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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

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Page 1: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Q-polynomial Association Schemes

Jason WillifordUniversity of Wyoming

Modern Trends in Algebraic Graph TheoryJune 4th, 2014

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 2: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Association Schemes

A d-class symmetric association scheme is a collection ofrelations R0, ...,Rd on a set X which satisfy:

The relations R0, . . .Rd partition X × X

R0 is the identity relation on X

Ri = RTi

there are constants pkij such that for all x

k∼y we have that

there are exactly pkij points z such that x

i∼z and zj∼y .

k

i j

kpij

x y

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 3: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Schurian Schemes

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

Page 4: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Imprimitivity

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

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 5: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

The Bose-Mesner Algebra

Let A0, . . .Ad be the adjacency matrices of the relationsR0, . . . ,Rd of an association scheme.These matrices satisfy:

AiAj =∑

k pkij Ak

Let A denote the span of the matrices A0, . . .Ad over the realnumbers.

Since the matrices are all symmetric, the matrix algebra A iscommutative, and is called the Bose-Mesner algebra of theassociation scheme.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 6: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

The Bose-Mesner Algebra

Let A0, . . .Ad be the adjacency matrices of the relationsR0, . . . ,Rd of an association scheme.These matrices satisfy:

AiAj =∑

k pkij Ak

Let A denote the span of the matrices A0, . . .Ad over the realnumbers.

Since the matrices are all symmetric, the matrix algebra A iscommutative, and is called the Bose-Mesner algebra of theassociation scheme.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 7: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Primitive Idempotents

The Bose-Mesner algebra A of an association scheme is acommutative algebra of real symmetric matrices, and so can besimultaneously diagonalized by a real orthogonal matrix.

This gives a decomposition of R|X | into d + 1 orthogonal commoneigenspaces V0,V1, . . .Vd .

Since all of the graphs are regular and J is in A, one of theseeigenspaces is the one-dimensional eigenspaceV0 =< (1, 1, . . . , 1) >.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 8: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Primitive Idempotents

The Bose-Mesner algebra A of an association scheme is acommutative algebra of real symmetric matrices, and so can besimultaneously diagonalized by a real orthogonal matrix.

This gives a decomposition of R|X | into d + 1 orthogonal commoneigenspaces V0,V1, . . .Vd .

Since all of the graphs are regular and J is in A, one of theseeigenspaces is the one-dimensional eigenspaceV0 =< (1, 1, . . . , 1) >.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 9: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Primitive Idempotents

The primitive idempotents E0,E1, . . . ,Ed which project onto theseeigenspaces can also be seen to be in A, and form another basis ofA.

Since the idempotents project onto orthogonal eigenspaces, wehave EiEj = 0 if i 6= j and EiEj = Ei if i = j .

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 10: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Primitive Idempotents

The primitive idempotents E0,E1, . . . ,Ed which project onto theseeigenspaces can also be seen to be in A, and form another basis ofA.

Since the idempotents project onto orthogonal eigenspaces, wehave EiEj = 0 if i 6= j and EiEj = Ei if i = j .

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 11: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Entrywise Multiplication

The Bose-Mesner Algebra of an association scheme is also closedunder entrywise multiplication, since for all i , j we have Ai ◦ Aj = 0if i 6= j , and Ai ◦ Aj = Ai if i = j .

We define the Krein parameters qkij to satisfy :

Ei ◦ Ej = 1|X |∑d

k=0 qkij Ek

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

ii are equal tothe multiplicities of the eigenvalues of the scheme, and so must bepositive integers.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 12: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Entrywise Multiplication

The Bose-Mesner Algebra of an association scheme is also closedunder entrywise multiplication, since for all i , j we have Ai ◦ Aj = 0if i 6= j , and Ai ◦ Aj = Ai if i = j .

We define the Krein parameters qkij to satisfy :

Ei ◦ Ej = 1|X |∑d

k=0 qkij Ek

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

ii are equal tothe multiplicities of the eigenvalues of the scheme, and so must bepositive integers.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 13: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Entrywise Multiplication

The Bose-Mesner Algebra of an association scheme is also closedunder entrywise multiplication, since for all i , j we have Ai ◦ Aj = 0if i 6= j , and Ai ◦ Aj = Ai if i = j .

We define the Krein parameters qkij to satisfy :

Ei ◦ Ej = 1|X |∑d

k=0 qkij Ek

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

ii are equal tothe multiplicities of the eigenvalues of the scheme, and so must bepositive integers.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 14: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Formal Duality

The matrices Ai satisfy:

A0 = I∑di=0 Ai = J

AiAj =∑

k pkij Ak

Ai ◦ Aj = δijAi

The matrices Ei satisfy:

E0 = 1|X |J∑d

i=0 Ei = I

Ei ◦ Ej = 1|X |∑

k qkij Ek

EiEj = δijEi

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 15: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Formal Duality

The matrices Ai satisfy:

A0 = I∑di=0 Ai = J

AiAj =∑

k pkij Ak

Ai ◦ Aj = δijAi

The matrices Ei satisfy:

E0 = 1|X |J∑d

i=0 Ei = I

Ei ◦ Ej = 1|X |∑

k qkij Ek

EiEj = δijEi

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 16: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Bipartite double

We can construct new association schemes from old usingKronecker products.

The bipartite double of a scheme {A0, . . . ,Ad} has matrices:(Ai 00 Ai

)and

(0 Ai

Ai 0

)for 0 ≤ i ≤ d .

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 17: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Polynomial Association Schemes

An association scheme is called P-polynomial (metric) providedthat, after suitably reordering the Ai , there are polynomials pk(x)of degree k for 0 ≤ k ≤ d such that Ak = pk(A1). This isequivalent to saying R1 is a distance-regular graph of diameter d .

An association scheme is called Q-polynomial (cometric) providedthat, after suitably reordering the Ei , there are polynomials qk(x)of degree k for 0 ≤ k ≤ d such that Ek = qk(E1), wheremultiplication is done entrywise.

Conjecture

(Bannai, Ito) For sufficiently large d, a primitive scheme isP-polynomial if and only if it is Q-polynomial.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 18: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Polynomial Association Schemes

An association scheme is called P-polynomial (metric) providedthat, after suitably reordering the Ai , there are polynomials pk(x)of degree k for 0 ≤ k ≤ d such that Ak = pk(A1). This isequivalent to saying R1 is a distance-regular graph of diameter d .

An association scheme is called Q-polynomial (cometric) providedthat, after suitably reordering the Ei , there are polynomials qk(x)of degree k for 0 ≤ k ≤ d such that Ek = qk(E1), wheremultiplication is done entrywise.

Conjecture

(Bannai, Ito) For sufficiently large d, a primitive scheme isP-polynomial if and only if it is Q-polynomial.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 19: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Polynomial Association Schemes

An association scheme is called P-polynomial (metric) providedthat, after suitably reordering the Ai , there are polynomials pk(x)of degree k for 0 ≤ k ≤ d such that Ak = pk(A1). This isequivalent to saying R1 is a distance-regular graph of diameter d .

An association scheme is called Q-polynomial (cometric) providedthat, after suitably reordering the Ei , there are polynomials qk(x)of degree k for 0 ≤ k ≤ d such that Ek = qk(E1), wheremultiplication is done entrywise.

Conjecture

(Bannai, Ito) For sufficiently large d, a primitive scheme isP-polynomial if and only if it is Q-polynomial.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 20: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

T -designs in Q-polynomial Schemes

Let Y be a subset of a Q-polynomial scheme, let χ be thecharacteristic vector of Y , and let T ⊂ {1, . . . d}. The set Y iscalled a T -design provided that Eiχ = 0 for all i ∈ T . IfT = {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 at − (v , k, λ) design for some λ.

Delsarte’s “conjecture”: T -designs will often have interestingproperties.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 21: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

T -designs in Q-polynomial Schemes

Let Y be a subset of a Q-polynomial scheme, let χ be thecharacteristic vector of Y , and let T ⊂ {1, . . . d}. The set Y iscalled a T -design provided that Eiχ = 0 for all i ∈ T . IfT = {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 at − (v , k , λ) design for some λ.

Delsarte’s “conjecture”: T -designs will often have interestingproperties.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 22: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

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 relationsoccurring in Y , and suppose Eiχ = 0 for all 1 ≤ i ≤ t, where χ isthe characteristic vector of Y . If t ≥ 2s − 2 then Y is aQ-polynomial subscheme.

Theorem

(Brouwer, Godsil, Koolen, Martin 2003) Let Y be a subset of aQ-polynomial scheme. Define the degree s of Y to be the numberof nontrivial relations occurring in Y , and supposew∗ = max{i : Eiχ 6= 0} where χ is the characteristic vector of Y .Then w∗ ≥ d − s. If equality holds, then Y is a Q-polynomialsubscheme.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 23: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

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 relationsoccurring in Y , and suppose Eiχ = 0 for all 1 ≤ i ≤ t, where χ isthe characteristic vector of Y . If t ≥ 2s − 2 then Y is aQ-polynomial subscheme.

Theorem

(Brouwer, Godsil, Koolen, Martin 2003) Let Y be a subset of aQ-polynomial scheme. Define the degree s of Y to be the numberof nontrivial relations occurring in Y , and supposew∗ = max{i : Eiχ 6= 0} where χ is the characteristic vector of Y .Then w∗ ≥ d − s. If equality holds, then Y is a Q-polynomialsubscheme.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 24: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Splitting Fields of Q-polynomial Schemes

The splitting field of an association scheme is the field generatedby the eigenvalues of the scheme.

Theorem

(Suzuki ’98) A Q-polynomial scheme can have at most 2Q-polynomial orderings.

A corollary of this is that the splitting field of a Q-polynomialscheme is at most a degree 2 extension of the rationals.

Theorem

(Martin, W ’09) For each integer m > 2 there are finitely manyQ-polynomial schemes with rk(E1) = m.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 25: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Splitting Fields of Q-polynomial Schemes

The splitting field of an association scheme is the field generatedby the eigenvalues of the scheme.

Theorem

(Suzuki ’98) A Q-polynomial scheme can have at most 2Q-polynomial orderings.

A corollary of this is that the splitting field of a Q-polynomialscheme is at most a degree 2 extension of the rationals.

Theorem

(Martin, W ’09) For each integer m > 2 there are finitely manyQ-polynomial schemes with rk(E1) = m.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 26: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Splitting Fields of Q-polynomial Schemes

The splitting field of an association scheme is the field generatedby the eigenvalues of the scheme.

Theorem

(Suzuki ’98) A Q-polynomial scheme can have at most 2Q-polynomial orderings.

A corollary of this is that the splitting field of a Q-polynomialscheme is at most a degree 2 extension of the rationals.

Theorem

(Martin, W ’09) For each integer m > 2 there are finitely manyQ-polynomial schemes with rk(E1) = m.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 27: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Parameters of Distance-Regular Graphs

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

i

(x)

i-1 (x)

c b

xy

i

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

i

The constants relate to the pkij of the resulting association scheme

by: bi = pi1,i+1, ci = pi

1,i−1, ai = pi1,i .

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 28: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Parameters of Distance-Regular Graphs

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

i

(x)

i-1 (x)

c b

xy

i

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

i

The constants relate to the pkij of the resulting association scheme

by: bi = pi1,i+1, ci = pi

1,i−1, ai = pi1,i .

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 29: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Imprimitivite Distance-Regular Graphs

An imprimitive distance-regular graph Γ of diameter d which is nota cycle falls into one of three types:

Γ is bipartite, meaning the vertices can be partitioned into twosets so that all edges have an endpoint in each set. This isequivalent to having pk

ij = 0 whenever i + j + k is odd. Thedistance two graph of Γ is called the halved graph, and is alsodistance-regular.

Γ is antipodal, meaning that the graph Γd is a union ofcomplete graphs (distance d in the graph Γ is an equivalencerelation). This is equivalent to having bj = cd−j for all jexcept possibly j = bd2 c. The antipodal quotient is alsodistance-regular, and called the folded graph.

Γ is both antipodal and bipartite.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 30: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Imprimitivite Distance-Regular Graphs

An imprimitive distance-regular graph Γ of diameter d which is nota cycle falls into one of three types:

Γ is bipartite, meaning the vertices can be partitioned into twosets so that all edges have an endpoint in each set. This isequivalent to having pk

ij = 0 whenever i + j + k is odd. Thedistance two graph of Γ is called the halved graph, and is alsodistance-regular.

Γ is antipodal, meaning that the graph Γd is a union ofcomplete graphs (distance d in the graph Γ is an equivalencerelation). This is equivalent to having bj = cd−j for all jexcept possibly j = bd2 c. The antipodal quotient is alsodistance-regular, and called the folded graph.

Γ is both antipodal and bipartite.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 31: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Imprimitivite Distance-Regular Graphs

An imprimitive distance-regular graph Γ of diameter d which is nota cycle falls into one of three types:

Γ is bipartite, meaning the vertices can be partitioned into twosets so that all edges have an endpoint in each set. This isequivalent to having pk

ij = 0 whenever i + j + k is odd. Thedistance two graph of Γ is called the halved graph, and is alsodistance-regular.

Γ is antipodal, meaning that the graph Γd is a union ofcomplete graphs (distance d in the graph Γ is an equivalencerelation). This is equivalent to having bj = cd−j for all jexcept possibly j = bd2 c. The antipodal quotient is alsodistance-regular, and called the folded graph.

Γ is both antipodal and bipartite.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 32: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Q-polynomial Association Schemes

An association scheme is called Q-polynomial provided that, aftersuitably reordering the Ei , there are polynomials qk(x) of degree kfor 0 ≤ k ≤ d such that Ek = qk(E1), where multiplication is doneentrywise.

We define constants b∗i and c∗i by: b∗i = qi1,i+1, c∗i = qi

1,i−1,

a∗i = qi1,i .

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

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 33: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

The Imprimitive case

Theorem

(Suzuki ’98) An imprimitive Q-polynomial association schemewhich is not a cycle is one of the following:

Q-bipartite, where qkij = 0 when i + j + k is odd.

Q-antipodal, where b∗j = c∗d−j for all j except possibly

j = bd2 c.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 ford = 4 do not exist.

Theorem

(Tanaka, Tanaka ’11) The hypothetical family of exceptions ford = 6 do not exist.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 34: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

The Imprimitive case

Theorem

(Suzuki ’98) An imprimitive Q-polynomial association schemewhich is not a cycle is one of the following:

Q-bipartite, where qkij = 0 when i + j + k is odd.

Q-antipodal, where b∗j = c∗d−j for all j except possibly

j = bd2 c.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 ford = 4 do not exist.

Theorem

(Tanaka, Tanaka ’11) The hypothetical family of exceptions ford = 6 do not exist.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 35: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

The Imprimitive case

Theorem

(Suzuki ’98) An imprimitive Q-polynomial association schemewhich is not a cycle is one of the following:

Q-bipartite, where qkij = 0 when i + j + k is odd.

Q-antipodal, where b∗j = c∗d−j for all j except possibly

j = bd2 c.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 ford = 4 do not exist.

Theorem

(Tanaka, Tanaka ’11) The hypothetical family of exceptions ford = 6 do not exist.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 36: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Q-antipodal schemes

Theorem

(Martin, Muzychuk, W ’07) A Q-antipodal scheme is“dismantlable”: it can be partitioned into w > 1 Q-antipodalclasses of equal size, with bd2 c nontrivial relations occurringbetween vertices in the same class, and the rest occurring betweenvertices in different classes, where any collection of theQ-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

Page 37: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Q-antipodal schemes

Theorem

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

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 38: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Extended Q-bipartite double

Theorem

An imprimitive Q-polynomial scheme is Q-bipartite if and only ifthere is a relation which is a perfect matching.

Theorem

(Martin, Muzychuk, W ’07) If a Q-polynomial scheme satisfiesb∗j + c∗j+1 = m1 + 1 for all j < d then there is a Q-bipartite schemewith d + 1 classes which can be built from the original scheme. Itis 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 00 A0

),

(Ai Ad+1−i

Ad+1−i Ai

), and

(0 A0

A0 0

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

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 39: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Extended Q-bipartite double

Theorem

An imprimitive Q-polynomial scheme is Q-bipartite if and only ifthere is a relation which is a perfect matching.

Theorem

(Martin, Muzychuk, W ’07) If a Q-polynomial scheme satisfiesb∗j + c∗j+1 = m1 + 1 for all j < d then there is a Q-bipartite schemewith d + 1 classes which can be built from the original scheme. Itis 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 00 A0

),

(Ai Ad+1−i

Ad+1−i Ai

), and

(0 A0

A0 0

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

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Extended Q-bipartite double

Theorem

An imprimitive Q-polynomial scheme is Q-bipartite if and only ifthere is a relation which is a perfect matching.

Theorem

(Martin, Muzychuk, W ’07) If a Q-polynomial scheme satisfiesb∗j + c∗j+1 = m1 + 1 for all j < d then there is a Q-bipartite schemewith d + 1 classes which can be built from the original scheme. Itis 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 00 A0

),

(Ai Ad+1−i

Ad+1−i Ai

), and

(0 A0

A0 0

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

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Schemes which are Q-antipodal and Q-bipartite

Theorem

(Martin, Le Compte, Owens ’10) An association scheme with 4classes which is Q-antipodal and Q-bipartite must be a schemegenerated by a set of mutually unbiased bases of a real vectorspace. Furthermore, any set of k > 1 mutually unbiased basesgives rise to such a scheme, and this scheme is not P-polynomialfor k > 2.

Theorem

(Bannai, Ito ’84) The bipartite double of an almost dual bipartitescheme (Q-polynomial, a∗1 = . . . a∗d−1 = 0 but a∗d 6= 0) isQ-bipartite and Q-antipodal. In particular, the bipartite doubles ofthe hermitian dual polar space graph is q-polynomial for all d.

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Schemes which are Q-antipodal and Q-bipartite

Theorem

(Martin, Le Compte, Owens ’10) An association scheme with 4classes which is Q-antipodal and Q-bipartite must be a schemegenerated by a set of mutually unbiased bases of a real vectorspace. Furthermore, any set of k > 1 mutually unbiased basesgives rise to such a scheme, and this scheme is not P-polynomialfor k > 2.

Theorem

(Bannai, Ito ’84) The bipartite double of an almost dual bipartitescheme (Q-polynomial, a∗1 = . . . a∗d−1 = 0 but a∗d 6= 0) isQ-bipartite and Q-antipodal. In particular, the bipartite doubles ofthe hermitian dual polar space graph is q-polynomial for all d.

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Examples of schemes that are P and Q-polynomial

distance-regular graphs with classical parameters (Hamming,Johnson, bilinear, quadratic forms, Grassmann, dual polarspace, 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

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Page 44: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

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 regulargraphs, 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) *

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Page 45: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

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 regulargraphs, 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

Page 46: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

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 regulargraphs, 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

Page 47: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

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 regulargraphs, 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) *

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Sporadic primitive strictly Q-polynomial schemes

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

4 is a primitive 5-class Q-polynomial scheme.The subgraph of the previous scheme formed by taking allvectors which have angles from a fixed vector satisfyingcos(θ) = −1

3 is a primitive 4-class Q-polynomial scheme.The subgraph of the previous 5-class scheme formed by takingall vectors which have angles from a fixed vector satisfyingcos(θ) = 7

15 is a primitive 3-class Q-polynomial scheme.The block scheme of the 4− (11, 5, 1) design is a primitive3-class Q-polynomial scheme.The block scheme of the 4− (47, 11, 8) design arising from aquadratic residue code is a primitive 3-class scheme.

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Sporadic strictly Q-polynomial schemes

ovoids 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 q111 = 0 (strongly regular

subconstituents)

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Linked systems of symmetric designs

A linked system of symmetric designs can be thought of as amultipartite graph on the vertex set V0 ∪ · · · ∪ Vl , l ≥ 1 such that:

The induced subgraph on Vi ∪ Vj for i 6= j is the incidencegraph 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 dependingon whether x and y are adjacent or not adjacent, respectively.

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

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Linked systems of symmetric designs

A linked system of symmetric designs can be thought of as amultipartite graph on the vertex set V0 ∪ · · · ∪ Vl , l ≥ 1 such that:

The induced subgraph on Vi ∪ Vj for i 6= j is the incidencegraph 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 dependingon whether x and y are adjacent or not adjacent, respectively.

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

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Linked systems of symmetric designs

Theorem

(van Dam ’99) Every Q-antipodal 3-class Q-polynomial associationscheme 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, withthe same design parameters.

Question

Which symmetric 2-designs can be extended to linked systems withl ≥ 2?

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Linked systems of symmetric designs

Theorem

(van Dam ’99) Every Q-antipodal 3-class Q-polynomial associationscheme 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, withthe same design parameters.

Question

Which symmetric 2-designs can be extended to linked systems withl ≥ 2?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 54: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Linked systems of symmetric designs

Theorem

(van Dam ’99) Every Q-antipodal 3-class Q-polynomial associationscheme 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, withthe same design parameters.

Question

Which symmetric 2-designs can be extended to linked systems withl ≥ 2?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 55: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Linked systems of symmetric designs

Theorem

(van Dam ’99) Every Q-antipodal 3-class Q-polynomial associationscheme 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, withthe same design parameters.

Question

Which symmetric 2-designs can be extended to linked systems withl ≥ 2?

Jason Williford University of Wyoming Q-polynomial Association Schemes

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Linked systems of symmetric designs

Some open cases:

2− (36, 15, 6), σ, τ = 8, 52− (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?

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Generalized Quadrangles

A generalized quadrangle of order (s, t) is a point-line incidencestructure satisfying:

Each point is on t + 1 lines, and each line contains s + 1points.

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 asthe lines and points, respectively, of the new quadrangle, withincidence reversed. This is called the dual of the GQ.

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Generalized Quadrangles

A generalized quadrangle of order (s, t) is a point-line incidencestructure satisfying:

Each point is on t + 1 lines, and each line contains s + 1points.

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 asthe lines and points, respectively, of the new quadrangle, withincidence reversed. This is called the dual of the GQ.

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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 lineswhich are totally isotropic with respect to a symplecticpolarity, s = t = q

Q(4, q), consisting of the points and totally isotropic lines of aparabolic quadric in PG (4, q), s = t = q

Q−(5, q), consisting of the points and totally isotropic lines ofa elliptic quadric in PG (5, q), s = q, t = q2

H(3, q2), consisting of the points and totally isotropic lines ofa hermitian variety in PG (3, q2), s = q2, t = q

The quadrangles Q(4, q) and W (q) are duals of one another, asare Q−(5, q) and H(3, q2).

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Hemisystems of GQ

Let S be a generalized quadrangle of order (q, q2). A hemisystemof S is a partition of the points into two sets of equal size suchthat each line has half of its points in each set.

A hemisystem was constructed in the quadrangle (3, 9) by Segre in1965, but no others were known until 2005.

Theorem

(Cossidente, Penttila ’05) Hemisystems exist in all classicalquadrangles of order (q, q2) for odd prime powers q.

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Hemisystems of GQ

Let S be a generalized quadrangle of order (q, q2). A hemisystemof S is a partition of the points into two sets of equal size suchthat each line has half of its points in each set.

A hemisystem was constructed in the quadrangle (3, 9) by Segre in1965, but no others were known until 2005.

Theorem

(Cossidente, Penttila ’05) Hemisystems exist in all classicalquadrangles of order (q, q2) for odd prime powers q.

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Schemes from hemisystems

Theorem

(Cameron, Goethals, Seidel ’78) Hemisystems give a stronglyregular subgraph of the collinearity graph of the GQ.

Haemers and Higman further defined the notion of a stronglyregular decomposition of a strongly regular graph, and thathemisystems give an example.

Theorem

(van Dam, Martin, Muzychuk ’13) Nonexceptional strongly regulardecompositions of strongly regular graphs into srg’s with the sameparameters give Q-antipodal schemes.

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Schemes from hemisystems

Theorem

(Cameron, Goethals, Seidel ’78) Hemisystems give a stronglyregular subgraph of the collinearity graph of the GQ.

Haemers and Higman further defined the notion of a stronglyregular decomposition of a strongly regular graph, and thathemisystems give an example.

Theorem

(van Dam, Martin, Muzychuk ’13) Nonexceptional strongly regulardecompositions of strongly regular graphs into srg’s with the sameparameters give Q-antipodal schemes.

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Schemes from hemisystems

Theorem

(Cameron, Goethals, Seidel ’78) Hemisystems give a stronglyregular subgraph of the collinearity graph of the GQ.

Haemers and Higman further defined the notion of a stronglyregular decomposition of a strongly regular graph, and thathemisystems give an example.

Theorem

(van Dam, Martin, Muzychuk ’13) Nonexceptional strongly regulardecompositions of strongly regular graphs into srg’s with the sameparameters give Q-antipodal schemes.

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Doubly subtended quadrangles

A set O of points of a GQ is called an ovoid if every line of the GQmeets the set O in one point.

Let S be a generalized quadrangle of order (q, q2), containing asubquadrangle 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 bythe point X .

We will call any two points which subtend the same ovoidantipodes. If an ovoid is subtended by two points we call it doublysubtended, and we say that S ′ is doubly subtended in S if all of thesubtended ovoids θX are doubly subtended.If two points are not antipodes, the ovoids they subtend meet in 1or q + 1 points.

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Doubly subtended quadrangles

A set O of points of a GQ is called an ovoid if every line of the GQmeets the set O in one point.

Let S be a generalized quadrangle of order (q, q2), containing asubquadrangle 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 bythe point X .

We will call any two points which subtend the same ovoidantipodes. If an ovoid is subtended by two points we call it doublysubtended, and we say that S ′ is doubly subtended in S if all of thesubtended ovoids θX are doubly subtended.If two points are not antipodes, the ovoids they subtend meet in 1or q + 1 points.

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Doubly subtended quadrangles

A set O of points of a GQ is called an ovoid if every line of the GQmeets the set O in one point.

Let S be a generalized quadrangle of order (q, q2), containing asubquadrangle 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 bythe point X .

We will call any two points which subtend the same ovoidantipodes. If an ovoid is subtended by two points we call it doublysubtended, and we say that S ′ is doubly subtended in S if all of thesubtended ovoids θX are doubly subtended.If two points are not antipodes, the ovoids they subtend meet in 1or q + 1 points.

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Doubly subtended quadrangles

A set O of points of a GQ is called an ovoid if every line of the GQmeets the set O in one point.

Let S be a generalized quadrangle of order (q, q2), containing asubquadrangle 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 bythe point X .

We will call any two points which subtend the same ovoidantipodes. If an ovoid is subtended by two points we call it doublysubtended, and we say that S ′ is doubly subtended in S if all of thesubtended ovoids θX are doubly subtended.If two points are not antipodes, the ovoids they subtend meet in 1or q + 1 points.

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Doubly subtended quadrangles

A set O of points of a GQ is called an ovoid if every line of the GQmeets the set O in one point.

Let S be a generalized quadrangle of order (q, q2), containing asubquadrangle 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 bythe point X .

We will call any two points which subtend the same ovoidantipodes. If an ovoid is subtended by two points we call it doublysubtended, and we say that S ′ is doubly subtended in S if all of thesubtended ovoids θX are doubly subtended.If two points are not antipodes, the ovoids they subtend meet in 1or q + 1 points.

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Doubly subtended quadrangles

Theorem

(Matt Brown, 1998) Let S be a quadrangle of order (q, q2) whichdoubly subtends a subquadrangle S ′ of order (q, q). Then the setof subtended ovoids of S ′ and the rosettes of S ′ (sets of ovoidssubtended by the points of a line of S\S ′) forms a semipartitalgeometry.

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A family of Q-bipartite schemes

(Penttila, Williford ’11) An association scheme can be constructedon 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. Thequotient scheme is the SRG of elliptic quadrics of Q(4, q).

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A family of Q-bipartite schemes

(Penttila, Williford ’11) An association scheme can be constructedon 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. Thequotient scheme is the SRG of elliptic quadrics of Q(4, q).

Jason Williford University of Wyoming Q-polynomial Association Schemes

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A family of Q-bipartite schemes

(Penttila, Williford ’11) An association scheme can be constructedon 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. Thequotient scheme is the SRG of elliptic quadrics of Q(4, q).

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 74: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

A family of Q-bipartite schemes

(Penttila, Williford ’11) An association scheme can be constructedon 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. Thequotient scheme is the SRG of elliptic quadrics of Q(4, q).

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 75: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

A family of Q-bipartite schemes

(Penttila, Williford ’11) An association scheme can be constructedon 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. Thequotient scheme is the SRG of elliptic quadrics of Q(4, q).

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 76: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

A family of Q-bipartite schemes

(Penttila, Williford ’11) An association scheme can be constructedon 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. Thequotient scheme is the SRG of elliptic quadrics of Q(4, q).

Jason Williford University of Wyoming Q-polynomial Association Schemes

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A family of Q-bipartite schemes

Examples : The GQ Q(4, q) consisting of the points and lines of aparabolic quadric in PG (4, q) is doubly subtended in Q−(5, q) (thepoints and lines of an elliptic quadric in PG (5, q)).

Certain flock GQ constructed by Kantor also doubly subtend aquadrangle isomorphic to Q(4, q) for q odd.

Jason Williford University of Wyoming Q-polynomial Association Schemes

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Relative Hemisystems

Let S be a GQ of order (q, q2) that doubly subtends asubquadrangle S ′ of order (q, q). A relative hemisystem of S is apartition of the points of S\S ′ into two sets such that every linemeets each set in q/2 points. In this case, q must be even.

Dualizing: Let S be a quadrangle of order (q2, q) containing aquadrangle S ′ of order (q, q). A set H of half of the lines of Swhich are disjoint from S ′ is called a relative hemisystem of Sprovided that for any point X of S exactly half of the lines throughX which are disjoint from S ′ are in H.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 79: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Relative Hemisystems

Let S be a GQ of order (q, q2) that doubly subtends asubquadrangle S ′ of order (q, q). A relative hemisystem of S is apartition of the points of S\S ′ into two sets such that every linemeets each set in q/2 points. In this case, q must be even.

Dualizing: Let S be a quadrangle of order (q2, q) containing aquadrangle S ′ of order (q, q). A set H of half of the lines of Swhich are disjoint from S ′ is called a relative hemisystem of Sprovided that for any point X of S exactly half of the lines throughX which are disjoint from S ′ are in H.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 80: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

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) withq = 2t and t > 1 give rise to primitive 3-class Q-polynomialschemes 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

Page 81: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

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) withq = 2t and t > 1 give rise to primitive 3-class Q-polynomialschemes 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

Page 82: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

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) withq = 2t and t > 1 give rise to primitive 3-class Q-polynomialschemes 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

Page 83: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Primitive Q-polynomial schemes

Theorem

(Cossidente ’13) Inequivalent relative hemisystems givenon-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 relativehemisystem of a quadrangle?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 84: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Primitive Q-polynomial schemes

Theorem

(Cossidente ’13) Inequivalent relative hemisystems givenon-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 relativehemisystem of a quadrangle?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 85: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Primitive Q-polynomial schemes

Theorem

(Cossidente ’13) Inequivalent relative hemisystems givenon-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 relativehemisystem of a quadrangle?

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 86: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

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

Page 87: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

A family of Schurian Q-polynomial schemes

Let V = F2nq with a non-degenerate alternating bilinear form B,

q ≡ 1 (mod 4). The maximal totally isotropic subspaces give anassociation scheme (symplectic dual polar space graph).

For each maximal U choose a dual basis, which gives rise to adeterminant δU . For each pair of maximals U,V , let k be theco-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

Page 88: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

A family of Schurian Q-polynomial schemes

Let V = F2nq with a non-degenerate alternating bilinear form B,

q ≡ 1 (mod 4). The maximal totally isotropic subspaces give anassociation scheme (symplectic dual polar space graph).

For each maximal U choose a dual basis, which gives rise to adeterminant δU . For each pair of maximals U,V , let k be theco-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

Page 89: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

A family of Schurian Q-polynomial schemes

Let V = F2nq with a non-degenerate alternating bilinear form B,

q ≡ 1 (mod 4). The maximal totally isotropic subspaces give anassociation scheme (symplectic dual polar space graph).

For each maximal U choose a dual basis, which gives rise to adeterminant δU . For each pair of maximals U,V , let k be theco-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

Page 90: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

A family of Schurian Q-polynomial schemes

Let X be the set of all ordered pairs (U, ε) where ε = ±1. Definethe following relations:

(U, ε1)i∼(V , ε2) if dim(U ∩ V ) = n − i and σ(U,V ) = ε1ε2 for

0 ≤ i ≤ n.

(U, ε1)2n+1−i∼ (V , ε2) if dim(U ∩ V ) = i and σ(U,V ) = −ε1ε2 for

0 ≤ i ≤ n.

Theorem

(Moorhouse, W) This scheme is Q-bipartite for all n > 1 whered = 2n + 1. These schemes have two Q-polynomial orderings; thesecond 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 secondordering is given by conjugation.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 91: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

A family of Schurian Q-polynomial schemes

Let X be the set of all ordered pairs (U, ε) where ε = ±1. Definethe following relations:

(U, ε1)i∼(V , ε2) if dim(U ∩ V ) = n − i and σ(U,V ) = ε1ε2 for

0 ≤ i ≤ n.

(U, ε1)2n+1−i∼ (V , ε2) if dim(U ∩ V ) = i and σ(U,V ) = −ε1ε2 for

0 ≤ i ≤ n.

Theorem

(Moorhouse, W) This scheme is Q-bipartite for all n > 1 whered = 2n + 1. These schemes have two Q-polynomial orderings; thesecond 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 secondordering is given by conjugation.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 92: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

A family of Schurian Q-polynomial schemes

Let X be the set of all ordered pairs (U, ε) where ε = ±1. Definethe following relations:

(U, ε1)i∼(V , ε2) if dim(U ∩ V ) = n − i and σ(U,V ) = ε1ε2 for

0 ≤ i ≤ n.

(U, ε1)2n+1−i∼ (V , ε2) if dim(U ∩ V ) = i and σ(U,V ) = −ε1ε2 for

0 ≤ i ≤ n.

Theorem

(Moorhouse, W) This scheme is Q-bipartite for all n > 1 whered = 2n + 1. These schemes have two Q-polynomial orderings; thesecond 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 secondordering is given by conjugation.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 93: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Not P-polynomial, but close

L1 =

0 b0 0 0 0 . . . 0 0 0 01 a1

2 b1 0 0 . . . 0 0 a12 0

0 c2a22

. . . 0 . . . 0 . . . 0 0

0 0. . .

. . . bd−1 0ad−1

2 0 0 00 0 0 cd

ad2

ad2 0 0 0 0

0 0 0 0 ad2

ad2 cd 0 0 0

0 0 0ad−1

2 0 bd−1. . .

. . . 0 0

0 0 . . . 0 0 0. . . a2

2 c2 00 a1

2 0 0 0 0 0 b1a12 1

0 0 0 0 0 0 0 0 b0 0

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 94: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

The Symplectic Group

Let q be a prime power and V = (F )2nq 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 BC D

)satisfying

(A BC D

)T (0 I−I 0

)(A BC D

)=

(0 I−I 0

)

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 95: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

The Symplectic Group

Let q be a prime power and V = (F )2nq 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 BC D

)satisfying

(A BC D

)T (0 I−I 0

)(A BC D

)=

(0 I−I 0

)

Jason Williford University of Wyoming Q-polynomial Association Schemes

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The Symplectic Group

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

)satisfying

AT C − CT A = BT D − DT B = 0 and AT D − CT B = I .

The stabilizer S of the maximal isotropic subspaceU =< e1, . . . , en > is(

A B0 D

)satisfying AT D = I and BT D − DT B = 0.

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 97: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

The Symplectic Group

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

)satisfying

AT C − CT A = BT D − DT B = 0 and AT D − CT B = I .

The stabilizer S of the maximal isotropic subspaceU =< e1, . . . , en > is(

A B0 D

)satisfying AT D = I and BT D − DT B = 0.

Jason Williford University of Wyoming Q-polynomial Association Schemes

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A family of Schurian Q-polynomial schemes

The permutation group given by the action of G on the cosets ofS is generously transitive, and yields the association scheme fromthe symplectic dual polar space graphs.

Now we choose q to be odd, and let S ′ be the subgroup of S ofmatrices(

A B0 D

)satisfying AT D = I , BT D − DT B = 0 and |A| is a

square.

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

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 99: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

A family of Schurian Q-polynomial schemes

The permutation group given by the action of G on the cosets ofS is generously transitive, and yields the association scheme fromthe symplectic dual polar space graphs.

Now we choose q to be odd, and let S ′ be the subgroup of S ofmatrices(

A B0 D

)satisfying AT D = I , BT D − DT B = 0 and |A| is a

square.

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

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 100: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

A Hypothetical Primitive Family

Question

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

Jason Williford University of Wyoming Q-polynomial Association Schemes

Page 101: Q-polynomial Association SchemesPolynomial Association Schemes An association scheme is called P-polynomial (metric) provided that, after suitably reordering the A i, there are polynomials

Parameters for q = 9

L1 =

0 60 0 0 01 3 2 54 00 4 2 0 540 8 0 28 240 0 5 30 25

L2 =

0 0 30 0 00 2 1 0 271 2 0 27 00 0 2 16 120 5 0 15 10

L3 =

0 0 0 405 00 54 0 189 1620 0 27 216 1621 28 16 216 1440 30 15 180 180

L4 =

0 0 0 0 3240 0 27 162 1350 54 0 162 1080 24 12 144 1441 25 10 180 108

Jason Williford University of Wyoming Q-polynomial Association Schemes