# Coxeter Groups

- [Introduction](introduction.md)

  - [The Normal Form for Words](introduction.md#the-normal-form-for-words)

- [Constructing Coxeter Groups](construct.md)

  - [`CoxeterGroup(grpcat, N): Cat, MonStgElt → grpcat`](construct.md#function-coxetergroup-cat-monstgelt)

  - [`CoxeterGroup(N): MonStgElt → Grp`](construct.md#function-coxetergroup-monstgelt)

  - [`IrreducibleCoxeterGroup(grpcat, X, n): Cat, MonStgElt, RngIntElt → grpcat`](construct.md#function-irreduciblecoxetergroup-cat-monstgelt-rngintelt)

  - [`IrreducibleCoxeterGroup(X, n): MonStgElt, RngIntElt → .`](construct.md#function-irreduciblecoxetergroup-monstgelt-rngintelt)

  - [`Example: Construct By Name`](construct.md#example-ex-fcfb78)

  - [`CoxeterGroup(grpcat, M): Cat, AlgMatElt → grpcat`](construct.md#function-coxetergroup-cat-algmatelt)

  - [`CoxeterGroup(grpcat, G): Cat, GrphUnd → grpcat`](construct.md#function-coxetergroup-cat-grphund)

  - [`CoxeterGroup(grpcat, D): Cat, GrphDir → grpcat`](construct.md#function-coxetergroup-cat-grphdir)

  - [`CoxeterGroup(M): AlgMatElt → Grp`](construct.md#function-coxetergroup-algmatelt)

  - [`CoxeterGroup(G): GrphUnd → Grp`](construct.md#function-coxetergroup-grphund)

  - [`CoxeterGroup(D): GrphDir → Grp`](construct.md#function-coxetergroup-grphdir)

  - [`Example: Construct From Matrix`](construct.md#example-ex-2b7d19)

  - [`CoxeterGroup(grpcat, R): Cat, RootStr → grpcat`](construct.md#function-coxetergroup-cat-rootstr)

  - [`CoxeterGroup(R): RootStr → GrpPermCox`](construct.md#function-coxetergroup-rootstr)

  - [`CoxeterGroup(A, B): Mtrx, Mtrx → GrpPermCox`](construct.md#function-coxetergroup-mtrx-mtrx)

  - [`CoxeterGroup(grpcat, A, B): Cat, Mtrx, Mtrx → grpcat`](construct.md#function-coxetergroup-cat-mtrx-mtrx)

  - [`Example: Construct By Root`](construct.md#example-ex-af5154)

- [Converting Between Types of Coxeter Group](construct-2.md)

  - [`CoxeterGroup(grpcat, W): Cat, Grp → grpcat, Map`](construct-2.md#function-coxetergroup-cat-grp)

  - [`Example: Construct By Group`](construct-2.md#example-ex-bd0d74)

  - [`ReflectionGroup(W): GrpFPCox → GrpMat, Map`](construct-2.md#function-reflectiongroup-grpfpcox)

  - [`ReflectionGroup(W): GrpPermCox → GrpMat, Map`](construct-2.md#function-reflectiongroup-grppermcox)

  - [`Example: Reflection Group Conversion`](construct-2.md#example-ex-576a6b)

- [Operations on Coxeter Groups](group-op.md)

  - [`IsIsomorphic(W1, W2): GrpPermCox, GrpPermCox → BoolElt`](group-op.md#function-isisomorphic-grppermcox-grppermcox)

  - [`IsCoxeterIsomorphic(W1, W2): GrpFPCox, GrpFPCox → BoolElt`](group-op.md#function-iscoxeterisomorphic-grpfpcox-grpfpcox)

  - [`IsCoxeterIsomorphic(W1, W2): GrpPermCox, GrpPermCox → BoolElt`](group-op.md#function-iscoxeterisomorphic-grppermcox-grppermcox)

  - [`IsCartanEquivalent(W1, W2): GrpPermCox, GrpPermCox → BoolElt`](group-op.md#function-iscartanequivalent-grppermcox-grppermcox)

  - [`Example: Coxeter Isomorphism`](group-op.md#example-ex-892fbe)

  - [`RootSystem(W): GrpPermCox → RootDtm`](group-op.md#function-rootsystem-grppermcox)

  - [`RootDatum(W): GrpPermCox → RootDtm`](group-op.md#function-rootdatum-grppermcox)

  - [`Example: Group To Root`](group-op.md#example-ex-e5a910)

  - [`CartanName(W): GrpFPCox → List`](group-op.md#function-cartanname-grpfpcox)

  - [`CartanName(W): GrpPermCox → MonStgElt`](group-op.md#function-cartanname-grppermcox)

  - [`CoxeterDiagram(W): GrpFPCox`](group-op.md#function-coxeterdiagram-grpfpcox)

  - [`CoxeterDiagram(W): GrpPermCox`](group-op.md#function-coxeterdiagram-grppermcox)

  - [`DynkinDiagram(W): GrpPermCox`](group-op.md#function-dynkindiagram-grppermcox)

  - [`Example: Names Diagrams`](group-op.md#example-ex-93c1ec)

  - [`CoxeterMatrix(W): GrpFPCox → AlgMatElt`](group-op.md#function-coxetermatrix-grpfpcox)

  - [`CoxeterMatrix(W): GrpPermCox → AlgMatElt`](group-op.md#function-coxetermatrix-grppermcox)

  - [`CoxeterGraph(W): GrpFPCox → GrphUnd`](group-op.md#function-coxetergraph-grpfpcox)

  - [`CoxeterGraph(W): GrpPermCox → GrphUnd`](group-op.md#function-coxetergraph-grppermcox)

  - [`CartanMatrix(W): GrpPermCox → AlgMatElt`](group-op.md#function-cartanmatrix-grppermcox)

  - [`DynkinDigraph(W): GrpPermCox → GrphDir`](group-op.md#function-dynkindigraph-grppermcox)

  - [`Rank(W): GrpFPCox → RngIntElt`](group-op.md#function-rank-grpfpcox)

  - [`Rank(W): GrpPermCox → RngIntElt`](group-op.md#function-rank-grppermcox)

  - [`NumberOfGenerators(W): GrpFPCox → RngIntElt`](group-op.md#function-numberofgenerators-grpfpcox)

  - [`NumberOfGenerators(W): GrpPermCox → RngIntElt`](group-op.md#function-numberofgenerators-grppermcox)

  - [`NumberOfPositiveRoots(W): GrpFPCox → RngIntElt`](group-op.md#function-numberofpositiveroots-grpfpcox)

  - [`NumberOfPositiveRoots(W): GrpPermCox → RngIntElt`](group-op.md#function-numberofpositiveroots-grppermcox)

  - [`NumPosRoots(W): GrpFPCox → RngIntElt`](group-op.md#function-numposroots-grpfpcox)

  - [`NumPosRoots(W): GrpPermCox → RngIntElt`](group-op.md#function-numposroots-grppermcox)

  - [`Dimension(W): GrpPermCox → RngIntElt`](group-op.md#function-dimension-grppermcox)

  - [`Example: Rank Dimension`](group-op.md#example-ex-6345f3)

  - [`ConjugacyClasses(W): GrpFPCox → [GrpFPCoxElt]`](group-op.md#function-conjugacyclasses-grpfpcox)

  - [`ConjugacyClasses(W): GrpPermCox → [GrpPermElt]`](group-op.md#function-conjugacyclasses-grppermcox)

  - [`FundamentalGroup(W): GrpPermCox → GrpAb`](group-op.md#function-fundamentalgroup-grppermcox)

  - [`IsogenyGroup(W): GrpPermCox → GrpAb`](group-op.md#function-isogenygroup-grppermcox)

  - [`CoisogenyGroup(W): GrpPermCox → GrpAb`](group-op.md#function-coisogenygroup-grppermcox)

  - [`BasicDegrees(W): GrpFPCox → RngIntElt`](group-op.md#function-basicdegrees-grpfpcox)

  - [`BasicDegrees(W): GrpPermCox → RngIntElt`](group-op.md#function-basicdegrees-grppermcox)

  - [`BasicCodegrees(W): GrpFPCox → RngIntElt`](group-op.md#function-basiccodegrees-grpfpcox)

  - [`BasicCodegrees(W): GrpPermCox → RngIntElt`](group-op.md#function-basiccodegrees-grppermcox)

  - [`Example: Basic Degrees`](group-op.md#example-ex-d80f8f)

  - [`BruhatLessOrEqual(x, y): GrpPermElt, GrpPermElt → BoolElt`](group-op.md#function-bruhatlessorequal)

  - [`BruhatDescendants(x): GrpPermElt → SetEnum`](group-op.md#function-bruhatdescendants)

  - [`BruhatDescendants(X): SetEnum → SetEnum`](group-op.md#function-bruhatdescendants-setenum)

  - [`Example: Bruhat Descendants`](group-op.md#example-ex-3232ff)

- [Properties of Coxeter Groups](group-prop.md)

  - [`IsFinite(W): GrpFPCox → BoolElt`](group-prop.md#function-isfinite-grpfpcox)

  - [`IsAffine(W): GrpFPCox → BoolElt`](group-prop.md#function-isaffine-grpfpcox)

  - [`IsHyperbolic(W): GrpFPCox → BoolElt`](group-prop.md#function-ishyperbolic-grpfpcox)

  - [`IsCompactHyperbolic(W): GrpFPCox → BoolElt`](group-prop.md#function-iscompacthyperbolic-grpfpcox)

  - [`IsIrreducible(W): GrpFPCox → BoolElt`](group-prop.md#function-isirreducible-grpfpcox)

  - [`IsIrreducible(W): GrpPermCox → BoolElt`](group-prop.md#function-isirreducible-grppermcox)

  - [`IsSemisimple(W): GrpPermCox → BoolElt`](group-prop.md#function-issemisimple-grppermcox)

  - [`IsCrystallographic(W): GrpPermCox → BoolElt`](group-prop.md#function-iscrystallographic-grppermcox)

  - [`IsSimplyLaced(W): GrpPermCox → BoolElt`](group-prop.md#function-issimplylaced-grppermcox)

  - [`IsSimplyLaced(W): GrpFPCox → BoolElt`](group-prop.md#function-issimplylaced-grpfpcox)

  - [`Example: Properties`](group-prop.md#example-ex-5b007e)

- [Operations on Elements](group-elt-op.md)

  - [`Example: Word Arithmetic`](group-elt-op.md#example-ex-8999bf)

  - [`# w: GrpFPCoxElt → RngIntElt`](group-elt-op.md#operation-operation-grpfpcoxelt-rngintelt)

  - [`Length(w): GrpFPCoxElt → RngIntElt`](group-elt-op.md#function-length-grpfpcoxelt)

  - [`Length(W, w): GrpPermCox, GrpPermElt → RngIntElt`](group-elt-op.md#function-length-grppermcox-grppermelt)

  - [`CoxeterLength(w): GrpFPCoxElt → RngIntElt`](group-elt-op.md#function-coxeterlength-grpfpcoxelt)

  - [`CoxeterLength(W, w): GrpPermCox, GrpPermElt → RngIntElt`](group-elt-op.md#function-coxeterlength-grppermcox-grppermelt)

  - [`LongestElement(W): GrpFPCox → SeqEnum`](group-elt-op.md#function-longestelement-grpfpcox)

  - [`LongestElement(W): GrpPermCox → GrpPermElt`](group-elt-op.md#function-longestelement-grppermcox)

  - [`CoxeterElement(W): GrpFPCox → SeqEnum`](group-elt-op.md#function-coxeterelement-grpfpcox)

  - [`CoxeterElement(W): GrpPermCox → GrpPermElt`](group-elt-op.md#function-coxeterelement-grppermcox)

  - [`CoxeterNumber(W): GrpFPCox → SeqEnum`](group-elt-op.md#function-coxeternumber-grpfpcox)

  - [`CoxeterNumber(W): GrpPermCox → GrpPermElt`](group-elt-op.md#function-coxeternumber-grppermcox)

  - [`Example: Longest Coxeter Elements`](group-elt-op.md#example-ex-ba43d4)

  - [`LeftDescentSet(W, w): GrpFPCox, GrpFPCoxElt → SetEnum`](group-elt-op.md#function-leftdescentset-grpfpcox-grpfpcoxelt)

  - [`LeftDescentSet(W, w): GrpPermCox, GrpPermElt → SetEnum`](group-elt-op.md#function-leftdescentset-grppermcox-grppermelt)

  - [`RightDescentSet(W, w): GrpFPCox, GrpFPCoxElt → SetEnum`](group-elt-op.md#function-rightdescentset-grpfpcox-grpfpcoxelt)

  - [`RightDescentSet(W, w): GrpPermCox, GrpPermElt → SetEnum`](group-elt-op.md#function-rightdescentset-grppermcox-grppermelt)

  - [`Example: Descent Sets`](group-elt-op.md#example-ex-e2a89d)

- [Roots, Coroots and Reflections](root-data-roots.md)

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

    - [`RootSpace(W): GrpPermCox → .`](root-data-roots.md#function-rootspace-grppermcox)

    - [`CorootSpace(W): GrpPermCox → .`](root-data-roots.md#function-corootspace-grppermcox)

    - [`SimpleRoots(W): GrpPermCox → Mtrx`](root-data-roots.md#function-simpleroots-grppermcox)

    - [`SimpleCoroots(W): GrpPermCox → Mtrx`](root-data-roots.md#function-simplecoroots-grppermcox)

    - [`Example: Root Space`](root-data-roots.md#example-ex-18ee8e)

    - [`NumberOfPositiveRoots(W): GrpPermCox → RngIntElt`](root-data-roots.md#function-numberofpositiveroots-grppermcox-2)

    - [`NumPosRoots(W): GrpPermCox → RngIntElt`](root-data-roots.md#function-numposroots-grppermcox-2)

    - [`Roots(W): GrpPermCox → SetIndx`](root-data-roots.md#function-roots-grppermcox)

    - [`Coroots(W): GrpPermCox → SetIndx`](root-data-roots.md#function-coroots-grppermcox)

    - [`PositiveRoots(W): GrpPermCox → SetIndx`](root-data-roots.md#function-positiveroots-grppermcox)

    - [`PositiveCoroots(W): GrpPermCox → SetIndx`](root-data-roots.md#function-positivecoroots-grppermcox)

    - [`Root(W, r): GrpPermCox, RngIntElt → SetIndx`](root-data-roots.md#function-root-grppermcox-rngintelt)

    - [`Coroot(W, r): GrpPermCox, RngIntElt → SetIndx`](root-data-roots.md#function-coroot-grppermcox-rngintelt)

    - [`RootPosition(W, v): GrpPermCox, . → SetIndx`](root-data-roots.md#function-rootposition-grppermcox)

    - [`CorootPosition(W, v): GrpPermCox, . → SetIndx`](root-data-roots.md#function-corootposition-grppermcox)

    - [`Example: Roots Coroots`](root-data-roots.md#example-ex-adcde8)

    - [`HighestRoot(W): GrpPermCox → .`](root-data-roots.md#function-highestroot-grppermcox)

    - [`HighestLongRoot(W): GrpPermCox → .`](root-data-roots.md#function-highestlongroot-grppermcox)

    - [`HighestShortRoot(W): GrpPermCox → .`](root-data-roots.md#function-highestshortroot-grppermcox)

    - [`Example: Heighest Roots`](root-data-roots.md#example-ex-3e3fcd)

    - [`CoxeterForm(W): GrpPermCox → AlgMatElt`](root-data-roots.md#function-coxeterform-grppermcox)

    - [`DualCoxeterForm(W): GrpPermCox → AlgMatElt`](root-data-roots.md#function-dualcoxeterform-grppermcox)

    - [`AdditiveOrder(W): GrpPermCox → SeqEnum`](root-data-roots.md#function-additiveorder-grppermcox)

    - [`PapiOrder(W,w): GrpPermCox, GrpPermElt → SeqEnum`](root-data-roots.md#function-papiorder-grppermcox-grppermelt)

    - [`PapiOrder(W, wd): GrpPermCox, SeqEnum[RngIntElt] → SeqEnum`](root-data-roots.md#function-papiorder-grppermcox-seqenum-rngintelt)

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

    - [`Sum(W, r, s): GrpPermCox, RngIntElt, RngIntElt → RngIntElt`](root-data-roots.md#function-sum-grppermcox-rngintelt-rngintelt)

    - [`IsPositive(W, r): GrpPermCox, RngIntElt → BoolElt`](root-data-roots.md#function-ispositive-grppermcox-rngintelt)

    - [`IsNegative(W, r): GrpPermCox, RngIntElt → BoolElt`](root-data-roots.md#function-isnegative-grppermcox-rngintelt)

    - [`Negative(W, r): GrpPermCox, RngIntElt → RngIntElt`](root-data-roots.md#function-negative-grppermcox-rngintelt)

    - [`LeftString(W, r, s): GrpPermCox, RngIntElt, RngIntElt → RngIntElt`](root-data-roots.md#function-leftstring-grppermcox-rngintelt-rngintelt)

    - [`RightString(W, r, s): GrpPermCox, RngIntElt, RngIntElt → RngIntElt`](root-data-roots.md#function-rightstring-grppermcox-rngintelt-rngintelt)

    - [`LeftStringLength(W, r, s): GrpPermCox, RngIntElt, RngIntElt → RngIntElt`](root-data-roots.md#function-leftstringlength-grppermcox-rngintelt-rngintelt)

    - [`RightStringLength(W, r, s): GrpPermCox, RngIntElt, RngIntElt → RngIntElt`](root-data-roots.md#function-rightstringlength-grppermcox-rngintelt-rngintelt)

    - [`Example: Root Arithmetic`](root-data-roots.md#example-ex-9ecc9b)

    - [`RootHeight(W, r): GrpPermCox, RngIntElt → RngIntElt`](root-data-roots.md#function-rootheight-grppermcox-rngintelt)

    - [`CorootHeight(W, r): GrpPermCox, RngIntElt → RngIntElt`](root-data-roots.md#function-corootheight-grppermcox-rngintelt)

    - [`RootNorms(W): GrpPermCox → [RngIntElt]`](root-data-roots.md#function-rootnorms-grppermcox)

    - [`CorootNorms(W): GrpPermCox → [RngIntElt]`](root-data-roots.md#function-corootnorms-grppermcox)

    - [`RootNorm(W, r): GrpPermCox, RngIntElt → RngIntElt`](root-data-roots.md#function-rootnorm-grppermcox-rngintelt)

    - [`CorootNorm(W, r): GrpPermCox, RngIntElt → RngIntElt`](root-data-roots.md#function-corootnorm-grppermcox-rngintelt)

    - [`IsLongRoot(W, r): GrpPermCox, RngIntElt → BoolElt`](root-data-roots.md#function-islongroot-grppermcox-rngintelt)

    - [`IsShortRoot(W, r): GrpPermCox, RngIntElt → BoolElt`](root-data-roots.md#function-isshortroot-grppermcox-rngintelt)

    - [`Example: Root Operations`](root-data-roots.md#example-ex-237223)

  - [Weights](root-data-roots.md#weights)

    - [`WeightLattice(W): GrpPermCox → Lat`](root-data-roots.md#function-weightlattice-grppermcox)

    - [`CoweightLattice(W): GrpPermCox → Lat`](root-data-roots.md#function-coweightlattice-grppermcox)

    - [`FundamentalWeights(W): GrpPermCox → SeqEnum`](root-data-roots.md#function-fundamentalweights-grppermcox)

    - [`FundamentalCoweights(W): GrpPermCox → SeqEnum`](root-data-roots.md#function-fundamentalcoweights-grppermcox)

    - [`IsDominant(R, v): RootDtm, . → ModTupFldElt, GrpFPCoxElt`](root-data-roots.md#function-isdominant-rootdtm)

    - [`DominantWeight(W, v): GrpPermCox, . → ModTupFldElt, GrpFPCoxElt`](root-data-roots.md#function-dominantweight-grppermcox)

    - [`WeightOrbit(W, v): GrpPermCox, . → {@ ModTupFldElt @}, [GrpFPCoxElt]`](root-data-roots.md#function-weightorbit-grppermcox)

    - [`Example: Dominant Weights`](root-data-roots.md#example-ex-181d61)

- [Reflections](rootrefl.md)

  - [`IsReflection(w): GrpFPElt → BoolElt`](rootrefl.md#function-isreflection-grpfpelt)

  - [`IsReflection(w): GrpPermElt → BoolElt, ., ., RngInt`](rootrefl.md#function-isreflection-grppermelt)

  - [`Reflections(W): GrpFPCox → [GrpFPCoxElt]`](rootrefl.md#function-reflections-grpfpcox)

  - [`Reflections(W): GrpPermCox → [GrpPermElt]`](rootrefl.md#function-reflections-grppermcox)

  - [`Example: Reflections`](rootrefl.md#example-ex-7fa1b2)

  - [`SimpleReflections(W): GrpFPCox → [GrpFPCoxElt]`](rootrefl.md#function-simplereflections-grpfpcox)

  - [`SimpleReflections(W): GrpPermCox → [GrpPermElt]`](rootrefl.md#function-simplereflections-grppermcox)

  - [`SimpleReflectionPermutations(W): GrpPermCox → [GrpPermElt]`](rootrefl.md#function-simplereflectionpermutations-grppermcox)

  - [`Reflection(W, r): GrpPermCox, RngIntElt → GrpPermElt`](rootrefl.md#function-reflection-grppermcox-rngintelt)

  - [`ReflectionPermutation(W, r): GrpPermCox, RngIntElt → GrpPermElt`](rootrefl.md#function-reflectionpermutation-grppermcox-rngintelt)

  - [`SimpleReflectionMatrices(W): GrpPermCox → []`](rootrefl.md#function-simplereflectionmatrices-grppermcox)

  - [`SimpleCoreflectionMatrices(W): GrpPermCox → []`](rootrefl.md#function-simplecoreflectionmatrices-grppermcox)

  - [`ReflectionMatrices(W): GrpPermCox → []`](rootrefl.md#function-reflectionmatrices-grppermcox)

  - [`CoreflectionMatrices(W): GrpPermCox → []`](rootrefl.md#function-coreflectionmatrices-grppermcox)

  - [`ReflectionMatrix(W, r): GrpPermCox, RngIntElt → []`](rootrefl.md#function-reflectionmatrix-grppermcox-rngintelt)

  - [`CoreflectionMatrix(W, r): GrpPermCox, RngIntElt → []`](rootrefl.md#function-coreflectionmatrix-grppermcox-rngintelt)

  - [`ReflectionWords(W): GrpPermCox → []`](rootrefl.md#function-reflectionwords-grppermcox)

  - [`ReflectionWord(W, r): GrpPermCox, RngIntElt → []`](rootrefl.md#function-reflectionword-grppermcox-rngintelt)

  - [`Example: Action`](rootrefl.md#example-ex-56c975)

- [Reflection Subgroups](subgroups.md)

  - [`ReflectionSubgroup(W, a): GrpPermCox, SetEnum → GrpPermCox`](subgroups.md#function-reflectionsubgroup-grppermcox-setenum)

  - [`ReflectionSubgroup(W, s): GrpPermCox, [] → GrpPermCox`](subgroups.md#function-reflectionsubgroup-grppermcox)

  - [`StandardParabolicSubgroup(W, J): GrpPermCox, SetEnum → GrpPermCox`](subgroups.md#function-standardparabolicsubgroup-grppermcox-setenum)

  - [`IsReflectionSubgroup(W, H): GrpPermCox, GrpPermCox → BoolElt`](subgroups.md#function-isreflectionsubgroup-grppermcox-grppermcox)

  - [`IsParabolicSubgroup(W, H): GrpPermCox, GrpPermCox → BoolElt`](subgroups.md#function-isparabolicsubgroup-grppermcox-grppermcox)

  - [`IsStandardParabolicSubgroup(W, H): GrpPermCox, GrpPermCox → BoolElt`](subgroups.md#function-isstandardparabolicsubgroup-grppermcox-grppermcox)

  - [`Overgroup(H): GrpPermCox → GrpPermCox`](subgroups.md#function-overgroup-grppermcox)

  - [`Overdatum(H): GrpPermCox → RootDtm`](subgroups.md#function-overdatum-grppermcox)

  - [`LocalCoxeterGroup(H): GrpPermCox → GrpPermCox, Map`](subgroups.md#function-localcoxetergroup-grppermcox)

  - [`Example: Reflection Subgroups`](subgroups.md#example-ex-0f53d0)

  - [`Transversal(W, H): GrpPermCox, GrpPermCox → SetIndx`](subgroups.md#function-transversal-grppermcox-grppermcox)

  - [`TransversalWords(W, H): GrpPermCox, GrpPermCox → SetIndx`](subgroups.md#function-transversalwords-grppermcox-grppermcox)

  - [`TransversalElt(W, H, x): GrpPermCox, GrpPermCox, GrpPermElt → GrpPermElt`](subgroups.md#function-transversalelt-grppermcox-grppermcox-grppermelt)

  - [`Example: Transversals`](subgroups.md#example-ex-a0c74a)

  - [`TransversalElt(W, x, H): GrpPermCox, GrpPermElt, GrpPermCox → GrpPermElt`](subgroups.md#function-transversalelt-grppermcox-grppermelt-grppermcox)

  - [`TransversalElt(W, H, x, J): GrpPermCox, GrpPermCox, GrpPermElt, GrpPermCox → GrpPermElt`](subgroups.md#function-transversalelt-grppermcox-grppermcox-grppermelt-grppermcox)

  - [`Transversal(W, J): GrpFPCox, \{RngIntElt\} → \{@ GrpFPCoxElt @\}`](subgroups.md#function-transversal-grpfpcox-rngintelt)

  - [`Transversal(W, J, L): GrpFPCox, \{RngIntElt\}, RngIntElt → \{@ GrpFPCoxElt @\}`](subgroups.md#function-transversal-grpfpcox-rngintelt-rngintelt)

  - [`Transversal(W, J, K): GrpFPCox, \{RngIntElt\}, \{RngIntElt\} → [ GrpFPCoxElt ], [ ]`](subgroups.md#function-transversal-grpfpcox-rngintelt-rngintelt-2)

  - [`DirectProduct(W1, W2): GrpPermCox, GrpPermCox → GrpPermCox`](subgroups.md#function-directproduct-grppermcox-grppermcox)

  - [`Dual(W): GrpPermCox → GrpPermCox`](subgroups.md#function-dual-grppermcox)

  - [`Example: Sum Dual`](subgroups.md#example-ex-4748b7)

- [Root Actions](action-root-coroot.md)

  - [`RootGSet(W): GrpPermCox → GSet`](action-root-coroot.md#function-rootgset-grppermcox)

  - [`CorootGSet(W): GrpPermCox → GSet`](action-root-coroot.md#function-corootgset-grppermcox)

  - [`Example: G Sets`](action-root-coroot.md#example-ex-90d28a)

  - [`RootAction(W): GrpPermCox → Map`](action-root-coroot.md#function-rootaction-grppermcox)

  - [`CorootAction(W): GrpPermCox → Map`](action-root-coroot.md#function-corootaction-grppermcox)

  - [`Example: Coroot Action`](action-root-coroot.md#example-ex-1fa494)

  - [`ReflectionGroup(W): GrpPermCox → GrpMat, Map`](action-root-coroot.md#function-reflectiongroup-grppermcox-2)

  - [`CoreflectionGroup(W): GrpPermCox → GrpMat, Map`](action-root-coroot.md#function-coreflectiongroup-grppermcox)

  - [`Example: Reflection Groups`](action-root-coroot.md#example-ex-2596af)

- [Standard Action](action-root-coroot-2.md)

  - [`StandardAction(W): GrpFPCox → Map`](action-root-coroot-2.md#function-standardaction-grpfpcox)

  - [`StandardAction(W): GrpPermCox → Map`](action-root-coroot-2.md#function-standardaction-grppermcox)

  - [`StandardActionGroup(W): GrpFPCox → GrpPerm, Map`](action-root-coroot-2.md#function-standardactiongroup-grpfpcox)

  - [`StandardActionGroup(W): GrpPermCox → GrpPerm, Map`](action-root-coroot-2.md#function-intrstandardactiongroup)

  - [`Example: Standard Action`](action-root-coroot-2.md#example-ex-eff773)

- [Braid Groups](group-braid.md)

  - [`BraidGroup(W): GrpFPCox → GrpFP, Map`](group-braid.md#function-braidgroup-grpfpcox)

  - [`PureBraidGroup(W): GrpFPCox → GrpFP, Map`](group-braid.md#function-purebraidgroup-grpfpcox)

  - [`Example: Braid Groups`](group-braid.md#example-ex-a9e54b)

- [$W$-graphs](group-wgraphs.md)

  - [`SetVerbose("WGraph", v): MonStgElt, RngIntElt`](group-wgraphs.md#function-setverbose-monstgelt-rngintelt)

  - [`Mij2EltRootTable(seq): SeqEnum → SeqEnum[SeqEnum[RngIntElt]]`](group-wgraphs.md#function-mij2eltroottable-seqenum)

  - [`Name2Mij(name): MonStgElt → SeqEnum`](group-wgraphs.md#function-name2mij-monstgelt)

  - [`Example: mijseq`](group-wgraphs.md#example-ex-320506)

  - [`Partition2WGtable(pi): SeqEnum → SeqEnum, GrpFPCox`](group-wgraphs.md#function-partition2wgtable-seqenum)

  - [`WGtable2WG(table): SeqEnum → GrphUnd`](group-wgraphs.md#function-wgtable2wg-seqenum)

  - [`TestWG(W,wg): GrpFPCox, GrphUnd → .`](group-wgraphs.md#function-testwg-grpfpcox-grphund)

  - [`TestWG(W,wg): GrpFPCox, GrphDir → .`](group-wgraphs.md#function-testwg-grpfpcox-grphdir)

  - [`TestWG(tp,wg): MonStgElt, GrphDir → .`](group-wgraphs.md#function-testwg-monstgelt-grphdir)

  - [`TestWG(tp,wg): MonStgElt, GrphDir → .`](group-wgraphs.md#function-testwg-monstgelt-grphdir-2)

  - [`Example: Specht Wgraph`](group-wgraphs.md#example-ex-8afb5c)

  - [`WGelement2WGtable(g,K): GrpFPCoxElt, SetEnum → SeqEnum, SeqEnum`](group-wgraphs.md#function-wgelement2wgtable-grpfpcoxelt-setenum)

  - [`Example: B5Wgraph`](group-wgraphs.md#example-ex-0bb421)

  - [`GetCells(wg): GrphUnd → SeqEnum`](group-wgraphs.md#function-getcells-grphund)

  - [`GetCells(wg): GrphDir → SeqEnum`](group-wgraphs.md#function-getcells-grphdir)

  - [`InduceWG(W,wg,seq): GrpFPCox, GrphUnd, SeqEnum → GrphUnd`](group-wgraphs.md#function-inducewg-grpfpcox-grphund-seqenum)

  - [`InduceWG(W,wg,seq): GrpFPCox, GrphDir, SeqEnum → GrphDir`](group-wgraphs.md#function-inducewg-grpfpcox-grphdir-seqenum)

  - [`InduceWGtable(J, table, W): SeqEnum, SeqEnum, GrpFPCox → SeqEnum[SeqEnum[RngIntElt]]`](group-wgraphs.md#function-inducewgtable-seqenum-seqenum-grpfpcox)

  - [`IsWGsymmetric(dwg): GrphDir → BoolElt, GrphDir`](group-wgraphs.md#function-iswgsymmetric-grphdir)

  - [`MakeDirected(uwg): GrphUnd → GrphDir`](group-wgraphs.md#function-makedirected-grphund)

  - [`TestHeckeRep(W,r): GrpFPCox, SeqEnum → .`](group-wgraphs.md#function-testheckerep-grpfpcox-seqenum)

  - [`WG2GroupRep(wg): GrphUnd → SeqEnum`](group-wgraphs.md#function-wg2grouprep-grphund)

  - [`WG2GroupRep(wg): GrphDir → SeqEnum`](group-wgraphs.md#function-wg2grouprep-grphdir)

  - [`WG2HeckeRep(W,wg): GrpFPCox, GrphUnd → SeqEnum`](group-wgraphs.md#function-wg2heckerep-grpfpcox-grphund)

  - [`WG2HeckeRep(W,wg): GrpFPCox, GrphDir → SeqEnum`](group-wgraphs.md#function-wg2heckerep-grpfpcox-grphdir)

  - [`WGidealgens2WGtable(dgens,K): SeqEnum, SetEnum → SeqEnum[SeqEnum[RngIntElt]], SetIndx`](group-wgraphs.md#function-wgidealgens2wgtable-seqenum-setenum)

  - [`Example: Wgraph Ideal`](group-wgraphs.md#example-ex-8bb5d1)

  - [`WriteWG(file,uwg): MonStgElt, GrphUnd → .`](group-wgraphs.md#function-writewg-monstgelt-grphund)

  - [`WriteWG(file,dwg): MonStgElt, GrphDir → .`](group-wgraphs.md#function-writewg-monstgelt-grphdir)

- [Related Structures](related.md)

  - [`CoxeterGroup(GrpFP, W): Cat, GrpPermCox → GrpFPCox`](related.md#function-coxetergroup-cat-grppermcox)

  - [`Presentation(W): GrpPermCox → GrpFPCox`](related.md#function-presentation-grppermcox)

  - [`ReflectionGroup(W): GrpPermCox → GrpMat`](related.md#function-reflectiongroup-grppermcox-3)

  - [`CoxeterGroup(GrpMat, W): Cat, GrpPermCox → GrpFPCox`](related.md#function-coxetergroup-cat-grppermcox-2)

  - [`LieAlgebra(W, R): GrpPermCox, Rng → AlgLie`](related.md#function-liealgebra-grppermcox-rng)

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