Groups of Lie Type
- Introduction
- Constructing Groups of Lie Type
- Split Groups
GroupOfLieType(N, k): MonStgElt, Rng → GrpLieGroupOfLieType(N, q): MonStgElt, RngIntElt → GrpLieGroupOfLieType(W, k): GrpPermCox, Rng → GrpLieGroupOfLieType(W, k): GrpMat, Rng → GrpLieGroupOfLieType(W, q): GrpPermCox, RngIntElt → GrpLieGroupOfLieType(W, q): GrpMat, RngIntElt → GrpLieGroupOfLieType(R, k): RootDtm, Rng → GrpLieGroupOfLieType(R, q): RootDtm, RngIntElt → GrpLieGroupOfLieType(C, k): Mtrx, Rng → GrpLieGroupOfLieType(D, k): GrphDir, Rng → GrpLieGroupOfLieType(C, q): Mtrx, RngIntElt → GrpLieGroupOfLieType(D, q): GrphDir, RngIntElt → GrpLieSimpleGroupOfLieType(X, n, k): MonStgElt, RngIntElt, Rng → GrpLieSimpleGroupOfLieType(X, n, q): MonStgElt, RngIntElt, RngIntElt → GrpLieGroupOfLieType(L): AlgLie → GrpLieIsNormalising(G): GrpLie → BoolElt- Example: Creating a Split Group of Lie Type
- Galois Cohomology
- Twisted Groups
TwistedGroupOfLieType(c): OneCoC → GrpLieTwistedGroupOfLieType(R, k, K): RootDtm, Rng, Rng → GrpLieTwistedGroupOfLieType(R, q, r): RootDtm, RngIntElt, RngIntElt → GrpLieTwistedGroupOfLieType(t, r, q): MonStgElt, RngIntElt, RngIntElt → GrpLie- Example: Twisted Group of Type 3D4
BaseRing(G): GrpLie → RngCoefficientRing(G): GrpLie → RngDefRing(G): GrpLie → RngUntwistedOvergroup(G): GrpLie → GrpLie- Example: Twisted Group 2A3 as a Subgroup
RelativeRootElement(G,delta,t): GrpLie, RngIntElt, [FldElt] → GrpLieElt- Example: Twisted Group via a Cocycle
- Operations on Groups of Lie Type
G eq H: GrpLie, GrpLie → BoolEltG subset H: GrpLie, GrpLie → BoolEltIsAlgebraicallyIsomorphic(G, H): GrpLie, GrpLie → BoolElt, MapIsIsogenous(G, H): GrpLie, GrpLie → BoolEltIsCartanEquivalent(G, H): GrpLie, GrpLie → BoolEltBaseRing(G): GrpLie → RngCoefficientRing(G): GrpLie → RngBaseExtend(G, K): GrpLie, Rng → GrpLie, MapChangeRing(G, K): GrpLie, Rng → GrpLieGenerators(G): GrpLie → {}NumberOfGenerators(G): GrpLie → RngIntEltNgens(G): GrpLie → RngIntEltAlgebraicGenerators(G): GrpLie → {}NumberOfAlgebraicGenerators(G): GrpLie → RngIntEltNalggens(G): GrpLie → RngIntElt- Example: Generators of a Group of Lie Type
Order(G): GrpLie → RngIntElt# G: GrpLie → RngIntEltFactoredOrder(G): GrpLie → RngIntEltDimension(G): GrpLie → RngIntElt- Example: Orders of Groups of Lie Type
CartanName(G): GrpLie → MtrxRootDatum(G): GrpLie → RootDtmDynkinDiagram(G): GrpLieCoxeterDiagram(G): GrpLieCoxeterMatrix(G): GrpLie → AlgMatEltCoxeterGraph(G): GrpLie → GrphUndCartanMatrix(G): GrpLie → GrphUndDynkinDigraph(G): GrpLie → GrphUndRank(G): GrpLie → RngIntEltReductiveRank(G): GrpLie → RngIntEltSemisimpleRank(G): GrpLie → RngIntEltCoxeterNumber(G): GrpLie → RngIntEltWeylGroup(G): GrpLie → GrpPermCoxWeylGroup(GrpPermCox, G): Cat, GrpLie → GrpPermCoxWeylGroup(GrpFPCox, G): Cat, GrpLie → GrpFPCoxWeylGroup(GrpMat, G): Cat, GrpLie → GrpMatFundamentalGroup(G): GrpLie → GrpAb, MapIsogenyGroup(G): GrpLie → GrpAb, MapCoisogenyGroup(G): GrpLie → GrpAb, Map
- Properties of Groups of Lie Type
- Constructing Elements
- Operations on Elements
- Properties of Elements
- Roots, Coroots, and Weights
- Accessing Roots and Coroots
- Reflections
- Operations and Properties for Root and Coroot Indices
RootHeight(G, r): GrpLie, RngIntElt → RngIntEltCorootHeight(G, r): GrpLie, RngIntElt → RngIntEltRootNorms(G): GrpLie → [RngIntElt]CorootNorms(G): GrpLie → [RngIntElt]RootNorm(G, r): GrpLie, RngIntElt → RngIntEltCorootNorm(G, r): GrpLie, RngIntElt → RngIntEltIsLongRoot(G, r): GrpLie, RngIntElt → BoolEltIsShortRoot(G, r): GrpLie, RngIntElt → BoolEltAdditiveOrder(G): GrpLie → SeqEnum- Example: Additive Order of Positive Roots
- Weights
- Building Groups of Lie Type
- Automorphisms
- Basic Functionality
- Constructing Special Automorphisms
InnerAutomorphism(G, x): GrpLie, GrpLieElt → MapDiagonalAutomorphism(G, v): GrpLie, ModTupRngElt → MapGraphAutomorphism(G, p): GrpLie, GrpPermElt → MapDiagramAutomorphism(G, p): GrpLie, GrpPermElt → MapFieldAutomorphism(G, sigma): GrpLie, Map → MapRandomAutomorphism(G): GrpLie → GrpLieAutoEltRandom(A): GrpLieAuto → GrpLieAutoEltDualityAutomorphism(G): GrpLie → GrpLieAutoEltFrobeniusMap(G,q): GrpLie, RngIntElt → GrpLieAutoElt
- Operations and Properties of Automorphisms
- Algebraic Homomorphisms
- Twisted Tori
- Sylow Subgroups
- Representations
- Curtis–Steinberg–Tits Presentations
CST_Generators(t,r,q,w): MonStgElt, RngIntElt, RngIntElt, SeqEnum → SeqEnum, SeqEnumCST_Presentation(t,r,q): MonStgElt, RngIntElt, RngIntElt → GrpSLP, SeqEnumCST_VerifyPresentation(t,r,q,X,Y): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum → BoolElt, RngIntElt- Example: Checking Curtis-Steinberg-Tits Relations
CSTtoChev(t,r,q,X,Y): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum → Map- Example: Map from a CST Presentation
ExtendGeneratorList(t,r,X,Y): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum → SeqEnum, SeqEnumIrreducibleHighestWeightRepresentation(G,w): GrpLie, SeqEnum → MapIrreducibleHighestWeightGenerators(G,w): GrpLie, SeqEnum → SeqEnum, SeqEnumIrreducibleHighestWeightFunction(G,w): GrpLie, SeqEnum → UserProgramVermaModule(G,w): GrpLie, SeqEnum → ModGrpUniversalHighWeightRepresentation(G,w): GrpLie, SeqEnum → Map, SeqEnum, SeqEnum- Chevalley Groups
StandardLieRepresentation(t,r): MonStgElt, RngIntElt → SeqEnum, SeqEnumAdjointChevalleyGroup(t,r,q): MonStgElt, RngIntElt, RngIntElt → GrpMat- Example: Adjoint Chevalley Groups
LieRootMatrix(R,α,B): RootDtm, ModTupFldElt, SetIndx → AlgMatEltLieRootMatrix(R,r,X): RootDtm, RngIntElt, SeqEnum → AlgMatEltLieTypeGenerators(t,k,q): MonStgElt, RngIntElt, RngIntElt → SeqEnum, SeqEnumLieTypeGenerators(t,k,K): MonStgElt, RngIntElt, FldFin → SeqEnum, SeqEnumLieTypeGenerators(G): GrpLie → SeqEnum, SeqEnumSLPGeneratorList(t,r,q): MonStgElt, RngIntElt, RngIntElt → SeqEnum, SeqEnum
- Morphisms and the Row Reduction Algorithm
Morphism(G,X,Y): GrpLie, SeqEnum, SeqEnum → MapChevalleyForm(ρ,A): Map[GrpLie,GrpMat], GrpMatElt → SeqEnum, FldFinElt- Example: Chevalley Normal Form of an Element
PrepareRewrite(t,r,q,X,Y): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum → UserProgram, MapTwistedPrepareRewrite(t,r,q,X,Y): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum → UserProgram, MapLieTypeRewrite(t,r,q,X,Y,g): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum, GrpMatElt → BoolElt, GrpSLPEltTwistedLieTypeRewrite(t,r,q,X,Y,g): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum, GrpMatElt → BoolElt, GrpSLPElt- Example: Rewriting in a Twisted Group of Lie Type
RowReductionMap(ρ): Map[GrpLie,GrpMat] → UserProgramTwistedRowReductionMap(ρ): Map[GrpLie,GrpMat] → UserProgram
- Bibliography