# Groups of Lie Type

- [Introduction](introduction.md)

  - [The Steinberg Presentation](introduction.md#the-steinberg-presentation)

  - [Bruhat Normalisation](introduction.md#bruhat-normalisation)

  - [Twisted Groups of Lie type](introduction.md#twisted-groups-of-lie-type)

- [Constructing Groups of Lie Type](construction.md)

  - [Split Groups](construction.md#split-groups)

    - [`GroupOfLieType(N, k): MonStgElt, Rng → GrpLie`](construction.md#function-groupoflietype-monstgelt-rng)

    - [`GroupOfLieType(N, q): MonStgElt, RngIntElt → GrpLie`](construction.md#function-groupoflietype-monstgelt-rngintelt)

    - [`GroupOfLieType(W, k): GrpPermCox, Rng → GrpLie`](construction.md#function-groupoflietype-grppermcox-rng)

    - [`GroupOfLieType(W, k): GrpMat, Rng → GrpLie`](construction.md#function-groupoflietype-grpmat-rng)

    - [`GroupOfLieType(W, q): GrpPermCox, RngIntElt → GrpLie`](construction.md#function-groupoflietype-grppermcox-rngintelt)

    - [`GroupOfLieType(W, q): GrpMat, RngIntElt → GrpLie`](construction.md#function-groupoflietype-grpmat-rngintelt)

    - [`GroupOfLieType(R, k): RootDtm, Rng → GrpLie`](construction.md#function-groupoflietype-rootdtm-rng)

    - [`GroupOfLieType(R, q): RootDtm, RngIntElt → GrpLie`](construction.md#function-groupoflietype-rootdtm-rngintelt)

    - [`GroupOfLieType(C, k): Mtrx, Rng → GrpLie`](construction.md#function-groupoflietype-mtrx-rng)

    - [`GroupOfLieType(D, k): GrphDir, Rng → GrpLie`](construction.md#function-groupoflietype-grphdir-rng)

    - [`GroupOfLieType(C, q): Mtrx, RngIntElt → GrpLie`](construction.md#function-groupoflietype-mtrx-rngintelt)

    - [`GroupOfLieType(D, q): GrphDir, RngIntElt → GrpLie`](construction.md#function-groupoflietype-grphdir-rngintelt)

    - [`SimpleGroupOfLieType(X, n, k): MonStgElt, RngIntElt, Rng → GrpLie`](construction.md#function-simplegroupoflietype-monstgelt-rngintelt-rng)

    - [`SimpleGroupOfLieType(X, n, q): MonStgElt, RngIntElt, RngIntElt → GrpLie`](construction.md#function-simplegroupoflietype-monstgelt-rngintelt-rngintelt)

    - [`GroupOfLieType(L): AlgLie → GrpLie`](construction.md#function-groupoflietype-alglie)

    - [`IsNormalising(G): GrpLie → BoolElt`](construction.md#function-isnormalising-grplie)

    - [`Example: Create`](construction.md#example-ex-108927)

  - [Galois Cohomology](construction.md#galois-cohomology)

    - [`GammaGroup(k, G): Fld, GrpLie → GGrp`](construction.md#function-gammagroup-fld-grplie)

    - [`GammaGroup(k, A): Fld, GrpLieAuto → GGrp`](construction.md#function-gammagroup-fld-grplieauto)

    - [`ActingGroup(G): GrpLie → Grp, Map`](construction.md#function-actinggroup-grplie)

    - [`ActingGroup(A): GrpLieAuto → Grp, Map`](construction.md#function-actinggroup-grplieauto)

    - [`ExtendGaloisCocycle(c): OneCoC → OneCoC`](construction.md#function-extendgaloiscocycle-onecoc)

    - [`GaloisCohomology(A): GGrp → SeqEnum`](construction.md#function-galoiscohomology-ggrp)

    - [`IsInTwistedForm(x, c): GrpLieElt, OneCoC → BoolElt`](construction.md#function-isintwistedform-grplieelt-onecoc)

    - [`Example: Gal Cohom`](construction.md#example-ex-914f01)

  - [Twisted Groups](construction.md#twisted-groups)

    - [`TwistedGroupOfLieType(c): OneCoC → GrpLie`](construction.md#function-twistedgroupoflietype-onecoc)

    - [`TwistedGroupOfLieType(R, k, K): RootDtm, Rng, Rng → GrpLie`](construction.md#function-twistedgroupoflietype-rootdtm-rng-rng)

    - [`TwistedGroupOfLieType(R, q, r): RootDtm, RngIntElt, RngIntElt → GrpLie`](construction.md#function-twistedgroupoflietype-rootdtm-rngintelt-rngintelt)

    - [`TwistedGroupOfLieType(t, r, q): MonStgElt, RngIntElt, RngIntElt → GrpLie`](construction.md#function-twistedgroupoflietype-monstgelt-rngintelt-rngintelt)

    - [`Example: Twisted Grp Lie Type1`](construction.md#example-ex-a0e6f9)

    - [`BaseRing(G): GrpLie → Rng`](construction.md#function-basering-grplie)

    - [`CoefficientRing(G): GrpLie → Rng`](construction.md#function-coefficientring-grplie)

    - [`DefRing(G): GrpLie → Rng`](construction.md#function-defring-grplie)

    - [`UntwistedOvergroup(G): GrpLie → GrpLie`](construction.md#function-untwistedovergroup-grplie)

    - [`Example: Twisted Grp Lie Type2`](construction.md#example-ex-61dd3f)

    - [`RelativeRootElement(G,delta,t): GrpLie, RngIntElt, [FldElt] → GrpLieElt`](construction.md#function-relativerootelement-grplie-rngintelt-fldelt)

    - [`Example: Relative Root Elts`](construction.md#example-ex-55c70c)

- [Operations on Groups of Lie Type](operations.md)

  - [`G eq H: GrpLie, GrpLie → BoolElt`](operations.md#operation-op-eq-grplie-grplie)

  - [`G subset H: GrpLie, GrpLie → BoolElt`](operations.md#operation-op-subset-grplie-grplie)

  - [`IsAlgebraicallyIsomorphic(G, H): GrpLie, GrpLie → BoolElt, Map`](operations.md#function-isalgebraicallyisomorphic-grplie-grplie)

  - [`IsIsogenous(G, H): GrpLie, GrpLie → BoolElt`](operations.md#function-isisogenous-grplie-grplie)

  - [`IsCartanEquivalent(G, H): GrpLie, GrpLie → BoolElt`](operations.md#function-iscartanequivalent-grplie-grplie)

  - [`BaseRing(G): GrpLie → Rng`](operations.md#function-basering-grplie-2)

  - [`CoefficientRing(G): GrpLie → Rng`](operations.md#function-coefficientring-grplie-2)

  - [`BaseExtend(G, K): GrpLie, Rng → GrpLie, Map`](operations.md#function-baseextend-grplie-rng)

  - [`ChangeRing(G, K): GrpLie, Rng → GrpLie`](operations.md#function-changering-grplie-rng)

  - [`Generators(G): GrpLie → {}`](operations.md#function-generators-grplie)

  - [`NumberOfGenerators(G): GrpLie → RngIntElt`](operations.md#function-numberofgenerators-grplie)

  - [`Ngens(G): GrpLie → RngIntElt`](operations.md#function-ngens-grplie)

  - [`AlgebraicGenerators(G): GrpLie → {}`](operations.md#function-algebraicgenerators-grplie)

  - [`NumberOfAlgebraicGenerators(G): GrpLie → RngIntElt`](operations.md#function-numberofalgebraicgenerators-grplie)

  - [`Nalggens(G): GrpLie → RngIntElt`](operations.md#function-nalggens-grplie)

  - [`Example: Generators`](operations.md#example-ex-a929c8)

  - [`Order(G): GrpLie → RngIntElt`](operations.md#function-order-grplie)

  - [`# G: GrpLie → RngIntElt`](operations.md#operation-operation-grplie-rngintelt)

  - [`FactoredOrder(G): GrpLie → RngIntElt`](operations.md#function-factoredorder-grplie)

  - [`Dimension(G): GrpLie → RngIntElt`](operations.md#function-dimension-grplie)

  - [`Example: Orders`](operations.md#example-ex-1128fd)

  - [`CartanName(G): GrpLie → Mtrx`](operations.md#function-cartanname-grplie)

  - [`RootDatum(G): GrpLie → RootDtm`](operations.md#function-rootdatum-grplie)

  - [`DynkinDiagram(G): GrpLie`](operations.md#function-dynkindiagram-grplie)

  - [`CoxeterDiagram(G): GrpLie`](operations.md#function-coxeterdiagram-grplie)

  - [`CoxeterMatrix(G): GrpLie → AlgMatElt`](operations.md#function-coxetermatrix-grplie)

  - [`CoxeterGraph(G): GrpLie → GrphUnd`](operations.md#function-coxetergraph-grplie)

  - [`CartanMatrix(G): GrpLie → GrphUnd`](operations.md#function-cartanmatrix-grplie)

  - [`DynkinDigraph(G): GrpLie → GrphUnd`](operations.md#function-dynkindigraph-grplie)

  - [`Rank(G): GrpLie → RngIntElt`](operations.md#function-rank-grplie)

  - [`ReductiveRank(G): GrpLie → RngIntElt`](operations.md#function-reductiverank-grplie)

  - [`SemisimpleRank(G): GrpLie → RngIntElt`](operations.md#function-semisimplerank-grplie)

  - [`CoxeterNumber(G): GrpLie → RngIntElt`](operations.md#function-coxeternumber-grplie)

  - [`WeylGroup(G): GrpLie → GrpPermCox`](operations.md#function-weylgroup-grplie)

  - [`WeylGroup(GrpPermCox, G): Cat, GrpLie → GrpPermCox`](operations.md#function-weylgroup-cat-grplie)

  - [`WeylGroup(GrpFPCox, G): Cat, GrpLie → GrpFPCox`](operations.md#function-weylgroup-cat-grplie-2)

  - [`WeylGroup(GrpMat, G): Cat, GrpLie → GrpMat`](operations.md#function-weylgroup-cat-grplie-3)

  - [`FundamentalGroup(G): GrpLie → GrpAb, Map`](operations.md#function-fundamentalgroup-grplie)

  - [`IsogenyGroup(G): GrpLie → GrpAb, Map`](operations.md#function-isogenygroup-grplie)

  - [`CoisogenyGroup(G): GrpLie → GrpAb, Map`](operations.md#function-coisogenygroup-grplie)

- [Properties of Groups of Lie Type](element-properties.md)

  - [`IsFinite(G): GrpLie → BoolElt`](element-properties.md#function-isfinite-grplie)

  - [`IsAbelian(G): GrpLie → BoolElt`](element-properties.md#function-isabelian-grplie)

  - [`IsSimple(G): GrpLie → BoolElt`](element-properties.md#function-issimple-grplie)

  - [`IsSimplyLaced(G): GrpLie → BoolElt`](element-properties.md#function-issimplylaced-grplie)

  - [`IsSemisimple(G): GrpLie → BoolElt`](element-properties.md#function-issemisimple-grplie)

  - [`IsAdjoint(G): GrpLie → BoolElt`](element-properties.md#function-isadjointg)

  - [`IsWeaklyAdjoint(G): GrpLie → BoolElt`](element-properties.md#function-isweaklyadjoint-grplie)

  - [`IsSimplyConnected(G): GrpLie → BoolElt`](element-properties.md#function-issimplyconnectedg)

  - [`IsWeaklySimplyConnected(G): GrpLie → BoolElt`](element-properties.md#function-isweaklysimplyconnected-grplie)

  - [`IsSplit(G): GrpLie → BoolElt`](element-properties.md#function-issplit-grplie)

  - [`IsTwisted(G): GrpLie → BoolElt`](element-properties.md#function-istwisted-grplie)

- [Constructing Elements](element-construction.md)

  - [`elt<G | L>: GrpLie, List → GrpMatElt`](element-construction.md#constructor-constructor-elt-grplie-list-grpmatelt)

  - [`Identity(G): GrpLie → GrpLieElt`](element-construction.md#function-identity-grplie)

  - [`Id(G): GrpLie → GrpLieElt`](element-construction.md#function-id-grplie)

  - [`G ! 1: GrpLie, RngIntElt → GrpLieElt`](element-construction.md#operation-op-grplie-rngintelt)

  - [`elt< G | >: GrpLie → GrpLieElt`](element-construction.md#constructor-constructor-elt-grplie-grplieelt)

  - [`Example: Element Create`](element-construction.md#example-ex-749aa7)

  - [`TorusTerm(G, r, t): GrpLie, RngIntElt, RngElt → GrpLieElt`](element-construction.md#function-torusterm-grplie-rngintelt-rngelt)

  - [`CoxeterElement(G): GrpLie → GrpPermElt`](element-construction.md#function-coxeterelement-grplie)

  - [`Random(G): GrpLie → GrpLieElt`](element-construction.md#function-random-grplie)

  - [`Eltlist(g): GrpLieElt → List`](element-construction.md#function-eltlist-grplieelt)

  - [`CentrePolynomials(G): GrpLie → {}`](element-construction.md#function-centrepolynomials-grplie)

  - [`CenterPolynomials(G): GrpLie → {}`](element-construction.md#function-centerpolynomials-grplie)

  - [`Example: Centre`](element-construction.md#example-ex-1715e7)

- [Operations on Elements](element-operators.md)

  - [Basic Operations](element-operators.md#basic-operations)

    - [`g * h: GrpLieElt, GrpLieElt → GrpLieElt`](element-operators.md#operation-op-times-grplieelt-grplieelt)

    - [`Example: Grp Lie Elt Product`](element-operators.md#example-ex-e7b37f)

    - [`g ^ -1: GrpLieElt → GrpLieElt`](element-operators.md#operation-op-pow-grplieelt)

    - [`Inverse(G): GrpLieElt → GrpLieElt`](element-operators.md#function-inverse-grplieelt)

    - [`g ^ n: GrpLieElt, RngIntElt → GrpLieElt`](element-operators.md#operation-op-pow-grplieelt-rngintelt)

    - [`g ^ h: GrpLieElt, GrpLieElt → GrpLieElt`](element-operators.md#operation-op-pow-grplieelt-grplieelt)

    - [`(g,  h): GrpLieElt, GrpLieElt → GrpLieElt`](element-operators.md#literal-literal-g-h-grplieelt-grplieelt-grplieelt)

    - [`Commutator(g, h): GrpLieElt, GrpLieElt → GrpLieElt`](element-operators.md#function-commutator-grplieelt-grplieelt)

    - [`Normalise(~g): GrpLieElt`](element-operators.md#function-normalise-grplieelt-ref)

    - [`Normalize(~g): GrpLieElt`](element-operators.md#function-normalize-grplieelt-ref)

    - [`Normalise(g): GrpLieElt → GrpLieElt`](element-operators.md#function-normalise-grplieelt)

    - [`Normalize(g): GrpLieElt → GrpLieElt`](element-operators.md#function-normalize-grplieelt)

    - [`Example: Grp Lie Elt Arith`](element-operators.md#example-ex-6ee091)

  - [Decompositions](element-operators.md#decompositions)

    - [`Bruhat(g): GrpLieElt → GrpLieElt, GrpLieElt, GrpLieElt, GrpLieElt`](element-operators.md#function-bruhat-grplieelt)

    - [`Example: Bruhat`](element-operators.md#example-ex-95f5c0)

    - [`MultiplicativeJordanDecomposition(x): GrpLieElt → GrpLieElt, GrpLieElt`](element-operators.md#function-multiplicativejordandecomposition-grplieelt)

  - [Conjugacy and Cohomology](element-operators.md#conjugacy-and-cohomology)

    - [`ConjugateIntoTorus(g): GrpLieElt → GrpLieElt, GrpLieElt`](element-operators.md#function-conjugateintotorus-grplieelt)

    - [`ConjugateIntoBorel(g): GrpLieElt → GrpLieElt, GrpLieElt`](element-operators.md#function-conjugateintoborel-grplieelt)

    - [`Lang(c, q): GrpLieElt, RngIntElt → GrpLieElt`](element-operators.md#function-lang-grplieelt-rngintelt)

- [Properties of Elements](more-element-operators.md)

  - [`IsSemisimple(x): GrpLieElt → BoolElt`](more-element-operators.md#function-issemisimple-grplieelt)

  - [`IsUnipotent(x): GrpLieElt → BoolElt`](more-element-operators.md#function-isunipotent-grplieelt)

  - [`IsCentral(x): GrpLieElt → BoolElt`](more-element-operators.md#function-iscentral-grplieelt)

- [Roots, Coroots and Weights](roots-coroots-weights.md)

  - [Accessing Roots and Coroots](roots-coroots-weights.md#accessing-roots-and-coroots)

    - [`RootSpace(G): GrpLie → Lat`](roots-coroots-weights.md#function-rootspace-grplie)

    - [`CorootSpace(G): GrpLie → Lat`](roots-coroots-weights.md#function-corootspace-grplie)

    - [`SimpleRoots(G): GrpLie → Mtrx`](roots-coroots-weights.md#function-simpleroots-grplie)

    - [`SimpleCoroots(G): GrpLie → Mtrx`](roots-coroots-weights.md#function-simplecoroots-grplie)

    - [`NumberOfPositiveRoots(G): GrpLie → RngIntElt`](roots-coroots-weights.md#function-numberofpositiveroots-grplie)

    - [`NumPosRoots(G): GrpLie → RngIntElt`](roots-coroots-weights.md#function-numposroots-grplie)

    - [`Roots(G): GrpLie → SetIndx`](roots-coroots-weights.md#function-roots-grplie)

    - [`Coroots(G): GrpLie → SetIndx`](roots-coroots-weights.md#function-coroots-grplie)

    - [`PositiveRoots(G): GrpLie → SetIndx`](roots-coroots-weights.md#function-positiveroots-grplie)

    - [`PositiveCoroots(G): GrpLie → SetIndx`](roots-coroots-weights.md#function-positivecoroots-grplie)

    - [`Root(G, r): GrpLie, RngIntElt → SetIndx`](roots-coroots-weights.md#function-root-grplie-rngintelt)

    - [`Coroot(G, r): GrpLie, RngIntElt → SetIndx`](roots-coroots-weights.md#function-coroot-grplie-rngintelt)

    - [`RootPosition(G, v): GrpLie, . → SetIndx`](roots-coroots-weights.md#function-rootposition-grplie)

    - [`CorootPosition(G, v): GrpLie, . → SetIndx`](roots-coroots-weights.md#function-corootposition-grplie)

    - [`Example: Roots Coroots`](roots-coroots-weights.md#example-ex-ac3db4)

    - [`HighestRoot(G): GrpLie → LatElt`](roots-coroots-weights.md#function-highestroot-grplie)

    - [`HighestLongRoot(G): GrpLie → LatElt`](roots-coroots-weights.md#function-highestlongroot-grplie)

    - [`HighestShortRoot(G): GrpLie → LatElt`](roots-coroots-weights.md#function-highestshortroot-grplie)

    - [`Example: Heighest Roots`](roots-coroots-weights.md#example-ex-69e0c7)

  - [Reflections](roots-coroots-weights.md#reflections)

    - [`Reflections(G): GrpLie → GrpLieElt`](roots-coroots-weights.md#function-reflections-grplie)

    - [`Reflection(G, r): GrpLie, RngIntElt → GrpLieElt`](roots-coroots-weights.md#function-reflection-grplie-rngintelt)

    - [`Example: Reflections`](roots-coroots-weights.md#example-ex-ff84ed)

  - [Operations and Properties for Root and Coroot Indices](roots-coroots-weights.md#operations-and-properties-for-root-and-coroot-indices)

    - [`RootHeight(G, r): GrpLie, RngIntElt → RngIntElt`](roots-coroots-weights.md#function-rootheight-grplie-rngintelt)

    - [`CorootHeight(G, r): GrpLie, RngIntElt → RngIntElt`](roots-coroots-weights.md#function-corootheight-grplie-rngintelt)

    - [`RootNorms(G): GrpLie → [RngIntElt]`](roots-coroots-weights.md#function-rootnorms-grplie)

    - [`CorootNorms(G): GrpLie → [RngIntElt]`](roots-coroots-weights.md#function-corootnorms-grplie)

    - [`RootNorm(G, r): GrpLie, RngIntElt → RngIntElt`](roots-coroots-weights.md#function-rootnorm-grplie-rngintelt)

    - [`CorootNorm(G, r): GrpLie, RngIntElt → RngIntElt`](roots-coroots-weights.md#function-corootnorm-grplie-rngintelt)

    - [`IsLongRoot(G, r): GrpLie, RngIntElt → BoolElt`](roots-coroots-weights.md#function-islongroot-grplie-rngintelt)

    - [`IsShortRoot(G, r): GrpLie, RngIntElt → BoolElt`](roots-coroots-weights.md#function-isshortroot-grplie-rngintelt)

    - [`AdditiveOrder(G): GrpLie → SeqEnum`](roots-coroots-weights.md#function-additiveorder-grplie)

    - [`Example: Additive Order`](roots-coroots-weights.md#example-ex-dabc7d)

  - [Weights](roots-coroots-weights.md#weights)

    - [`WeightLattice(G): GrpLie → Lat`](roots-coroots-weights.md#function-weightlattice-grplie)

    - [`CoweightLattice(G): GrpLie → Lat`](roots-coroots-weights.md#function-coweightlattice-grplie)

    - [`FundamentalWeights(G): GrpLie → Mtrx`](roots-coroots-weights.md#function-fundamentalweights-grplie)

    - [`FundamentalCoweights(G): GrpLie → Mtrx`](roots-coroots-weights.md#function-fundamentalcoweights-grplie)

    - [`DominantWeight(G, v): GrpLie, . → ModTupFldElt, GrpFPCoxElt`](roots-coroots-weights.md#function-dominantweight-grplie)

- [Building Groups of Lie Type](create-new-root-datum.md)

  - [`SubsystemSubgroup(G, a): GrpLie, SetEnum → RootDtm`](create-new-root-datum.md#function-subsystemsubgroup-grplie-setenum)

  - [`SubsystemSubgroup(G, s): GrpLie, SeqEnum → RootDtm`](create-new-root-datum.md#function-subsystemsubgroup-grplie-seqenum)

  - [`Example: Root Subdata`](create-new-root-datum.md#example-ex-0494af)

  - [`DirectProduct(G1, G2): GrpLie, GrpLie → GrpLie`](create-new-root-datum.md#function-directproduct-grplie-grplie)

  - [`Dual(G): GrpLie → GrpLie`](create-new-root-datum.md#function-dual-grplie)

  - [`SolubleRadical(G): GrpLie → GrpLie`](create-new-root-datum.md#function-solubleradical-grplie)

  - [`StandardMaximalTorus(G): GrpLie → GrpLie`](create-new-root-datum.md#function-standardmaximaltorus-grplie)

  - [`Example: Direct Product Dual Radical`](create-new-root-datum.md#example-ex-c4d20b)

- [Automorphisms](automorphisms.md)

  - [Basic Functionality](automorphisms.md#basic-functionality)

    - [`AutomorphismGroup(G): GrpLie → GrpLieAuto`](automorphisms.md#function-automorphismgroup-grplie)

    - [`IdentityAutomorphism(G): GrpLie → GrpLieAutoElt`](automorphisms.md#function-identityautomorphism-grplie)

    - [`One(A): GrpLieAuto → GrpLieAutoElt`](automorphisms.md#function-one-grplieauto)

    - [`Id(A): GrpLieAuto → GrpLieAutoElt`](automorphisms.md#function-id-grplieauto)

    - [`Mapping(a): GrpLieAutoElt → Map`](automorphisms.md#function-mapping-grplieautoelt)

    - [`Automorphism(m): Map → GrpLieAutoElt`](automorphisms.md#function-automorphism-map)

    - [`h * g: GrpLieAutoElt, GrpLieAutoElt → GrpLieAutoElt`](automorphisms.md#operation-op-times-grplieautoelt-grplieautoelt)

    - [`h ^ n: GrpLieAutoElt, RngIntElt → GrpLieAutoElt`](automorphisms.md#operation-op-pow-grplieautoelt-rngintelt)

    - [`g ^ h: GrpLieAutoElt, GrpLieAutoElt → GrpLieAutoElt`](automorphisms.md#operation-op-pow-grplieautoelt-grplieautoelt)

    - [`Domain(A): GrpLieAuto → GrpLie`](automorphisms.md#function-domain-grplieauto)

    - [`Codomain(A): GrpLieAuto → GrpLie`](automorphisms.md#function-codomain-grplieauto)

    - [`Domain(h): GrpLieAutoElt → GrpLie`](automorphisms.md#function-domain-grplieautoelt)

    - [`Codomain(h): GrpLieAutoElt → GrpLie`](automorphisms.md#function-codomain-grplieautoelt)

  - [Constructing Special Automorphisms](automorphisms.md#constructing-special-automorphisms)

    - [`InnerAutomorphism(G, x): GrpLie, GrpLieElt → Map`](automorphisms.md#function-innerautomorphism-grplie-grplieelt)

    - [`DiagonalAutomorphism(G, v): GrpLie, ModTupRngElt → Map`](automorphisms.md#function-diagonalautomorphism-grplie-modtuprngelt)

    - [`GraphAutomorphism(G, p): GrpLie, GrpPermElt → Map`](automorphisms.md#function-graphautomorphism-grplie-grppermelt)

    - [`DiagramAutomorphism(G, p): GrpLie, GrpPermElt → Map`](automorphisms.md#function-diagramautomorphism-grplie-grppermelt)

    - [`FieldAutomorphism(G, sigma): GrpLie, Map → Map`](automorphisms.md#function-fieldautomorphism-grplie-map)

    - [`RandomAutomorphism(G): GrpLie → GrpLieAutoElt`](automorphisms.md#function-randomautomorphism-grplie)

    - [`Random(A): GrpLieAuto → GrpLieAutoElt`](automorphisms.md#function-random-grplieauto)

    - [`DualityAutomorphism(G): GrpLie → GrpLieAutoElt`](automorphisms.md#function-dualityautomorphism-grplie)

    - [`FrobeniusMap(G,q): GrpLie, RngIntElt → GrpLieAutoElt`](automorphisms.md#function-frobeniusmap-grplie-rngintelt)

  - [Operations and Properties of Automorphisms](automorphisms.md#operations-and-properties-of-automorphisms)

    - [`DecomposeAutomorphism(h): GrpLieAutoElt → GrpLieAutoElt, GrpLieAutoElt, GrpLieAutoElt, Rec`](automorphisms.md#function-decomposeautomorphism-grplieautoelt)

    - [`IsAlgebraic(h): GrpLieAutoElt → BoolElt`](automorphisms.md#function-isalgebraic-grplieautoelt)

    - [`Example: Automorphism`](automorphisms.md#example-ex-6e4baf)

- [Algebraic Homomorphisms](homomorphism.md)

  - [`GroupOfLieTypeHomomorphism(phi, k): Map, Rng → .`](homomorphism.md#function-groupoflietypehomomorphism-map-rng)

  - [`Example: Creating Root Data Homomorphisms`](homomorphism.md#example-ex-42434f)

- [Twisted Tori](twisted-tori.md)

  - [`TwistedTorusOrder(R, w): RootDtm, GrpPermElt → SeqEnum`](twisted-tori.md#function-twistedtorusorder-rootdtm-grppermelt)

  - [`TwistedToriOrders(G): GrpLie → SeqEnum`](twisted-tori.md#function-twistedtoriorders-grplie)

  - [`TwistedToriOrders(R): RootDtm → SeqEnum`](twisted-tori.md#function-twistedtoriorders-rootdtm)

  - [`TwistedTorus(G, w): GrpLie, GrpPermElt → List`](twisted-tori.md#function-twistedtorus-grplie-grppermelt)

  - [`TwistedTori(G): GrpLie → SeqEnum`](twisted-tori.md#function-twistedtori-grplie)

  - [`Example: Grp Lie Tori`](twisted-tori.md#example-ex-5e0bd4)

  - [`Example: Grp Lie Tori2`](twisted-tori.md#example-ex-bc3840)

- [Sylow Subgroups](sylow-subgroups.md)

  - [`PrintSylowSubgroupStructure(G): GrpLie`](sylow-subgroups.md#function-printsylowsubgroupstructure-grplie)

  - [`SylowSubgroup(G, p): GrpLie, RngIntElt → List`](sylow-subgroups.md#function-sylowsubgroup-grplie-rngintelt)

  - [`Example: Grp Lie Sylow`](sylow-subgroups.md#example-ex-dd465e)

- [Representations](representations.md)

  - [`StandardRepresentation(G): GrpLie → Map`](representations.md#function-standardrepresentation-grplie)

  - [`AdjointRepresentation(G): GrpLie → Map, AlgLie`](representations.md#function-adjointrepresentation-grplie)

  - [`LieAlgebra(G): GrpLie → AlgLie, Map`](representations.md#function-liealgebra-grplie)

  - [`HighestWeightRepresentation(G, v): GrpLie, . → Map`](representations.md#function-highestweightrepresentation-grplie)

  - [`Example: Standard Representation`](representations.md#example-ex-be5644)

  - [`ContravariantForm(ρ): Map[GrpLie,GrpMat] → AlgMatElt`](representations.md#function-contravariantform-map-grplie-grpmat)

  - [`GeneralisedRowReduction(ρ): Map → Map`](representations.md#function-generalisedrowreduction-map)

  - [`RowReductionHomomorphism(ρ): Map → Map`](representations.md#function-rowreductionhomomorphism-map)

  - [`Inverse(ρ): Map → Map`](representations.md#function-inverse-map)

- [Curtis–Steinberg–Tits Presentations](cst-presentations.md)

  - [`CST_Generators(t,r,q,w): MonStgElt, RngIntElt, RngIntElt, SeqEnum → SeqEnum, SeqEnum`](cst-presentations.md#function-cst-generators-monstgelt-rngintelt-rngintelt-seqenum)

  - [`CST_Presentation(t,r,q): MonStgElt, RngIntElt, RngIntElt → GrpSLP, SeqEnum`](cst-presentations.md#function-cst-presentation-monstgelt-rngintelt-rngintelt)

  - [`CST_VerifyPresentation(t,r,q,X,Y): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum → BoolElt, RngIntElt`](cst-presentations.md#function-cst-verifypresentation-monstgelt-rngintelt-rngintelt-seqenum-seqenum)

  - [`Example: CST Pres`](cst-presentations.md#example-ex-edd24d)

  - [`CSTtoChev(t,r,q,X,Y): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum → Map`](cst-presentations.md#function-csttochev-monstgelt-rngintelt-rngintelt-seqenum-seqenum)

  - [`Example: CS Tto Chev`](cst-presentations.md#example-ex-430d2a)

  - [`ExtendGeneratorList(t,r,X,Y): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum → SeqEnum, SeqEnum`](cst-presentations.md#function-extendgeneratorlist-monstgelt-rngintelt-rngintelt-seqenum-seqenum)

  - [`IrreducibleHighestWeightRepresentation(G,w): GrpLie, SeqEnum → Map`](cst-presentations.md#function-irreduciblehighestweightrepresentation-grplie-seqenum)

  - [`IrreducibleHighestWeightGenerators(G,w): GrpLie, SeqEnum → SeqEnum, SeqEnum`](cst-presentations.md#function-irreduciblehighestweightgenerators-grplie-seqenum)

  - [`IrreducibleHighestWeightFunction(G,w): GrpLie, SeqEnum → UserProgram`](cst-presentations.md#function-irreduciblehighestweightfunction-grplie-seqenum)

  - [`VermaModule(G,w): GrpLie, SeqEnum → ModGrp`](cst-presentations.md#function-vermamodule-grplie-seqenum)

  - [`UniversalHighWeightRepresentation(G,w): GrpLie, SeqEnum → Map, SeqEnum, SeqEnum`](cst-presentations.md#function-universalhighweightrepresentation-grplie-seqenum)

  - [Chevalley Groups](cst-presentations.md#chevalley-groups)

    - [`StandardLieRepresentation(t,r): MonStgElt, RngIntElt → SeqEnum, SeqEnum`](cst-presentations.md#function-standardlierepresentation-monstgelt-rngintelt)

    - [`AdjointChevalleyGroup(t,r,q): MonStgElt, RngIntElt, RngIntElt → GrpMat`](cst-presentations.md#function-adjointchevalleygroup-monstgelt-rngintelt-rngintelt)

    - [`Example: Adjoint Chev`](cst-presentations.md#example-ex-1c5530)

    - [`LieRootMatrix(R,α,B): RootDtm, ModTupFldElt, SetIndx → AlgMatElt`](cst-presentations.md#function-lierootmatrix-rootdtm-modtupfldelt-setindx)

    - [`LieRootMatrix(R,r,X): RootDtm, RngIntElt, SeqEnum → AlgMatElt`](cst-presentations.md#function-lierootmatrix-rootdtm-rngintelt-seqenum)

    - [`LieTypeGenerators(t,k,q): MonStgElt, RngIntElt, RngIntElt → SeqEnum, SeqEnum`](cst-presentations.md#function-lietypegenerators-monstgelt-rngintelt-rngintelt)

    - [`LieTypeGenerators(t,k,K): MonStgElt, RngIntElt, FldFin → SeqEnum, SeqEnum`](cst-presentations.md#function-lietypegenerators-monstgelt-rngintelt-fldfin)

    - [`LieTypeGenerators(G): GrpLie → SeqEnum, SeqEnum`](cst-presentations.md#function-lietypegenerators-grplie)

    - [`SLPGeneratorList(t,r,q): MonStgElt, RngIntElt, RngIntElt → SeqEnum, SeqEnum`](cst-presentations.md#function-slpgeneratorlist-monstgelt-rngintelt-rngintelt)

  - [Morphisms and the Row Reduction Algorithm](cst-presentations.md#morphisms-and-the-row-reduction-algorithm)

    - [`Morphism(G,X,Y): GrpLie, SeqEnum, SeqEnum → Map`](cst-presentations.md#function-morphism-grplie-seqenum-seqenum)

    - [`ChevalleyForm(ρ,A): Map[GrpLie,GrpMat], GrpMatElt → SeqEnum, FldFinElt`](cst-presentations.md#function-chevalleyform-map-grplie-grpmat-grpmatelt)

    - [`Example: Chev Form`](cst-presentations.md#example-ex-6fae1c)

    - [`PrepareRewrite(t,r,q,X,Y): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum → UserProgram, Map`](cst-presentations.md#function-preparerewrite-monstgelt-rngintelt-rngintelt-seqenum-seqenum)

    - [`TwistedPrepareRewrite(t,r,q,X,Y): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum → UserProgram, Map`](cst-presentations.md#function-twistedpreparerewrite-monstgelt-rngintelt-rngintelt-seqenum-seqenum)

    - [`LieTypeRewrite(t,r,q,X,Y,g): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum, GrpMatElt → BoolElt, GrpSLPElt`](cst-presentations.md#function-lietyperewrite-monstgelt-rngintelt-rngintelt-seqenum-seqenum-grpmatelt)

    - [`TwistedLieTypeRewrite(t,r,q,X,Y,g): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum, GrpMatElt → BoolElt, GrpSLPElt`](cst-presentations.md#function-twistedlietyperewrite-monstgelt-rngintelt-rngintelt-seqenum-seqenum-grpmatelt)

    - [`Example: Lie Rewrite`](cst-presentations.md#example-ex-22cf73)

    - [`RowReductionMap(ρ): Map[GrpLie,GrpMat] → UserProgram`](cst-presentations.md#function-rowreductionmap-map-grplie-grpmat)

    - [`TwistedRowReductionMap(ρ): Map[GrpLie,GrpMat] → UserProgram`](cst-presentations.md#function-twistedrowreductionmap-map-grplie-grpmat)