hemisystems

Let S be a GQ of order (q, q2) that doubly subtends a subquadrangle S′ of order (q, q). A relative hemisystem of S is a partition of the points of S\S′ into two sets such that every line meets each set in q/2 points. In this case, q must be even. Dualizing: Let S be a quadrangle of order (q2, q) containing a quadrangle S′ of order (q, q). A set H of half of the lines of S which are disjoint from S′ is called a relative hemisystem of S provided that for any point X of S exactly half of the lines through X which are disjoint from S′ are in H.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Primitive Q-polynomial schemes

The GQ H(3, q2) is dual to Q−(5, q). The subquadrangle Q(4, q) in Q−(5, q) corresponds to a copy of W (q) embedded in H(3, q2), and the points of the set H correspond to the lines L of H(3, q2) which do not meet the fixed copy of W (q). Theorem (Penttila, Williford ’11) Relative hemisystems of H(3, q2) with q = 2t and t > 1 give rise to primitive 3-class Q-polynomial schemes which are not generated by distance-regular graphs. For q = 2t the quadrangle H(3, q2) has a relative hemisystem.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Primitive Q-polynomial schemes

The GQ H(3, q2) is dual to Q−(5, q). The subquadrangle Q(4, q) in Q−(5, q) corresponds to a copy of W (q) embedded in H(3, q2), and the points of the set H correspond to the lines L of H(3, q2) which do not meet the fixed copy of W (q). Theorem (Penttila, Williford ’11) Relative hemisystems of H(3, q2) with q = 2t and t > 1 give rise to primitive 3-class Q-polynomial schemes which are not generated by distance-regular graphs. For q = 2t the quadrangle H(3, q2) has a relative hemisystem.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Primitive Q-polynomial schemes

The GQ H(3, q2) is dual to Q−(5, q). The subquadrangle Q(4, q) in Q−(5, q) corresponds to a copy of W (q) embedded in H(3, q2), and the points of the set H correspond to the lines L of H(3, q2) which do not meet the fixed copy of W (q). Theorem (Penttila, Williford ’11) Relative hemisystems of H(3, q2) with q = 2t and t > 1 give rise to primitive 3-class Q-polynomial schemes which are not generated by distance-regular graphs. For q = 2t the quadrangle H(3, q2) has a relative hemisystem.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Primitive Q-polynomial schemes

Theorem (Cossidente ’13) Inequivalent relative hemisystems give non-isomorphic association schemes. Theorem (Cossidente ’13) There are relative hemisystems of H(3, q2) admitting PSL(2, q) as their full automorphism group. Question Must a scheme with these parameters come from a relative hemisystem of a quadrangle?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Primitive Q-polynomial schemes

Theorem (Cossidente ’13) Inequivalent relative hemisystems give non-isomorphic association schemes. Theorem (Cossidente ’13) There are relative hemisystems of H(3, q2) admitting PSL(2, q) as their full automorphism group. Question Must a scheme with these parameters come from a relative hemisystem of a quadrangle?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Primitive Q-polynomial schemes

Theorem (Cossidente ’13) Inequivalent relative hemisystems give non-isomorphic association schemes. Theorem (Cossidente ’13) There are relative hemisystems of H(3, q2) admitting PSL(2, q) as their full automorphism group. Question Must a scheme with these parameters come from a relative hemisystem of a quadrangle?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Question Which Schurian schemes are Q-polynomial? Question Classify families of Q-polynomial schemes with unbounded d. (Hard even if P-polynomial assumed as well!)

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Schurian Q-polynomial schemes

Let V = F2n

q with a non-degenerate alternating bilinear form B,

q ≡ 1 (mod 4). The maximal totally isotropic subspaces give an association scheme (symplectic dual polar space graph). For each maximal U choose a dual basis, which gives rise to a determinant δU. For each pair of maximals U, V , let k be the co-dimension of their intersection in U and V . Let u1, . . . , un, v1, . . . , vn be bases of U and V with ui = vi for k + 1 ≤ i ≤ n. The Maslov index on pairs of maximals is given by: σ(U, V ) = χ(δU(u1, . . . , un)δV (v1, . . . , vn)det[B(ui, vj) : 1 ≤ i, j ≤ k]) .

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Schurian Q-polynomial schemes

Let V = F2n

q with a non-degenerate alternating bilinear form B,

q ≡ 1 (mod 4). The maximal totally isotropic subspaces give an association scheme (symplectic dual polar space graph). For each maximal U choose a dual basis, which gives rise to a determinant δU. For each pair of maximals U, V , let k be the co-dimension of their intersection in U and V . Let u1, . . . , un, v1, . . . , vn be bases of U and V with ui = vi for k + 1 ≤ i ≤ n. The Maslov index on pairs of maximals is given by: σ(U, V ) = χ(δU(u1, . . . , un)δV (v1, . . . , vn)det[B(ui, vj) : 1 ≤ i, j ≤ k]) .

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Schurian Q-polynomial schemes

Let V = F2n

q with a non-degenerate alternating bilinear form B,

q ≡ 1 (mod 4). The maximal totally isotropic subspaces give an association scheme (symplectic dual polar space graph). For each maximal U choose a dual basis, which gives rise to a determinant δU. For each pair of maximals U, V , let k be the co-dimension of their intersection in U and V . Let u1, . . . , un, v1, . . . , vn be bases of U and V with ui = vi for k + 1 ≤ i ≤ n. The Maslov index on pairs of maximals is given by: σ(U, V ) = χ(δU(u1, . . . , un)δV (v1, . . . , vn)det[B(ui, vj) : 1 ≤ i, j ≤ k]) .

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Schurian Q-polynomial schemes

Let X be the set of all ordered pairs (U, ǫ) where ǫ = ±1. Define the following relations: (U, ǫ1) i ∼(V , ǫ2) if dim(U ∩ V ) = n − i and σ(U, V ) = ǫ1ǫ2 for 0 ≤ i ≤ n. (U, ǫ1)2n+1−i ∼ (V , ǫ2) if dim(U ∩ V ) = i and σ(U, V ) = −ǫ1ǫ2 for 0 ≤ i ≤ n. Theorem (Moorhouse, W) This scheme is Q-bipartite for all n > 1 where d = 2n + 1. These schemes have two Q-polynomial orderings; the second is: E0, Ed, E2, Ed−2, E4, Ed−4, . . . ,Ed−5, E5, Ed−3, E3, Ed−1, E1 (Suzuki type 3). For q nonsquare, the splitting field is quadratic and the second

• rdering is given by conjugation.

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Schurian Q-polynomial schemes

Let X be the set of all ordered pairs (U, ǫ) where ǫ = ±1. Define the following relations: (U, ǫ1) i ∼(V , ǫ2) if dim(U ∩ V ) = n − i and σ(U, V ) = ǫ1ǫ2 for 0 ≤ i ≤ n. (U, ǫ1)2n+1−i ∼ (V , ǫ2) if dim(U ∩ V ) = i and σ(U, V ) = −ǫ1ǫ2 for 0 ≤ i ≤ n. Theorem (Moorhouse, W) This scheme is Q-bipartite for all n > 1 where d = 2n + 1. These schemes have two Q-polynomial orderings; the second is: E0, Ed, E2, Ed−2, E4, Ed−4, . . . ,Ed−5, E5, Ed−3, E3, Ed−1, E1 (Suzuki type 3). For q nonsquare, the splitting field is quadratic and the second

• rdering is given by conjugation.

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Schurian Q-polynomial schemes

Let X be the set of all ordered pairs (U, ǫ) where ǫ = ±1. Define the following relations: (U, ǫ1) i ∼(V , ǫ2) if dim(U ∩ V ) = n − i and σ(U, V ) = ǫ1ǫ2 for 0 ≤ i ≤ n. (U, ǫ1)2n+1−i ∼ (V , ǫ2) if dim(U ∩ V ) = i and σ(U, V ) = −ǫ1ǫ2 for 0 ≤ i ≤ n. Theorem (Moorhouse, W) This scheme is Q-bipartite for all n > 1 where d = 2n + 1. These schemes have two Q-polynomial orderings; the second is: E0, Ed, E2, Ed−2, E4, Ed−4, . . . ,Ed−5, E5, Ed−3, E3, Ed−1, E1 (Suzuki type 3). For q nonsquare, the splitting field is quadratic and the second

• rdering is given by conjugation.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Not P-polynomial, but close

L1 =                     b0 . . . 1

a1 2

b1 . . .

a1 2

c2

a2 2

... . . . . . . ... ... bd−1

ad−1 2

cd

ad 2 ad 2 ad 2 ad 2

cd

ad−1 2

bd−1 ... ... . . . ...

a2 2

c2

a1 2

b1

a1 2

1 b0                    

Jason Williford University of Wyoming Q-polynomial Association Schemes

The Symplectic Group

Let q be a prime power and V = (F)2n

q with non-degenerate

alternating bilinear form B, B(x, y) = x1yn+1 − xn+1y1 + · · · + xny2n − x2nyn. The symplectic group PSp(V ) consists of the block matrices A B C D

• satisfying

A B C D T 0 I −I A B C D

• =

I −I

• Jason Williford University of Wyoming

Q-polynomial Association Schemes

The Symplectic Group

Let q be a prime power and V = (F)2n

q with non-degenerate

alternating bilinear form B, B(x, y) = x1yn+1 − xn+1y1 + · · · + xny2n − x2nyn. The symplectic group PSp(V ) consists of the block matrices A B C D

• satisfying

A B C D T 0 I −I A B C D

• =

I −I

• Jason Williford University of Wyoming

Q-polynomial Association Schemes

The Symplectic Group

The symplectic group G = PSp(V ) consists of the block matrices A B C D

• satisfying

ATC − C TA = BTD − DTB = 0 and ATD − C TB = I. The stabilizer S of the maximal isotropic subspace U =< e1, . . . , en > is A B D

• satisfying ATD = I and BTD − DTB = 0.

Jason Williford University of Wyoming Q-polynomial Association Schemes

The Symplectic Group

The symplectic group G = PSp(V ) consists of the block matrices A B C D

• satisfying

ATC − C TA = BTD − DTB = 0 and ATD − C TB = I. The stabilizer S of the maximal isotropic subspace U =< e1, . . . , en > is A B D

• satisfying ATD = I and BTD − DTB = 0.

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Schurian Q-polynomial schemes

The permutation group given by the action of G on the cosets of S is generously transitive, and yields the association scheme from the symplectic dual polar space graphs. Now we choose q to be odd, and let S′ be the subgroup of S of matrices A B D

• satisfying ATD = I, BTD − DTB = 0 and |A| is a

square. Now we let G act on S′. For q ≡ 1 (mod 4) we obtain the aforementioned scheme.

Jason Williford University of Wyoming Q-polynomial Association Schemes

A family of Schurian Q-polynomial schemes

The permutation group given by the action of G on the cosets of S is generously transitive, and yields the association scheme from the symplectic dual polar space graphs. Now we choose q to be odd, and let S′ be the subgroup of S of matrices A B D

• satisfying ATD = I, BTD − DTB = 0 and |A| is a

square. Now we let G act on S′. For q ≡ 1 (mod 4) we obtain the aforementioned scheme.

Jason Williford University of Wyoming Q-polynomial Association Schemes

A Hypothetical Primitive Family

Question Are these schemes the extended Q-bipartite doubles of primitive Q-polynomial schemes for square q?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Parameters for q = 9

L1 =       60 1 3 2 54 4 2 54 8 28 24 5 30 25       L2 =       30 2 1 27 1 2 27 2 16 12 5 15 10       L3 =       405 54 189 162 27 216 162 1 28 16 216 144 30 15 180 180       L4 =       324 27 162 135 54 162 108 24 12 144 144 1 25 10 180 108      

Jason Williford University of Wyoming Q-polynomial Association Schemes