‣ HasCyclotomicSplittingField( CC ) | ( property ) |
Returns: true or false
Returns if the CC has a cyclotomic splitting field
‣ IntersectionAlgebra( B ) | ( operation ) |
Returns: homogeneous coherent configuration
Takes a list of intersection matrices, B, and returns the Intersection Algebra. The intersection matrix (B_i)_{jk} = p_{ij}^k must be for valid intersection numbers p_{ij}^k for some homogeneous coherent configuration.
‣ IntersectionMatrices( CC ) | ( attribute ) |
Returns: L
Returns a list L of the intersection matrices of an intersecgion algebra object, CC, where the i-th entry of L is B_{i-1} and (B_{i})_{jk} = p_{ij}^k.
‣ IntersectionNumber( CC, i, j, k ) | ( operation ) |
Returns: p_{ij}^k
Returns the intersection number p_{ij}^k for a intersection algebra.
‣ NumberOfClasses( CC ) | ( attribute ) |
Returns: d
Returns d for a d-class intersection algebra.
‣ Valencies( CC ) | ( attribute ) |
Returns: L
Returns a list L of valencies of a coherent configuration CC. The i-th entry of L is k_{i-1}.
‣ Order( CC ) | ( attribute ) |
Returns: n
Returns the order n (number of vertices) of the intersection algebra.
‣ SplittingField( CC ) | ( attribute ) |
Returns: F
Returns the splitting field of the CC
‣ HasCyclotomicSplittingField( CC ) | ( property ) |
Returns: true or false
Returns if the CC has a cyclotomic splitting field
‣ HasRationalSplittingField( CC ) | ( property ) |
Returns: true or false
Returns true if the splitting field is the rationals, false otherwise.
‣ KreinParameter( CC, i, j, k ) | ( operation ) |
Returns: q_{i,j}^k
Compute the krein parameter q_{i,j}^k of a commutative intersection algebra.
‣ KreinParameters( CC ) | ( attribute ) |
Returns: L
Return a list L of all Krein parameters of a commutative intersection algebra, where L[i][j,k] = q_{i,j}^k.
‣ IsQBipartite( CC ) | ( property ) |
Returns: true or false
Returns if the intersection algebra CC is Q-bipartite.
‣ IsPBipartite( CC ) | ( property ) |
Returns: true or false
Returns if the intersection algebra CC is bipartite.
‣ IsQAntipodal( CC ) | ( property ) |
Returns: true or false
Returns if the intersection algebra CC is Q-antipodal.
‣ IsPAntipodal( CC ) | ( property ) |
Returns: true or false
Returns if the intersection algebra CC is antipodal.
‣ ReorderRelations( CC, L ) | ( operation ) |
Returns: coherent configuration
Takes a intersection algebra CC and a list L, where L is a reordering of the relations. Returns an intersection algebra where the i-th relation of the CC becomes the j-th relation in the intersection algebra, where j = L_i. Note that L_i must be equal to \{0, \ldots, d \} as a set, and additionally requires that L_1 = 0.
‣ ReorderMinimalIdempotents( CC, L ) | ( operation ) |
Returns: coherent configuration
Takes an intersection algebra CC and a list L, where L is a reordering of the minimal idempotents. Returns an intersection algebra where the i-th idempotent of the CC becomes the j-th idempotent in the new intersection algebra, where j = L_i. Note that L_i must be equal to \{0, \ldots, d \} as a set, and additionally requires that L_1 = 0.
‣ ViewRelationDistributionDiagram( CC ) | ( operation ) |
Returns: true (Displays relation distribution diagram)
Take a CC and display the relation-distribution diagram with respect to R_1.
‣ IsCommutative( CC ) | ( property ) |
Returns: true or false
Checks if the input is a commutative intersection algebra.
‣ IsSymmetricIntersectionAlgebra( CC ) | ( property ) |
Returns: true or false
Checks if the input is a symmetric intersection algebra.
‣ NumberOfCharacters( CC ) | ( attribute ) |
Returns: n
Returns the number n of characters of CC.
‣ IntersectionAlgebraFromMatrixOfEigenvalues( P ) | ( operation ) |
Returns: intersection algebra object
Returns the intersection algebra determined by a matrix of eigenvalues.
‣ HammingSchemeIntersectionAlgebra( n, q ) | ( operation ) |
Returns: intersection algebra objectn
Returns the intersection algebra of the Hamming scheme, H(n, q).
‣ GrassmanSchemeIntersectionAlgebra( n, k, q ) | ( operation ) |
Returns: intersection algebra object
Returns the intersection algebra of the Grassmann scheme, J_q(n, k).
‣ CyclotomicSchemeIntersectionAlgebra( n, d ) | ( operation ) |
Returns: intersection algebra object
Returns the intersection algebra of the Cyclotomic scheme, Cyc(n, d).
‣ IntersectionAlgebraFromIntersectionArray( n, k, q ) | ( operation ) |
Returns: intersection algebra object
Returns the intersection algebra of a DRG given by its intersection array.
‣ IntersectionAlgebraFromClassicalParameters( n, k, q ) | ( operation ) |
Returns: intersection algebra object
Returns the intersection algebra of a DRG given by its classical parameters.
‣ IntersectionAlgebraFromStronglyRegularGraphParameters( n, k, q ) | ( operation ) |
Returns: intersection algebra object
Returns the intersection algebra of a SRG given by its parameters [n, k, \lambda, \mu].
‣ IsFusionOfHomogeneousCoherentConfiguration( CC, L ) | ( operation ) |
Returns: true or false
Takes the intersection algebra object of a d-class homogeneous coherent configuration CC, and checks if the partion L of \{0, \ldots, d\} corresponds to a valid fusion.
‣ SchurianSchemeIntersectionAlgebra( G ) | ( operation ) |
Returns: intersection algebra
Returns the Schurian scheme defined by G, where G is a transitive permutation group. A Schurian scheme is a special case of CoherentConfigurationByOrbitals and is symmetric.
‣ MapFromIntersectionMatricesToCentralIdempotents( I ) | ( attribute ) |
Returns: M
Takes an intersection algebra object, I, and returns a matrix, M, which maps the intersection matrices to the central idempotents of the intersection algebra. The central idempotent F_i = \sum_{i=1}^{d+1} M_{ij} B_j.
‣ MapFromIntersectionMatricesToCentralIdempotentsOverRationals( I ) | ( attribute ) |
Returns: M
Takes an intersection algebra object, I, and returns a matrix, M, which maps the intersection matrices to the central idempotents of the intersection algebra over the rationals. The central idempotent F_i = \sum_{i=1}^{d+1} M_{ij} B_j.
‣ ImageOfIntersectionAlgebra( $A$, $\sigma$ ) | ( operation ) |
Returns: true A^\sigma
Take a d-class intersection algebra A and return its image under the permutation \sigma \in Sym([1 .. d]). If p_{ij}^k is an intersection number of A, then in the image the intersection number is p_{i^\sigma j^\sigma}^{k^\sigma}
‣ IsomorphismIntersectionAlgebras( $A$, $B$ ) | ( operation ) |
Returns: \sigma
Take two d-class intersection algebras A and B and return \sigma \in Sym([1 .. d]) such that A^\sigma = B.
‣ AreIsomorphicIntersectionAlgebras( $A$, $B$ ) | ( operation ) |
Returns: \sigma
Take two d-class intersection algebras A and B and return true if they are isomorphic. Return false otherwise.
‣ CanonisingMap( A ) | ( attribute ) |
Returns: perm
Returns two permutations which will produce the canonical form of the intersection algebra A. The canonical form can be obtained by ImageOfIntersectionAlgebra(A, perm) Any intersection algebra which is isomorphic to A will the same canonical form.
‣ CanonicalFormOfIntersectionAlgebra( A ) | ( operation ) |
Returns: B
Returns the canonical form, B, of the intersection algebra A. Any intersection algebra which is isomorphic to A will have B as the canonical form.
‣ AutomorphismGroup( A ) | ( attribute ) |
Returns: G
Returns the automorphism group G of the intersection algebra object A. G is a permutation group acting on the relations, such that for all g \in G, p_{i^g j^g}^{k^g} = p_{ij}^k.
‣ IsFusionOfIntersectionAlgebra( CC, L ) | ( operation ) |
Returns: true or false
Takes a d-class homogeneous coherent configuration CC, and checks if the partion L of \{0, \ldots, d\} corresponds to a valid fusion.
‣ FusionOfIntersectionAlgebra( CC, L ) | ( operation ) |
Returns: homogeneous coherent configuration
Takes a d-class homogeneous coherent configuration CC and returns a fusion scheme corresponding to L, where L is a partion of \{0, \ldots, d\}. Note that the ordering of the cells of L is irrelevant. The method will sort the fused relations according to the smallest value in each cell.
‣ ConverseRelationPairs( CC ) | ( attribute ) |
Returns: L
Returns a list L of either tuples or singletons, corresponding to relations and their converses or symmetric relations.
‣ ConverseRelation( CC, i ) | ( operation ) |
Returns: j
Returns j such that R_j = R_i^\top, the converse relation of i .
‣ IsStratifiable( CC ) | ( attribute ) |
Returns: true or false
If the fusion of transposed relations produces a valid association scheme, then CC is stratifiable.
‣ SymmetrisationOfIntersectionAlgebra( CC ) | ( operation ) |
Returns: Association scheme
Given a homogeneous coherent configuration, CC, the symmetrisation is computed if possible, otherwise fail is returned. The symmetrisation of a homogeneous coherent configuration takes any non-symmetric relations and fuses them together. The result may or may not be a valid homogeneous coherent configuration. If it is valid, then it is an association scheme (Symmetric coherent configuration). If CC is commutative, then it can be symmetrised.
‣ FeasibleNonTrivialFusionsOfIntersectionAlgebra( CC[, k[, flag]] ) | ( attribute ) |
Returns: list of feasibly fusionable relations
Returns a list where each entry is a collection of relations which may be fused to form a feasible homogeneous coherent configuration Trivial means either no relations are fused, or all non-identity relations are fused. If the additional argument k is given, only fusions with k-classes are returned. If flag is also given and is equal to true, then all fusions with at most k-classes are returned.
‣ AllNonTrivialFusionsOfIntersectionAlgebra( CC ) | ( operation ) |
Returns: List of all non-trivial fusions of CC
Returns a list of all homogeneous coherent configurations such that each element of the list is a non-trivial fusion of CC. Trivial means either no relations are fused, or all non-identity relations are fused. If the additional argument k is given, only fusions with k-classes are returned. If flag is also given and is equal to true, then all fusions with at most k-classes are returned.
‣ AllFusionsOfIntersectionAlgebra( CC ) | ( operation ) |
Returns: List of all fusions of CC
Returns a list of all homogeneous coherent configurations such that each element of the list is a fusion of CC. Includes trivial fusions, i.e the original homogeneous coherent configuration, and the coherent configuration resulting from the fusion of all non-identity relations
‣ FeasibleNonTrivialSymmetricFusionsOfIntersectionAlgebra( CC ) | ( attribute ) |
Returns: list of feasibly fusionable relations
Returns a list where each entry is a collection of relations which may be fused to form a feasible association scheme (i.e. relations are symmetric) Trivial means either no relations are fused, or all non-identity relations are fused. If the additional argument k is given, only fusions with k-classes are returned. If flag is also given and is equal to true, then all fusions with at most k-classes are returned.
‣ AllNonTrivialSymmetricFusionsOfIntersectionAlgebra( CC ) | ( operation ) |
Returns: List of all non-trivial fusions of CC
Returns a list of all association schemes (i.e symmetric relations) such that each element of the list is a non-trivial fusion of CC. Trivial means either no relations are fused, or all non-identity relations are fused. If the additional argument k is given, only fusions with k-classes are returned. If flag is also given and is equal to true, then all fusions with at most k-classes are returned.
‣ AllSymmetricFusionsOfIntersectionAlgebra( CC ) | ( operation ) |
Returns: List of all fusions of CC
Returns a list of all association schemes (i.e symmetric relations) such that each element of the list is a fusion of CC. Includes trivial fusions, i.e the original homogeneous coherent configuration (if it is an association scheme), and the coherent configuration resulting from the fusion of all non-identity relations
‣ FirstFeasibleNonTrivialSymmetricFusionOfIntersectionAlgebra( CC ) | ( attribute ) |
Returns: list of feasibly fusionable relations
Returns a list of relations which may be fused to form a feasible association scheme (i.e. relations are symmetric), if possible. Returns fail if no nontrivial fusion exists. Trivial means either no relations are fused, or all non-identity relations are fused.
‣ CentralIdempotentsOfIntersectionAlgebra( I ) | ( attribute ) |
Returns: central idempotents
Returns the central idempotents of the intersection algebra.
‣ CentralIdempotentsOfIntersectionAlgebraOverRationals( I ) | ( attribute ) |
Returns: central idempotents
Returns the central idempotents of the intersection algebra over the rationals.
‣ MatrixOfDualEigenvalues( CC ) | ( attribute ) |
Returns: Q
Returns a the dual matrix of eigenvalues, Q, for a homogeneous coherent configuration CC.
‣ MatrixOfEigenvalues( CC ) | ( attribute ) |
Returns: P
Returns a the matrix of eigenvalues (or character table), P, for the intersection algebra of a homogeneous coherent configuration CC.
‣ FitMatrixOfEigenvalues( A, P ) | ( operation ) |
Returns: P2
Checks if P is the matrix of eigenvalues of intersection algebra A, upto some reordering of the columns. In such a case, P2, the reordered matrix is returned. If not, returns fail.
‣ IsPPolynomial( CC ) | ( property ) |
Returns: true or false
Returns if the homogeneous coherent configuration CC is P-polynomial.
‣ FirstPPolynomialOrdering( CC ) | ( attribute ) |
Returns: P-polynomial ordering or fail
Returns the first P-polynomial ordering admitted by the homogeneous coherent configuration CC, and fail otherwise.
‣ AdmitsPPolynomialOrdering( CC ) | ( property ) |
Returns: true or false
Returns if the homogeneous coherent configuration CC admits a P-polynomial ordering.
‣ IsMetric( CC ) | ( operation ) |
Returns: true or false
Alias for IsPPolynomial.
‣ FirstMetricOrdering( CC ) | ( operation ) |
Returns: metric ordering or fail
Alias for FirstPPolynomialOrdering.
‣ AdmitsMetricOrdering( CC ) | ( operation ) |
Returns: true or false
Alias for AdmitsPPolynomialOrdering.
‣ AllPPolynomialOrderings( CC ) | ( attribute ) |
Returns: L
Calculate the list L of all P-polynomial orderings of a homogeneous coherent configuration.
‣ AllMetricOrderings( CC ) | ( operation ) |
Returns: L
Alias for AllPPolynomialOrderings.
‣ IntersectionArray( CC ) | ( attribute ) |
Returns: List
Returns the intersection array if CC is P-polynomial.
‣ ClassicalParameters( CC ) | ( attribute ) |
Returns: [d, b, \alpha, \beta]
Returns the classical parameters if the CC is metric with classical parameters.
‣ StronglyRegularGraphParameters( CC ) | ( attribute ) |
Returns: [d, b, \alpha, \beta]
Returns the parameters \{n, k; \lambda, \mu \} if the CC is an association scheme with 2 classes.
‣ AdmitsQPolynomialOrdering( CC ) | ( property ) |
Returns: true or false
Returns if the homogeneous coherent configuration CC admits a Q-polynomial ordering.
‣ AdmitsCometricOrdering( CC ) | ( operation ) |
Returns: true or false
Alias for AdmitsQPolynomialOrdering.
‣ IsQPolynomial( CC ) | ( property ) |
Returns: true or false
Returns if the commutative coherent configuration CC is Q-polynomial.
‣ IsCometric( CC ) | ( operation ) |
Returns: true or false
Alias for is Q-polynomial.
‣ AllQPolynomialOrderings( CC ) | ( attribute ) |
Returns: L
Calculate a list L of all Q-polynomial orderings of a homogeneous coherent configuration.
‣ AllCometricOrderings( CC ) | ( operation ) |
Returns: L
Alias for AllQPolynomialOrderings.
‣ KreinArray( CC ) | ( attribute ) |
Returns: List
Returns the Krein (or dual intersection) array if CC is Q-polynomial.
‣ DualIntersectionArray( CC ) | ( operation ) |
Returns: List
Alias for KreinArray.
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