‣ HomogeneousCoherentConfiguration( M ) | ( operation ) |
Returns: homogeneous coherent configuration
Takes the relationship matrix, \(M\), describing a homogeneous coherent configuration and returns a HomogeneousCoherentConfiguration object. The matrix \(M = \sum_{i=0}^d i A_i\), where \(A_i\) are the adjacency matrices describing a coherent configuration. Checks that the matrix satisfies the axioms of a homogeneous coherent configuration.
‣ HomogeneousCoherentConfigurationNC( M ) | ( operation ) |
Returns: homogeneous coherent configuration
Same as HomogeneousCoherentConfiguration but without performing any checks. Use this method only if you know with certainty that \(M\) describes a coherent configuration.
‣ AssociationScheme( M ) | ( operation ) |
Returns: homogeneous coherent configuration
Takes the relationship matrix, \(M\), describing an associatioin scheme and returns an association scheme (symmetric coherent configuration). This is simply a HomogeneousCoherentConfiguration object, but with the known property of being symmetric. The matrix \(M = \sum_{i=0}^d i A_i\), where \(A_i\) are the adjacency matrices describing an association scheme. Checks that the matrix satisfies the association scheme axioms.
‣ AssociationSchemeNC( M ) | ( operation ) |
Returns: homogeneous coherent configuration
Same as AssociationScheme but without performing any checks. Use this method only if you know with certainty that \(M\) describes an association scheme (symmetric coherent configuration).
‣ ReorderRelations( CC, L ) | ( operation ) |
Returns: coherent configuration
Takes a homogeneous coherent configuration CC and a list L, where L is a reordering of the relations. Returns a homogeneous coherent configuration where the \(i\)-th relation of the CC becomes the \(j\)-th relation in the new homogeneous coherent configuration, where \(j = L_i\). Note that \(L\) must be equal to \(\{0, \ldots, d \}\) as a set, and additionally requires that \(L_1 = 0\).
‣ RelationMatrix( CC ) | ( operation ) |
Returns: \(M\)
Takes a homogeneous coherent configuration and returns the underlying relation matrix \(M = \sum_{i=0}^d i A_i\), where \(A_i\) are the adjacency matrices of the coherent configuration
‣ Relation( CC, x, y ) | ( operation ) |
Returns: i
Takes a CC and two points, x and y, and returns i such that \((x, y) \in R_i\).
‣ Neighbours( CC, p, k ) | ( operation ) |
Returns: L
Returns a list \(L\) of all the points \(y\) of CC such that \((p,y) \in R_k\).
‣ Neighbours( CC, p, L ) | ( operation ) |
Returns: L
Returns a list \(L\) of all the points \(y\) of CC such that \((p,y) \in R_k\) for some \(k \in L\).
‣ IsCommutative( CC ) | ( property ) |
Returns: true or false
Checks if the input is a commutative coherent configuration.
‣ IsSymmetricCoherentConfiguration( CC ) | ( property ) |
Returns: true or false
Checks if the input is a symmetric coherent configuration.
‣ IsAssociationScheme( CC ) | ( operation ) |
Returns: true or false
Alias for IsSymmetricCoherentConfiguration
‣ NumberOfClasses( CC ) | ( attribute ) |
Returns: d
Returns \(d\) for a \(d\)-class homogeneous coherent configuration.
‣ Rank( CC ) | ( operation ) |
Returns: d
Returns \(d\) for a \(d\)-class homogeneous coherent configuration.
‣ AdjacencyMatrices( CC ) | ( attribute ) |
Returns: L
Returns a list \(L\), where the \(i\)-th entry of \(L\) is the adjacency matrix \(A_{i-1}\), where \((A_i)_{xy} =1\) if \((x,y) \in R_i\) and \((A_i)_{xy} =0\) otherwise.
‣ Order( CC ) | ( attribute ) |
Returns: n
Returns the order \(n\) (number of vertices) of the coherent configuration.
‣ IntersectionNumber( CC, i, j, k ) | ( operation ) |
Returns: \(p_{ij}^k\)
Returns the intersection number \(p_{ij}^k\) for a coherent configuration CC.
‣ Valencies( CC ) | ( attribute ) |
Returns: L
Returns a list L of valencies of a coherent configuration CC. The \(i\)-th entry of \(L\) is \(k_{i-1}\).
‣ IntersectionAlgebraOfHomogeneousCoherentConfiguration( CC ) | ( attribute ) |
Returns: L
Returns an IntersectionAlgebra object for CC
‣ IntersectionMatrices( CC ) | ( attribute ) |
Returns: L
Returns a list L of the intersection matrices of a homogeneous coherent configuration \(CC\), where the \(i\)-th entry of \(L\) is \(B_{i-1}\) and \((B_{i})_{jk} = p_{ij}^k\).
‣ NumberOfCharacters( CC ) | ( attribute ) |
Returns: n
Returns the number \(n\) of characters of CC.
‣ SplittingField( CC ) | ( attribute ) |
Returns: F
Returns the splitting field of the CC
‣ 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 homogeneous coherent configuration.
‣ KreinParameters( CC ) | ( attribute ) |
Returns: L
Return a list \(L\) of all Krein parameters of a commutative homogeneous coherent configuration, where \(L[i][j,k] = q_{i,j}^k\).
‣ SaveHomogeneousCoherentConfigurationWithCertainAttributes( file, A, L ) | ( operation ) |
Returns: true
Saves homogeneous coherent configuration A to file F with the attributes listed in L. Note that L must be a list of strings, where each entry is an attribute known for A. Note that Print or PrintTo will only return the relation matrix of a homogeneous coherent configuration, which contains all necessary information about the homogeneous coherent configuration, but may require a lot of computation to reobtain its attributes. Hence this method is intended to alow saving of computationally difficult or time consuming attributes directly. It also alows the user to choose which attributes to save, since some attributes are very large, but easily recomputed. For example, it is often desirable to save the matrix of eigenvalues, and perhaps the automorphism group and intersection matrices, while it is not generally desirable to also save the adjacency matrices or minimal idempotents.
‣ ReadHomogeneousCoherentConfigurationWithCertainAttributes( file, A, L ) | ( operation ) |
Returns: homogeneous coherent configuration
Reads in a homogenous coherent configuration from file and sets it to have the attributes stored in the file. This reads files of the type formed by SaveHomogeneousCoherentConfigurationWithCertainAttributes.
‣ IsThin( CC ) | ( property ) |
Returns: true or false
Check if the homogeneous coherent configuration is thin.
‣ IsQuasiThin( CC ) | ( property ) |
Returns: true or false
Check if the homogeneous coherent configuration is quasi thin.
‣ IsPrimitive( CC ) | ( property ) |
Returns: true or false
Check if the homogeneous coherent configuration is primitve.
‣ ReorderMinimalIdempotents( CC, L ) | ( operation ) |
Returns: coherent configuration
Takes a homogeneous coherent configuration CC and a list L, where L is a reordering of the minimal idempotents. Returns a homogeneous coherent configuration where the \(i\)-th idempotent of the CC becomes the \(j\)-th idempotent in the new homogeneous coherent configuration, 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 diistribution diagram)
Take a CC and display the relation-distribution diagram with respect to \(R_1\).
‣ Description( CC ) | ( attribute ) |
Returns: lis
Take a CC and returns a list containing various descriptions/names of the CC, if available. Note that most homogeneous coherent configurations will not have a description. Some famous homogeneous coherent configurations, association schemes, or distance regular graphs in the library, as well as families that have constructor methods, will have names. Some will have multiple descriptions, hence they are given as a list. You can check if a homogeneous coherent configuration has assigned descriptions with HasDescription, or set one with SetDescription.
‣ DistanceRegularGraphScheme( A ) | ( operation ) |
Returns: homogeneous coherent configuration
Constructs an association scheme from the adjacency matrix A of a distance regular graph.
‣ DistanceRegularGraphSchemeNC( A ) | ( operation ) |
Returns: homogeneous coherent configuration
Same as DistanceRegularGraphScheme but without checking that a valid association scheme arises.
‣ StronglyRegularGraphScheme( A ) | ( operation ) |
Returns: homogeneous coherent configuration
Constructs an association scheme from the adjacency matrix A of a strongly regular graph.
‣ StronglyRegularGraphSchemeNC( A ) | ( operation ) |
Returns: homogeneous coherent configuration
Same as StronglyRegularGraphScheme but without checking that a valid association scheme arises.
‣ BilinearFormsScheme( F, n, k ) | ( operation ) |
Returns: homogeneous coherent configuration
Returns the bilinear forms scheme for the finite field \(F\) with a bilinear form from \(F^n \times F^n\) to \(F^k\).
‣ CyclotomicScheme( q, d ) | ( operation ) |
Returns: homogeneous coherent configuration
Let \(q\) be a prime power, and \(d\) a divisor of \(q-1\). Returns the cyclotomic scheme \(Cyc(q,d)\).
‣ GrassmannScheme( n, k, q ) | ( operation ) |
Returns: homogeneous coherent configuration
Returns the Grassmann scheme, \(J_q(n, k)\).
‣ GroupCoherentConfiguration( G ) | ( operation ) |
Returns: homogeneous coherent configuration
Returns the coherent configuration on the conjugacy classes of a group \(G\).
‣ HammingScheme( n, q ) | ( operation ) |
Returns: homogeneous coherent configuration
Returns the Hamming scheme, \(H(n, q)\).
‣ JohnsonScheme( n, k ) | ( operation ) |
Returns: homogeneous coherent configuration
Returns the Johnson scheme, \(J(n, k)\).
‣ DirectProductOfHomogeneousCoherentConfigurations( CC1, CC2 ) | ( operation ) |
Returns: homogeneous coherent configuration
Takes two homogeneous coherent configurations CC1 and CC2 and returns their direct product.
‣ WreathProductOfHomogeneousCoherentConfigurations( CC1, CC2 ) | ( operation ) |
Returns: homogeneous coherent configuration
Takes two homogeneous coherent configurations CC1 and CC2 and returns their wreath product.
‣ BipartiteDoubleOfAssociationScheme( A ) | ( operation ) |
Returns: Association scheme
Returns the bipartite double of an association scheme.
‣ ExtendedQBipartiteDouble( A ) | ( operation ) |
Returns: Association scheme
Given a cometric association scheme satisfying \(b_j^{*} + c_{j+1}^{*} = m +1\) for \(0 \leq j \leq d-1 \), returns the extended Q-bipartite double.
‣ HomogeneousCoherentConfigurationByOrbitals( G ) | ( operation ) |
Returns: homogeneous coherent configuration
Constructs a "group-case" coherent configuration, where the relations are defined by the orbitals of \(G\) on \(\{1, \ldots, n\} \times \{1, \ldots, n\}\). \(G\) must be a permutation group which is transitive on \(\{1, \ldots, n\}\).
‣ HomogeneousCoherentConfigurationByOrbitals( G, H ) | ( operation ) |
Returns: homogeneous coherent configuration
Constructs a "group-case" coherent configuration, where the relations are defined by the orbitals of \(G\) on \(G/H\). \(G\) is a group, \(H\) is a subgroup of \(G\), \(G/H\) is the set of right cosets of \(G\) on \(H\), and \(G\) must be transitive on \(G/H\).
‣ SchurianAssociationScheme( G ) | ( operation ) |
Returns: homogeneous coherent configuration
Returns the Schurian scheme defined by \(G\), where \(G\) is a generously transitive permutation group. A Schurian scheme is a special case of CoherentConfigurationByOrbitals and is symmetric.
‣ SchurianCoherentConfiguration( G ) | ( operation ) |
Returns: homogeneous coherent configuration
Alias for HomogeneousCoherentConfigurationByOrbitals
‣ HomogeneousCoherentConfiguration( n, k ) | ( operation ) |
Returns: homogeneous coherent configuration
Returns the \(k\)-th homogeneous coherent configuration of order \(n\). Library is complete for \(5 \le n \le 38\) excluding \(n \in \{35, 36, 37\}\), corresponding to [HM]. Thin schemes are are assigned numbers after those in [HM].
‣ NumberOfHomogeneousCoherentConfigurations( n ) | ( operation ) |
Returns: m
Returns the number \(m\) of homogeneous coherent configuration of order \(n\) contained in the library.
‣ AvailableHomogeneousCoherentConfigurations( ) | ( operation ) |
Returns: L
Returns a list \(L\) of the orders for which the library contains homogeneous coherent configurations.
‣ AllHomogeneousCoherentConfigurations( n ) | ( operation ) |
Returns: L
Returns a list \(L\) of all homogeneous coherent configuration of order \(n\).
‣ SmallSchemeIdentification( CC ) | ( attribute ) |
Returns: id
Returns the identification, id, of the homogeneous coherent configuration in the library which is is isomorphic to CC. Thus HomogeneousCoherentConfiguration(n, id) will be isomorphic to CC, where n is the order of CC.
‣ Digraph( CC, k ) | ( operation ) |
Returns: homogeneous coherent configuration
Returns the digraph object associated with the k-th relation of a homogeneous coherent configuration CC. Note that the identity relation is excluded.
‣ Digraph( CC, S ) | ( operation ) |
Returns: homogeneous coherent configuration
Returns the digraph object which is a union of the relations of a homogeneous coherent configuration CC given by the set S. Note that the identity relation is excluded.
‣ AutomorphismGroup( CC ) | ( attribute ) |
Returns: G
Returns the automorphism group \(G\) of the coherent configuration CC. \(G\) is a permutation group acting on the index set of the vertices.
‣ AlgebraicAutomorphismGroup( CC ) | ( attribute ) |
Returns: G
Returns the algebraic automorphism group \(G\) of the coherent configuration CC. \(G\) is a permutation group acting on the set of non-trivial relations.
‣ ImageOfHomogeneousCoherentConfiguration( CC, p, $\sigma$ ) | ( operation ) |
Returns: true or false
For a \(d\)-class homogeneous coherent configuration with relation matrix \(M\) and order \(n\), takes a permutation \(p\) on the set \(\{1 .. n\}\) and permutation \(\sigma\) on the set \(\{1 .. d\}\) and returns the \(d\)-class homogenous coherent configuration with relation matrix \(P^{-1} M^\sigma P\), where \(P\) is the permutation matrix defined by \(\sigma\).
‣ IsomorphismHomogeneousCoherentConfigurations( A, B ) | ( operation ) |
Returns: true or false
If there exists a permutation matrix \(P\) and permutation \(\sigma\) such that if \(M\) is the relation matrix of \(A\), then \(P^{-1} M^\sigma P\) is the relation matrix of \(B\), then the operation will return \([p, \sigma]\), where \(p\) is the permutation defining \(P\). If no such \(P\) and \(\sigma\) exist, then the operation will return fail.
‣ AlgebraicIsomorphismHomogeneousCoherentConfigurations( A, B ) | ( operation ) |
Returns: true or false
Returns a permutation which maps the set of relations of \(A\) to \(B\) such that the intersection numbers are preserved. Returns fail if such a bijection between relations does not exist.
‣ AreIsomorphicHomogeneousCoherentConfigurations( A, B ) | ( operation ) |
Returns: true or false
If there exists a permutation matrix \(P\) and permutation \(\sigma\) such that if \(M\) is the relation matrix of \(A\), then \(P^{-1} M^\sigma P\) is the relation matrix of \(B\), then the operation will return true. Returns false otherwise.
‣ AreAlgebraicallyIsomorphicHomogeneousCoherentConfigurations( A, B ) | ( operation ) |
Returns: true or false
Returns true if there exists a permutation which maps the set of relations of \(A\) to \(B\) such that the intersection numbers are preserved. Returns false otherwise.
‣ CanonisingMap( CC ) | ( attribute ) |
Returns: [perm1, perm2]
Returns two permutations which will produce the canonical form of the homogeneous coherent configuration CC. The canonical form can be obtained by ImageOfHomogeneousCoherentConfiguration(CC, perm1, perm2) Any homogenouse coherent configuration which is isomorphic to CC will the same canonical form.
‣ CanonicalFormOfHomogeneousCoherentConfiguration( CC ) | ( operation ) |
Returns: CC2
Returns the canonical form, CC2, of the homogeneous coherent configuration CC. Any homogenouse coherent configuration which is isomorphic to CC will have CC2 as the canonical form.
‣ ConstructorGroup( CC ) | ( attribute ) |
Returns: group or false
Checks if the coherent configuration was constructed by a group and returns it if it was, or returns false otherwise.
‣ IsGenerouslyTransitive( G ) | ( property ) |
Returns: true or false
Checks if the permutation group \(G\) is generously transitive.
‣ IsGenerouslyTransitive( G, L ) | ( operation ) |
Returns: true or false
Checks that the permutation group \(G\) acts generously transitive on the set \(L\).
‣ IsSchurian( CC ) | ( property ) |
Returns: true or false
Checks if the input is a Schurian scheme, that is, if the automorphism group is transitive.
‣ IsFusionOfHomogeneousCoherentConfiguration( 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.
‣ FusionOfHomogeneousCoherentConfiguration( 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.
‣ SymmetrisationOfHomogeneousCoherentConfiguration( 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.
‣ FusingPartitionOfHomogeneousCoherentConfigurations( CC1, CC2 ) | ( operation ) |
Returns: partition
Takes two homogeneous coherent configurations, CC1 and CC2, where CC1 is \(d\)-class. If CC2 is equal to a homogeneous coherent configuration formed by fusing the relations of CC2, this will return the partition of \({0, \ldots, d}\) corresponding to this fusion. If CC2 cannot be produced as a fusion, then "fail" is returned. This operation does not consider isomorphic homogeneous coherent configurations - CC2 must be exactly equal to a fusion.
‣ FeasibleNonTrivialFusionsOfHomgeneousCoherentConfiguration( 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.
‣ AllNonTrivialFusionsOfHomgeneousCoherentConfiguration( 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.
‣ AllFusionsOfHomgeneousCoherentConfiguration( 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
‣ FeasibleNonTrivialSymmetricFusionsOfHomgeneousCoherentConfiguration( 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.
‣ AllNonTrivialSymmetricFusionsOfHomgeneousCoherentConfiguration( 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.
‣ AllSymmetricFusionsOfHomgeneousCoherentConfiguration( 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
‣ IsomorphismToFusionScheme( A, B ) | ( operation ) |
Returns: p1, p2, f
If A is isomorphic to a fusion of B, then f is the fusion of \(B\), and [p1, p2] is the map which carries \(B\) to this fusion.
‣ IsIsomorphicToFusionScheme( A, B ) | ( operation ) |
Returns: p1, p2, f
Returns true if \(A\) is isomorphic to a fusion of \(B\). Returns false otherwise.
‣ FirstFeasibleNonTrivialSymmetricFusionOfHomogeneousCoherentConfiguration( 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.
‣ MapFromAdjacencyMatricesToMinimalIdempotents( CC ) | ( attribute ) |
Returns: M
Takes a homogeneous coherent configuration object, CC, and returns a matrix, M, which maps the adjacency matrices to the minimal idempotents of the adjacency algebra. The central idempotent \(E_i = \sum_{i=0}^d M_{ij} A_j\).
‣ MapFromAdjacencyMatricesToMinimalIdempotentsOverRationals( CC ) | ( attribute ) |
Returns: M
Takes a homogeneous coherent configuration, CC, and returns a matrix, M, which maps the adjacency matrices to the minimal idempotents of the adjacency algebra over the rationals. The minimal idempotent \(E_i = \sum_{i=0}^d M_{ij} A_j\).
‣ MinimalIdempotents( CC ) | ( attribute ) |
Returns: minimal idempotents
Returns the minimal idempotents of the homogeneous coherent configuration.
‣ MinimalIdempotentsOverRationals( CC ) | ( attribute ) |
Returns: minimal idempotents
Returns the minimal idempotents of the homogeneous coherent configuration over the rationals.
‣ MatrixOfEigenvalues( CC ) | ( attribute ) |
Returns: P
Returns a the matrix of eigenvalues (or character table), \(P\), for a homogeneous coherent configuration CC.
‣ MatrixOfDualEigenvalues( CC ) | ( attribute ) |
Returns: Q
Returns a the matrix of dual eigenvalues, \(Q\), for a homogeneous coherent configuration CC.
‣ Multiplicities( CC ) | ( attribute ) |
Returns: n
Returns the multiplicities of characters of CC.
‣ CharacterTableOfHomogeneousCoherentConfiguration( CC ) | ( attribute ) |
Returns: P
TODO.
‣ FitMatrixOfEigenvalues( CC, P ) | ( operation ) |
Returns: P2
Checks if P is the matrix of eigenvalues of homogeneous coherent configuration CC, upto some reordering of the columns. In such a case, P2, the reordered matrix is returned. If not, returns fail.
‣ CharacterTableOfSchurianHomogeneousCoherentConfiguration( CC, P ) | ( operation ) |
Returns: P2
Computes the character table of a Schurian coherent configuration by using the group. The ordering of the columns does not respect the ordering of the coherent configuration, so "FitMatrixOfEigenvalues" must be used. Sometimes the group theoretic method is much faster and sometimes it is much slower than the other methods.
‣ IsCharacterTableOfHomogeneousCoherentConfiguration( CC, T, mults ) | ( operation ) |
Returns: P2
Checks if T is the matrix of eigenvalues of homogeneous coherent configuration CC, given multiplicities mults.
‣ MatrixOfEigenvaluesOfCyclotomicScheme( n, k, q ) | ( operation ) |
Returns: P
Returns the matrix of eigenvalues \(P\) of the scheme \(Cyc(q,d)\).
‣ MatrixOfEigenvaluesOfGrassmannScheme( n, k, q ) | ( operation ) |
Returns: P
Returns the matrix of eigenvalues \(P\) of the Grassmann scheme \(J_q(n, k)\).
‣ MatrixOfEigenvaluesOfHammingScheme( n, q ) | ( operation ) |
Returns: P
Returns matrix of eigenvalue \(P\) for the Hamming scheme, \(H(n, q)\).
‣ MatrixOfEigenvaluesOfJohnsonScheme( n, k ) | ( operation ) |
Returns: P
Returns the matrix of eigenvalues \(P\) of the Johnson scheme \(J(n, k)\).
‣ 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.
‣ IsStronglyRegularGraph( CC ) | ( property ) |
Returns: true or false
Check if a coherent configuration is a strongly regular graph (a \(2\)-class primitive association scheme).
‣ 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.
‣ IsPBipartite( CC ) | ( property ) |
Returns: true or false
Returns if the homogeneous coherent configuration CC is bipartite.
‣ IsPAntipodal( CC ) | ( property ) |
Returns: true or false
Returns if the homogeneous coherent configuration CC is antipodal.
‣ 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.
‣ IsQBipartite( CC ) | ( property ) |
Returns: true or false
Returns if the homogeneous coherent configuration CC is Q-bipartite.
‣ IsQAntipodal( CC ) | ( property ) |
Returns: true or false
Returns if the homogeneous coherent configuration CC is Q-antipodal.
‣ CharacteristicVector( Omega, X ) | ( operation ) |
Returns: \(\chi_X\)
Takes a subset X of Omega and returns the characteristic vector. The characteristic vector is a 0,1-vector indexed by the entries of Omega, with a 1 at position x if x is in X, and 0 otherwise.
‣ CharacteristicVector( X, n ) | ( operation ) |
Returns: \(\chi_X\)
Takes a subset X of of [1 .. n] and returns the characteristic vector \(chi_X\).
‣ InnerDistribution( v, CC ) | ( operation ) |
Returns: a
Returns the inner distribution \(a\) of a vector v with respect to the adjacency matrices of the coherent configuration CC. Note that v must be a vector over R^n where n is the order of CC. CC must be commutative.
‣ MacWilliamsTransform( v, CC ) | ( operation ) |
Returns: aQ
Returns the MacWilliams transform \(aQ\) of a vector \(v\) with respect to a coherent confiiguration CC. Takes either a vector \(v\) in \(R^n\) and converts it to its inner distribution vector first, or takes the inner distribution directly.
‣ DualBoseMesnerBasis( CC, p ) | ( operation ) |
Returns: L
Returns a list \(L\) with the dual Bose-Mesner basis of a homogeneous coherent configuration with respect to the point p, such that \(L_i = \tilde E_{i-1}\).
‣ DualBoseMesnerBasis( CC ) | ( operation ) |
Returns: L
Returns a list \(L\) with the dual Bose-Mesner basis of a homogeneous coherent configuration with respect to the first point, such that \(L_i = \tilde E_{i-1}\).
‣ OuterDistribution( v, CC ) | ( operation ) |
Returns: B
Returns the outer distribution \(B\) of a vector \(v\) with respect to the adjacency matrices of the coherent configuration CC. Note that \(v\) must be a vector over \(R^n\) where n is the order of CC. CC must be commutative.
‣ DelsarteDesignType( CC, S ) | ( operation ) |
Returns: T
Returns T such that S is a Delsarte T-design, that is, \(\chi_S E_i =0\) for all \(i \in T\). S must be a subset of the vertices.
‣ WeightedDelsarteDesignType( CC, S ) | ( operation ) |
Returns: T
Returns T such that v is a (weighted) Delsarte T-design, that is, \(v E_i =0\) for all \(i \in T\). v must be a weighted characteristic vector with respect to the vertices.
‣ IsDelsarteTDesign( CC, S ) | ( operation ) |
Returns: true or false
Checks that S is a (weighted) Delsarte T-design, that is, \(\chi_S E_i =0\) for all \(i \in T\). S must be either a subset of the vertices, or a weighted characteristic vector with respect to the vertices.
‣ DualDegreeSet( CC, S ) | ( operation ) |
Returns: K
Returns the dual degree set K for a set S, that is, \(\chi_S E_i \neq 0\) for all \(i \in K\). S must be a subset of the vertices.
‣ WeightedDualDegreeSet( CC, S ) | ( operation ) |
Returns: K
Returns the dual degree set K for a set S, that is, \(v E_i \neq 0\) for all \(i \in K\). v must be a (weighted) characteristic vector with restpect to the vertices.
‣ AreDesignOrthogonal( CC, S1, S2 ) | ( operation ) |
Returns: true or false
If S1 and S2 are either subsets of vertices, or (weighted) characteristic vectors, this checks that they are design orthogonal, that is, their dual degree sets are disjoint.
‣ ApproximateRealMinimalIdempotent( CC, i ) | ( operation ) |
Returns: approximation of \(E_i\)
Returns the \(i\)-th idempotent with entries approximated by floats. All entries must be real values.
‣ ApproximateRealMinimalIdempotents( CC ) | ( operation ) |
Returns: approximation of minimal idempotents
Returns a list of idempotents with entries approximated by floats. All entries must be real values.
‣ ApproximateRealMatrixOfEigenvalues( CC ) | ( operation ) |
Returns: approximation of matrix of eigenvalues
Returns the matrix of eigenvalues with entries approximated by floats. All entries must be real values.
‣ BoseMesnerAlgebra( CC ) | ( operation ) |
Returns: A
Returns the Bose-Mesner algebra \(A\) of a homogeneous coherent configuration.
‣ AdjacencyAlgebra( CC ) | ( operation ) |
Returns: A
Returns the adjacency algebra \(A\) of a homogeneous coherent configuration. This is an alias for BoseMesnerAlgebra.
‣ TerwilligerAlgebra( CC, p ) | ( operation ) |
Returns: T
Returns the Terwilliger algebra \(T\) of a homogeneous coherent configuration with respect to the point p.
‣ TerwilligerAlgebra( CC ) | ( operation ) |
Returns: T
Returns the Terwilliger algebra \(T\) of a homogeneous coherent configuration with respect to the first point.
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