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 E i χ = 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 occurring in Y , and suppose E i χ = 0 for all 1 ≤ i ≤ t, where χ is the characteristic vector of Y . If t ≥ 2 s − 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 of nontrivial relations occurring in Y , and suppose w ∗ = max { i : E i χ � = 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 occurring in Y , and suppose E i χ = 0 for all 1 ≤ i ≤ t, where χ is the characteristic vector of Y . If t ≥ 2 s − 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 of nontrivial relations occurring in Y , and suppose w ∗ = max { i : E i χ � = 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 ( E 1 ) = 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 ( E 1 ) = 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 ( E 1 ) = 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 b 0 , . . . b d − 1 , c 1 , . . . , c d 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 c i and the number of z which are distance i + 1 from x and adjacent to y is b i . c b i i y x i-1 (x) i (x) i+1 (x) The constants relate to the p k ij of the resulting association scheme by: b i = p i 1 , i +1 , c i = p i 1 , i − 1 , a i = p i 1 , i . Jason Williford University of Wyoming Q -polynomial Association Schemes i (x) i-1 (x)
Parameters of Distance-Regular Graphs A graph Γ of diameter d is distance-regular if and only if there are constants b 0 , . . . b d − 1 , c 1 , . . . , c d 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 c i and the number of z which are distance i + 1 from x and adjacent to y is b i . c b i i y x i-1 (x) i (x) i+1 (x) The constants relate to the p k ij of the resulting association scheme by: b i = p i 1 , i +1 , c i = p i 1 , i − 1 , a i = p i 1 , i . Jason Williford University of Wyoming Q -polynomial Association Schemes i (x) i-1 (x)
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 p k 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 Γ d is a union of complete graphs (distance d in the graph Γ is an equivalence relation). This is equivalent to having b j = c d − 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 p k 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 Γ d is a union of complete graphs (distance d in the graph Γ is an equivalence relation). This is equivalent to having b j = c d − 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 p k 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 Γ d is a union of complete graphs (distance d in the graph Γ is an equivalence relation). This is equivalent to having b j = c d − 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 E i , there are polynomials q k ( x ) of degree k for 0 ≤ k ≤ d such that E k = q k ( E 1 ), where multiplication is done entrywise. We define constants b ∗ i and c ∗ i by: b ∗ i = q i 1 , i +1 , c ∗ i = q i 1 , i − 1 , a ∗ i = q i 1 , i . These are analogous to the b i , c i and a i 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 q k 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 q k 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 q k 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 ( E 1 ) if d is odd, and w ≤ rk ( E 2 ) 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 = m 1 + 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: � 0 � A 0 � � � � 0 A i A d +1 − i A 0 , , and 0 A 0 A d +1 − i A i A 0 0 where A 0 , . . . , A d 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 = m 1 + 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: � 0 � A 0 � � � � 0 A i A d +1 − i A 0 , , and 0 A 0 A d +1 − i A i A 0 0 where A 0 , . . . , A d 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 = m 1 + 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: � 0 � A 0 � � � � 0 A i A d +1 − i A 0 , , and 0 A 0 A d +1 − i A i A 0 0 where A 0 , . . . , A d 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 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 E 6 , E 7 , E 8 , Leech lattice, Martinet lattice bipartite doubles of SRG’s with q 1 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 V 0 ∪ · · · ∪ V l , l ≥ 1 such that: The induced subgraph on V i ∪ V j 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 ∈ Γ i , y ∈ Γ j we have | Γ( x ) ∩ Γ( y ) ∩ V k | = σ or τ depending on 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 V 0 ∪ · · · ∪ V l , l ≥ 1 such that: The induced subgraph on V i ∪ V j 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 ∈ Γ i , y ∈ Γ j we have | Γ( x ) ∩ Γ( y ) ∩ V k | = σ or τ depending on 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 (2 2 t +2 , 2 2 t +1 − 2 t , 2 2 t − 2 t ) designs with l = 2 2 t +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 (2 2 t +2 , 2 2 t +1 − 2 t , 2 2 t − 2 t ) designs with l = 2 2 t +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 (2 2 t +2 , 2 2 t +1 − 2 t , 2 2 t − 2 t ) designs with l = 2 2 t +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 (2 2 t +2 , 2 2 t +1 − 2 t , 2 2 t − 2 t ) designs with l = 2 2 t +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 = q 2 H (3 , q 2 ), consisting of the points and totally isotropic lines of a hermitian variety in PG (3 , q 2 ), s = q 2 , t = q The quadrangles Q (4 , q ) and W ( q ) are duals of one another, as are Q − (5 , q ) and H (3 , q 2 ). Jason Williford University of Wyoming Q -polynomial Association Schemes
Hemisystems of GQ Let S be a generalized quadrangle of order ( q , q 2 ). A hemisystem of 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 , q 2 ) 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 , q 2 ). A hemisystem of 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 , q 2 ) 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 , q 2 ), containing a subquadrangle S ′ of order ( q , q ). For any X ∈ S \ S ′ we have that the set θ X of all points in S ′ collinear with X is an ovoid of S ′ . We say that θ X 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 θ X are doubly subtended. If two points are not antipodes, the ovoids they subtend meet in 1 or 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 , q 2 ), containing a subquadrangle S ′ of order ( q , q ). For any X ∈ S \ S ′ we have that the set θ X of all points in S ′ collinear with X is an ovoid of S ′ . We say that θ X 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 θ X are doubly subtended. If two points are not antipodes, the ovoids they subtend meet in 1 or 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 , q 2 ), containing a subquadrangle S ′ of order ( q , q ). For any X ∈ S \ S ′ we have that the set θ X of all points in S ′ collinear with X is an ovoid of S ′ . We say that θ X 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 θ X are doubly subtended. If two points are not antipodes, the ovoids they subtend meet in 1 or 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 , q 2 ), containing a subquadrangle S ′ of order ( q , q ). For any X ∈ S \ S ′ we have that the set θ X of all points in S ′ collinear with X is an ovoid of S ′ . We say that θ X 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 θ X are doubly subtended. If two points are not antipodes, the ovoids they subtend meet in 1 or 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 , q 2 ), containing a subquadrangle S ′ of order ( q , q ). For any X ∈ S \ S ′ we have that the set θ X of all points in S ′ collinear with X is an ovoid of S ′ . We say that θ X 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 θ X are doubly subtended. If two points are not antipodes, the ovoids they subtend meet in 1 or 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 , q 2 ) which doubly subtends a subquadrangle S ′ of order ( q , q ) . Then the set of 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 on the points of S \ S ′ as follows: R 1 : All pairs ( X , Y ) where X and Y are not collinear and | θ X ∩ θ Y | = 1. R 2 : All pairs ( X , Y ) where X and Y are not collinear and | θ X ∩ θ Y | = q + 1. R 3 : All pairs ( X , Y ) where X and Y are collinear (here | θ X ∩ θ Y | = 1). R 4 : 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 on the points of S \ S ′ as follows: R 1 : All pairs ( X , Y ) where X and Y are not collinear and | θ X ∩ θ Y | = 1. R 2 : All pairs ( X , Y ) where X and Y are not collinear and | θ X ∩ θ Y | = q + 1. R 3 : All pairs ( X , Y ) where X and Y are collinear (here | θ X ∩ θ Y | = 1). R 4 : 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 on the points of S \ S ′ as follows: R 1 : All pairs ( X , Y ) where X and Y are not collinear and | θ X ∩ θ Y | = 1. R 2 : All pairs ( X , Y ) where X and Y are not collinear and | θ X ∩ θ Y | = q + 1. R 3 : All pairs ( X , Y ) where X and Y are collinear (here | θ X ∩ θ Y | = 1). R 4 : 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 on the points of S \ S ′ as follows: R 1 : All pairs ( X , Y ) where X and Y are not collinear and | θ X ∩ θ Y | = 1. R 2 : All pairs ( X , Y ) where X and Y are not collinear and | θ X ∩ θ Y | = q + 1. R 3 : All pairs ( X , Y ) where X and Y are collinear (here | θ X ∩ θ Y | = 1). R 4 : 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 on the points of S \ S ′ as follows: R 1 : All pairs ( X , Y ) where X and Y are not collinear and | θ X ∩ θ Y | = 1. R 2 : All pairs ( X , Y ) where X and Y are not collinear and | θ X ∩ θ Y | = q + 1. R 3 : All pairs ( X , Y ) where X and Y are collinear (here | θ X ∩ θ Y | = 1). R 4 : 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 on the points of S \ S ′ as follows: R 1 : All pairs ( X , Y ) where X and Y are not collinear and | θ X ∩ θ Y | = 1. R 2 : All pairs ( X , Y ) where X and Y are not collinear and | θ X ∩ θ Y | = q + 1. R 3 : All pairs ( X , Y ) where X and Y are collinear (here | θ X ∩ θ Y | = 1). R 4 : 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 , q 2 ) 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 ( q 2 , 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 , q 2 ) 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 ( q 2 , 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 , q 2 ) 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 , q 2 ), and the points of the set H correspond to the lines L of H (3 , q 2 ) which do not meet the fixed copy of W ( q ). Theorem (Penttila, Williford ’11) Relative hemisystems of H (3 , q 2 ) with q = 2 t and t > 1 give rise to primitive 3-class Q-polynomial schemes which are not generated by distance-regular graphs. For q = 2 t the quadrangle H (3 , q 2 ) has a relative hemisystem. Jason Williford University of Wyoming Q -polynomial Association Schemes
Primitive Q -polynomial schemes The GQ H (3 , q 2 ) 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 , q 2 ), and the points of the set H correspond to the lines L of H (3 , q 2 ) which do not meet the fixed copy of W ( q ). Theorem (Penttila, Williford ’11) Relative hemisystems of H (3 , q 2 ) with q = 2 t and t > 1 give rise to primitive 3-class Q-polynomial schemes which are not generated by distance-regular graphs. For q = 2 t the quadrangle H (3 , q 2 ) has a relative hemisystem. Jason Williford University of Wyoming Q -polynomial Association Schemes
Primitive Q -polynomial schemes The GQ H (3 , q 2 ) 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 , q 2 ), and the points of the set H correspond to the lines L of H (3 , q 2 ) which do not meet the fixed copy of W ( q ). Theorem (Penttila, Williford ’11) Relative hemisystems of H (3 , q 2 ) with q = 2 t and t > 1 give rise to primitive 3-class Q-polynomial schemes which are not generated by distance-regular graphs. For q = 2 t the quadrangle H (3 , q 2 ) 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 , q 2 ) 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 , q 2 ) 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 , q 2 ) 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 = F 2 n 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 δ U . For each pair of maximals U , V , let k be the co-dimension of their intersection in U and V . Let u 1 , . . . , u n , v 1 , . . . , v n be bases of U and V with u i = v i for k + 1 ≤ i ≤ n . The Maslov index on pairs of maximals is given by: σ ( U , V ) = χ ( δ U ( u 1 , . . . , u n ) δ V ( v 1 , . . . , v n ) det [ B ( u i , v j ) : 1 ≤ i , j ≤ k ]) . Jason Williford University of Wyoming Q -polynomial Association Schemes
A family of Schurian Q -polynomial schemes Let V = F 2 n 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 δ U . For each pair of maximals U , V , let k be the co-dimension of their intersection in U and V . Let u 1 , . . . , u n , v 1 , . . . , v n be bases of U and V with u i = v i for k + 1 ≤ i ≤ n . The Maslov index on pairs of maximals is given by: σ ( U , V ) = χ ( δ U ( u 1 , . . . , u n ) δ V ( v 1 , . . . , v n ) det [ B ( u i , v j ) : 1 ≤ i , j ≤ k ]) . Jason Williford University of Wyoming Q -polynomial Association Schemes
A family of Schurian Q -polynomial schemes Let V = F 2 n 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 δ U . For each pair of maximals U , V , let k be the co-dimension of their intersection in U and V . Let u 1 , . . . , u n , v 1 , . . . , v n be bases of U and V with u i = v i for k + 1 ≤ i ≤ n . The Maslov index on pairs of maximals is given by: σ ( U , V ) = χ ( δ U ( u 1 , . . . , u n ) δ V ( v 1 , . . . , v n ) det [ B ( u i , v j ) : 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 , ǫ 1 ) i ∼ ( V , ǫ 2 ) if dim ( U ∩ V ) = n − i and σ ( U , V ) = ǫ 1 ǫ 2 for 0 ≤ i ≤ n . ( U , ǫ 1 ) 2 n +1 − i ∼ ( V , ǫ 2 ) if dim ( U ∩ V ) = i and σ ( U , V ) = − ǫ 1 ǫ 2 for 0 ≤ i ≤ n . Theorem (Moorhouse, W) This scheme is Q-bipartite for all n > 1 where d = 2 n + 1 . These schemes have two Q-polynomial orderings; the second is: E 0 , E d , E 2 , E d − 2 , E 4 , E d − 4 , . . . ,E d − 5 , E 5 , E d − 3 , E 3 , E d − 1 , E 1 (Suzuki type 3). For q nonsquare, the splitting field is quadratic and the second ordering 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 , ǫ 1 ) i ∼ ( V , ǫ 2 ) if dim ( U ∩ V ) = n − i and σ ( U , V ) = ǫ 1 ǫ 2 for 0 ≤ i ≤ n . ( U , ǫ 1 ) 2 n +1 − i ∼ ( V , ǫ 2 ) if dim ( U ∩ V ) = i and σ ( U , V ) = − ǫ 1 ǫ 2 for 0 ≤ i ≤ n . Theorem (Moorhouse, W) This scheme is Q-bipartite for all n > 1 where d = 2 n + 1 . These schemes have two Q-polynomial orderings; the second is: E 0 , E d , E 2 , E d − 2 , E 4 , E d − 4 , . . . ,E d − 5 , E 5 , E d − 3 , E 3 , E d − 1 , E 1 (Suzuki type 3). For q nonsquare, the splitting field is quadratic and the second ordering 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 , ǫ 1 ) i ∼ ( V , ǫ 2 ) if dim ( U ∩ V ) = n − i and σ ( U , V ) = ǫ 1 ǫ 2 for 0 ≤ i ≤ n . ( U , ǫ 1 ) 2 n +1 − i ∼ ( V , ǫ 2 ) if dim ( U ∩ V ) = i and σ ( U , V ) = − ǫ 1 ǫ 2 for 0 ≤ i ≤ n . Theorem (Moorhouse, W) This scheme is Q-bipartite for all n > 1 where d = 2 n + 1 . These schemes have two Q-polynomial orderings; the second is: E 0 , E d , E 2 , E d − 2 , E 4 , E d − 4 , . . . ,E d − 5 , E 5 , E d − 3 , E 3 , E d − 1 , E 1 (Suzuki type 3). For q nonsquare, the splitting field is quadratic and the second ordering is given by conjugation. Jason Williford University of Wyoming Q -polynomial Association Schemes
Not P -polynomial, but close 0 b 0 0 0 0 . . . 0 0 0 0 a 1 a 1 1 b 1 0 0 . . . 0 0 0 2 2 ... a 2 0 c 2 0 . . . 0 . . . 0 0 2 ... ... a d − 1 0 0 b d − 1 0 0 0 0 2 a d a d 0 0 0 c d 0 0 0 0 2 2 L 1 = a d a d 0 0 0 0 c d 0 0 0 2 2 ... ... a d − 1 0 0 0 0 b d − 1 0 0 2 ... a 2 0 0 . . . 0 0 0 c 2 0 2 a 1 a 1 0 0 0 0 0 0 b 1 1 2 2 0 0 0 0 0 0 0 0 b 0 0 Jason Williford University of Wyoming Q -polynomial Association Schemes
The Symplectic Group Let q be a prime power and V = ( F ) 2 n q with non-degenerate alternating bilinear form B , B ( x , y ) = x 1 y n +1 − x n +1 y 1 + · · · + x n y 2 n − x 2 n y n . The symplectic group PSp ( V ) consists of the block matrices � A � B satisfying C D � T � 0 � 0 � A � � A � � B I B I = C D − I 0 C D − I 0 Jason Williford University of Wyoming Q -polynomial Association Schemes
The Symplectic Group Let q be a prime power and V = ( F ) 2 n q with non-degenerate alternating bilinear form B , B ( x , y ) = x 1 y n +1 − x n +1 y 1 + · · · + x n y 2 n − x 2 n y n . The symplectic group PSp ( V ) consists of the block matrices � A � B satisfying C D � T � 0 � 0 � A � � A � � B I B I = C D − I 0 C D − I 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 satisfying C D A T C − C T A = B T D − D T B = 0 and A T D − C T B = I . The stabilizer S of the maximal isotropic subspace U = < e 1 , . . . , e n > is � A B � satisfying A T D = I and B T D − D T B = 0. 0 D 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 satisfying C D A T C − C T A = B T D − D T B = 0 and A T D − C T B = I . The stabilizer S of the maximal isotropic subspace U = < e 1 , . . . , e n > is � A B � satisfying A T D = I and B T D − D T B = 0. 0 D 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 satisfying A T D = I , B T D − D T B = 0 and | A | is a 0 D 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 satisfying A T D = I , B T D − D T B = 0 and | A | is a 0 D 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
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