Matrix Groups over General Rings
- Introduction
- Creation of a Matrix Group
- Construction of the General Linear Group
- Construction of a Matrix Group Element
- Construction of a General Matrix Group
- Changing Rings
ChangeRing(G, S): GrpMat, Rng → GrpMat, MapChangeRing(G, S, f): GrpMat, Rng, Map → GrpMat, MapRestrictField(G, S): GrpMat, FldFin → GrpMat, MapExtendField(G, L): GrpMat, FldFin → GrpMat, Map
- Coercion between Matrix Structures
- Accessing Associated Structures
- Homomorphisms
hom<G -> H | L>: GrpMat, Grp, List → MapDomain(f): Map → GrpCodomain(f): Map → GrpImage(f): Map → GrpKernel(f): Map → GrpIsHomomorphism(G, H, Q): GrpMat, GrpMat, SeqEnum[GrpMatElt] → Bool, Map- Example: Dihedral Group to Symmetric Group
PermutationRepresentation(G): GrpMat → Map, GrpPerm, GrpMat- Example: Permutation Representation of SL(4, 5)
- Construction of Extensions
DirectProduct(G, H): GrpMat, GrpMat → GrpMatDirectProduct(Q): [ GrpMat ] → GrpMatSemiLinearGroup(G, S): GrpMat, FldFin → GrpMatTensorWreathProduct(G, H): GrpMat, GrpPerm → GrpMatWreathProduct(G, H): GrpMat, GrpPerm → GrpMat- Example: Direct, Tensor, and Wreath Products
- Operations on Matrices
- Arithmetic with Matrices
g * h: GrpMatElt, GrpMatElt → GrpMatEltg ^ n: GrpMatElt, RngIntElt → GrpMatEltg / h: GrpMatElt, GrpMatElt → GrpMatEltg ^ h: GrpMatElt, GrpMatElt → GrpMatElt(g, h): GrpMatElt, GrpMatElt → GrpMatElt(g₁, ..., gᵣ): GrpMatElt, ..., GrpMatElt → GrpMatElt- Example: Arithmetic with Matrices in GL(3, 4)
- Predicates for Matrices
- Matrix Invariants
- Global Properties
- Group Order
- Membership and Equality
g in G: GrpMatElt, GrpMat → BoolEltg notin G: GrpMatElt, GrpMat → BoolEltS subset G: { GrpMatElt }, GrpMat → BoolEltH subset G: GrpMat, GrpMat → BoolEltIsSubgroup(H,G): GrpMat, GrpMat → BoolEltS notsubset G: { GrpMatElt }, GrpMat → BoolEltH notsubset G: GrpMat, GrpMat → BoolEltH eq G: GrpMat, GrpMat → BoolEltH ne G: GrpMat, GrpMat → BoolElt
- Set Operations
- Abstract Group Predicates
- Conjugacy
Class(H, x): GrpMat, GrpMatElt → { GrpMatElt }Conjugates(H, x): GrpMat, GrpMatElt → { GrpMatElt }ClassMap(G): GrpMat → MapConjugacyClasses(G: parameters): GrpMat → [ < RngIntElt, RngIntElt, GrpMatElt > ]Classes(G: parameters): GrpMat → [ < RngIntElt, RngIntElt, GrpMatElt > ]ClassRepresentative(G, x): GrpMat, GrpMatElt → GrpMatEltClassRepresentative(G, i): GrpMat, RngIntElt → GrpMatEltClassCentraliser(G, i): GrpMat, RngIntElt → GrpMatClassCentralizer(G, i): GrpMat, RngIntElt → GrpMatClassRepresentativeFromInvariants(G, p, h, t): GrpMat, SeqEnum, SeqEnum, FldFinElt → GrpMatEltIsConjugate(G, g, h): GrpMat, GrpMatElt, GrpMatElt → BoolElt, GrpMatElt | UnassNumberOfClasses(G): GrpMat → RngIntEltNclasses(G): GrpMat → RngIntEltPowerMap(G): GrpMat → MapAssertAttribute(G, "Classes", Q): GrpMat, MonStgElt, SeqEnum- Example: Conjugacy Classes from the Rational Database
- Conjugacy in Classical Groups
DualPolynomial(f): RngUPolElt → RngUPolEltStarIrreduciblePolynomials(F,d): FldFin, RngIntElt → SeqEnumPhiDual(f,phi): RngUPolElt → RngUPolEltPhiIrreduciblePolynomials(F,d): FldFin, RngIntElt → SeqEnum[Tup]ExtendedSymplecticGroup(n,q,m): RngIntElt, RngIntElt, RngIntElt → GrpMatExtendedSp(n,q,m): RngIntElt, RngIntElt, RngIntElt → GrpMatIndexOfSp(G): GrpMat → RngIntEltTildeDualPolynomial(f): RngUPolElt → RngUPolEltTildeIrreduciblePolynomials(q,d): RngIntElt, RngIntElt → SeqEnumExtendedSpecialUnitaryGroup(n,q,m): RngIntElt, RngIntElt, RngIntElt → GrpMatExtendedSU(n,q,m): RngIntElt, RngIntElt, RngIntElt → GrpMatIndexOfSU(G): GrpMat → RngIntEltClassicalConjugacyClasses(G): GrpMat → SeqEnum, SetIndxClassicalClasses(G): GrpMat → SeqEnum, SetIndxClassicalConjugacyClasses(type,d,q): MonStgElt, RngIntElt, RngIntElt → SeqEnum, SetIndxClassicalClasses(type,d,q): MonStgElt, RngIntElt, RngIntElt → SeqEnum, SetIndxClassicalCentralizer(G,g): GrpMat, GrpMatElt → GrpMatClassicalCentraliserOrder(G,g): GrpMat, GrpMatElt → RngIntEltFactClassicalClassSize(G,g): GrpMat, GrpMatElt → RngIntEltClassicalIsConjugate(G,g,h): GrpMat, GrpMatElt, GrpMatElt → BoolElt, GrpMatElt- Example: Centralisers in an Orthogonal Group
ClassicalClassMap(G): GrpMat → MapClassicalClassMap(G,C,L): GrpMat, SeqEnum, SetIndx → MapClassesForFixedSemisimple(G,x): GrpMat, GrpMatElt → SeqEnum, SetIndxIsometryGroupClassLabel(type, g): MonStgElt, GrpMatElt → SetMulti- Example: Classical Class Map for SO+(6, 9)
- Example: Conjugacy Classes of GO(7, 9)
- Example: Products of Involutions in Sp(d, q)
UnipotentClasses(type,d,q): MonStgElt, RngIntElt, RngIntElt → SeqEnum, SeqEnumUnipotentClasses(type,d,F): MonStgElt, RngIntElt, FldFin → SeqEnum, SeqEnumSemisimpleClasses(type,d,q): MonStgElt, RngIntElt, RngIntElt → SeqEnumSemisimpleClasses(type,d,F): MonStgElt, RngIntElt, FldFin → SeqEnumIsometryGroupNumberOfClasses(type, n): MonStgElt, RngIntElt → RngUPolElt- Example: Unipotent Classes of SU(12, 9)
ProjectiveClassicalClasses(type,d,q): MonStgElt, RngIntElt, RngIntElt → SeqEnum, GrpPerm, HomGrp, SeqEnumProjectiveClassicalCentraliser(G, g): GrpMat, GrpMatElt → GrpMatProjectiveClassicalIsConjugate(G, g, h): GrpMat, GrpMatElt, GrpMatElt → BoolElt, GrpMatElt- Example: Fast Projective Conjugacy Classes
- Example: Projective Class Preimages as Matrices
SpinConjugacyClasses(G): GrpMat → SeqEnum, SeqEnumSpinClasses(G): GrpMat → SeqEnum, SeqEnumSpinCentralizer(G,g): GrpMat, GrpMatElt → GrpMatSpinIsConjugate(G,g,h): GrpMat, GrpMatElt, GrpMatElt → BoolElt, GrpMatEltSpinClassMap(G): GrpMat → Map
- Subgroups
- Construction of Subgroups
- Elementary Properties of Subgroups
Index(G, H): GrpMat, GrpMat → RngIntEltFactoredIndex(G, H): GrpMat, GrpMat → [ <RngIntElt, RngIntElt> ]IsCentral(G, H): GrpMat, GrpMat → BoolEltIsMaximal(G, H): GrpMat, GrpMat → BoolEltIsNormal(G, H): GrpMat, GrpMat → BoolEltIsSubnormal(G, H): GrpMat, GrpMat → BoolElt
- Standard Subgroups
H ^ g: GrpMat, GrpMatElt → GrpMatConjugate(H, g): GrpMat, GrpMatElt → GrpMatH meet K: GrpMat, GrpMat → GrpMatCommutatorSubgroup(G, H, K): GrpMat, GrpMat, GrpMat → GrpMatCommutatorSubgroup(H, K): GrpMat, GrpMat → GrpMatCentraliser(G, g): GrpMat, GrpMatElt → GrpMatCentralizer(G, g): GrpMat, GrpMatElt → GrpMatCentraliser(G, H): GrpMat, GrpMat → GrpMatCentralizer(G, H): GrpMat, GrpMat → GrpMatCore(G, H): GrpMat, GrpMat → GrpMatH ^ G: GrpMat, GrpMat → GrpMatNormalClosure(G, H): GrpMat, GrpMat → GrpMatNormalizer(G, H): GrpMat, GrpMat → GrpMatGLNormalizer(H : parameter): GrpMat → GrpMatSylowSubgroup(G, p): GrpMat, RngIntElt → GrpMatSylow(G, p): GrpMat, RngIntElt → GrpMatpCore(G, p): GrpMat, RngIntElt → GrpMat
- Low Index Subgroups
LowIndexSubgroups(G,n: parameters): GrpMat, RngIntElt → SeqEnumLowIndexSubgroups(G,t: parameters): GrpMat, Tup → SeqEnumLowIndexSubgroups(G, N, n: parameters): GrpMat, RngIntElt → SeqEnumLowIndexSubgroups(G, N, t: parameters): GrpMat, Tup → SeqEnumLowIndexSubgroupsCT(G, R : parameters): GrpMat, RngIntElt → [ GrpMat ]LowIndexSubgroupsCT(G, R: parameters): GrpMat, <RngIntElt, RngIntElt> → [ GrpMat ]- Example: Low Index Subgroups of GL(4, 5)
- Conjugacy Classes of Subgroups
SubgroupClasses(G: parameters): GrpMat → [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]Subgroups(G: parameters): GrpMat → [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]MaximalSubgroups(G: parameters): GrpMat → [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]MaximalSubgroups(G,N: parameters): GrpMat, GrpMat → [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]SubgroupsLift(G, A, B, Q: parameters): GrpMat, GrpMat, GrpMat, SeqEnum → SeqEnumIsConjugate(G, H, K): GrpMat, GrpMat, GrpMat → BoolElt, GrpMatElt | UnassIsGLConjugate(H, K): GrpMat, GrpMat → BoolElt, GrpMatElt | Unass
- Quotient Groups
- Construction of Quotient Groups
- Abelian, Nilpotent, and Soluble Quotients
AbelianQuotient(G): GrpMat → GrpAb, MapElementaryAbelianQuotient(G, p): GrpMat, RngIntElt → GrpAb, MappQuotient(G, p, c): GrpMat, RngIntElt, RngIntElt → GrpPC, Map, SeqEnum, BoolEltNilpotentQuotient(G, c): GrpMat, RngIntElt → GrpGPC, MapSolvableQuotient(G): GrpMat → GrpPC, MapSolubleQuotient(G): GrpMat → GrpPC, MapPCGroup(G): GrpMat → GrpPC, Map- Example: Abelian and Soluble Quotients
- Matrix Group Actions
- Orbits and Stabilizers
u * g: ModTupRngElt, GrpMatElt → ModTupRngElty ^ g: Elt, GrpMatElt → Elty ^ G: Elt, GrpMat → SetEnumOrbit(G, y): GrpMat, Elt → SetEnumOrbitBounded(G, y, b): GrpMat, Elt, RngIntElt → BoolElt, SetEnumOrbits(G): GrpMat → [ SetIndx ]LineOrbits(G): GrpMat → [ SetIndx ]OrbitClosure(G, S): GrpMat, { Elt } → GSetStabilizer(G, y): GrpMat, Elt → GrpMat- Example: Orbits of Vectors and Subspaces
- Orbit and Stabilizer Functions for Large Groups
OrbitsOfSpaces(G, k): GrpMat, RngIntElt → SeqEnumNumberOfFixedSpaces(x, s): GrpMatElt, RngIntElt → RngIntEltNumberOfFixedSpaces(x, s): AlgMatElt, RngIntElt → RngIntElt- Example: Orbits of 2-Spaces under an Exterior Square
EstimateOrbit(G, v: parameters): GrpMat, ModTupFldElt → RngIntElt, RngIntElt, RngIntEltEstimateOrbit(G, U: parameters): GrpMat, ModTupFld → RngIntElt, RngIntElt, RngIntEltApproximateStabiliser(G, A, U: parameters): GrpMat, GrpMat, ModTupFld → GrpMat, GrpMat, RngIntElt, RngIntElt, RngIntElt- Example: Estimating an Orbit and Stabiliser
StabiliserOfSpaces(Q): SeqEnum → GrpMat, SeqEnum- Example: Stabiliser of Spaces
IsUnipotent(G): GrpMat → BoolEltUnipotentStabiliser(G, U: parameters): GrpMat, ModTupFld → GrpMat, ModTupFld, GrpMatElt, GrpSLPElt- Example: Unipotent Stabiliser of Spaces
- Action on Orbits
OrbitAction(G, T): GrpMat, Elt → Hom(Grp), GrpPerm, GrpMatOrbitActionBounded(G, T, b): GrpMat, Elt, RngIntElt → BoolElt, Hom(Grp), GrpPerm, GrpMatOrbitImage(G, T): GrpMat, Set → GrpPerm, SetIndxOrbitImageBounded(G, T, b): GrpMat, Set, RngIntElt → BoolElt, GrpPerm, SetIndxOrbitKernel(G, T): GrpMat, Set → GrpMatOrbitKernelBounded(G, T, b): GrpMat, Set, RngIntElt → BoolElt, GrpMat- Example: Permutation Representation from Eigenspaces
- Action on a Coset Space
- Action on the Natural \(G\)-Module
- Normal and Subnormal Subgroups
- Coset Tables and Transversals
- Presentations
- Automorphism Groups
- Representation Theory
LinearCharacters(G): GrpMat → [ Chtr ]CharacterTable(G: parameters): GrpMat → TabChtrPermutationCharacter(G, H): GrpMat, GrpMat → AlgChtrEltGModule(G): GrpMat → ModGrpGModule(G, A): GrpMat, AlgMat → ModGrpGModule(G, Q): GrpMat, [ AlgMatElt ] → ModGrpGModule(G, A, B): GrpMat, GrpMat, GrpMat → ModGrp, MapPermutationModule(G, H, R): GrpMat, GrpMat, Rng → ModGrpChangeOfBasisMatrix(G, S): GrpMat, ModGrp → AlgMatElt- Example: Refining a Normal Subgroup with Modules
- Base and Strong Generating Set
- Introduction
- Controlling Selection of a Base
GoodBasePoints(G: parameters): GrpMat → []AssertAttribute(G, "Base", B): GrpMat, MonStgElt, TupHasAttribute(G, "Base"): GrpMat, MonStgElt → BoolElt, TupAssertAttribute(GrpMat, "FirstBasicOrbitBound", n): Cat, MonStgElt, RngIntEltHasAttribute(GrpMat, "FirstBasicOrbitBound"): Cat, MonStgElt → BoolElt, RngIntElt
- Construction of a Base and Strong Generating Set
- Defining Values for Attributes
AssertAttribute(G, "Order", n): GrpMat, MonStgElt, RngIntEltAssertAttribute(G, "Order", Q): GrpMat, MonStgElt, [Tup(RngIntElt, RngIntElt)]AssertAttribute(G, "IsVerified", b): GrpMat, MonStgElt, BoolEltHasAttribute(G, "Order"): GrpMat, MonStgElt → RngIntEltHasAttribute(G, "FactoredOrder"): GrpMat, MonStgElt → [Tup(RngIntElt, RngIntElt)]HasAttribute(G, "IsVerified"): GrpMat, MonStgElt → BoolElt
- Accessing the Base and Strong Generating Set
Base(G): GrpMat → [Elt]BasePoint(G, i): GrpMat, RngIntElt → EltBasicOrbit(G, i): GrpMat, RngIntElt → SetIndxBasicOrbitLength(G, i): GrpMat, RngIntElt → RngIntEltBasicOrbitLengths(G): GrpMat → [RngIntElt]BasicStabilizer(G, i): GrpMat, RngIntElt → GrpMatBasicStabiliser(G, i): GrpMat, RngIntElt → GrpMatBasicStabilizerChain(G): GrpMat → [GrpMat]BasicStabiliserChain(G): GrpMat → [GrpMat]NumberOfStrongGenerators(G): GrpMat → RngIntEltNsgens(G): GrpMat → RngIntEltStrongGenerators(G): GrpMat → SetIndx(GrpMat)
- Soluble Matrix Groups
- Bibliography