Introduction

      The Normal Form for Words

 
Constructing Coxeter Groups
      CoxeterGroup(grpcat, N) : Cat, MonStgElt -> grpcat
      CoxeterGroup(N) : MonStgElt -> Grp
      IrreducibleCoxeterGroup(grpcat, X, n) : Cat, MonStgElt, RngIntElt -> grpcat
      Example GrpCox_ConstructByName (H108E1)
      CoxeterGroup(grpcat, M) : Cat, AlgMatElt -> grpcat
      CoxeterGroup(grpcat, G) : Cat, GrphUnd -> grpcat
      CoxeterGroup(grpcat, D) : Cat, GrphDir -> grpcat
      CoxeterGroup(M) : AlgMatElt -> Grp
      CoxeterGroup(G) : GrphUnd -> Grp
      CoxeterGroup(D) : GrphDir -> Grp
      Example GrpCox_ConstructFromMatrix (H108E2)
      CoxeterGroup(grpcat, R) : Cat, RootStr -> grpcat
      CoxeterGroup(R) : RootStr -> GrpPermCox
      CoxeterGroup(A, B) : Mtrx, Mtrx -> GrpPermCox
      CoxeterGroup(grpcat, A, B) : Cat, Mtrx, Mtrx -> grpcat
      Example GrpCox_ConstructByRoot (H108E3)

 
Converting Between Types of Coxeter Group
      CoxeterGroup(grpcat, W) : Cat, Grp -> grpcat, Map
      Example GrpCox_ConstructByGroup (H108E4)
      ReflectionGroup(W) : GrpFPCox -> GrpMat, Map
      ReflectionGroup(W) : GrpPermCox -> GrpMat, Map
      Example GrpCox_ReflectionGroupConversion (H108E5)

 
Operations on Coxeter Groups
      IsIsomorphic(W1, W2) : GrpPermCox, GrpPermCox -> BoolElt
      IsCoxeterIsomorphic(W1, W2) : GrpFPCox, GrpFPCox -> BoolElt
      IsCartanEquivalent(W1, W2) : GrpPermCox, GrpPermCox -> BoolElt
      Example GrpCox_CoxeterIsomorphism (H108E6)
      RootSystem(W) : GrpPermCox -> RootDtm
      RootDatum(W) : GrpPermCox -> RootDtm
      Example GrpCox_GroupToRoot (H108E7)
      CartanName(W) : GrpFPCox -> List
      CoxeterDiagram(W) : GrpFPCox ->
      DynkinDiagram(W) : GrpPermCox ->
      Example GrpCox_NamesDiagrams (H108E8)
      CoxeterMatrix(W) : GrpFPCox -> AlgMatElt
      CoxeterGraph(W) : GrpFPCox -> GrphUnd
      CartanMatrix(W) : GrpPermCox -> AlgMatElt
      DynkinDigraph(W) : GrpPermCox -> GrphDir
      Rank(W) : GrpFPCox -> RngIntElt
      NumberOfPositiveRoots(W) : GrpFPCox -> RngIntElt
      Dimension(W) : GrpPermCox -> RngIntElt
      Example GrpCox_RankDimension (H108E9)
      ConjugacyClasses(W) : GrpFPCox -> [GrpFPCoxElt]
      FundamentalGroup(W) : GrpPermCox -> GrpAb
      IsogenyGroup(W) : GrpPermCox -> GrpAb
      CoisogenyGroup(W) : GrpPermCox -> GrpAb
      BasicDegrees(W) : GrpFPCox -> RngIntElt
      BasicCodegrees(W) : GrpFPCox -> RngIntElt
      Example GrpCox_BasicDegrees (H108E10)
      BruhatLessOrEqual(x, y) : GrpPermElt, GrpPermElt -> BoolElt
      BruhatDescendants(x) : GrpPermElt -> SetEnum
      BruhatDescendants(X) : SetEnum -> SetEnum
      Example GrpCox_BruhatDescendants (H108E11)

 
Properties of Coxeter Groups
      IsFinite(W) : GrpFPCox -> BoolElt
      IsAffine(W) : GrpFPCox -> BoolElt
      IsHyperbolic(W) : GrpFPCox -> BoolElt
      IsCompactHyperbolic(W) : GrpFPCox -> BoolElt
      IsIrreducible(W) : GrpFPCox -> BoolElt
      IsSemisimple(W) : GrpPermCox -> BoolElt
      IsCrystallographic(W) : GrpPermCox -> BoolElt
      IsSimplyLaced(W) : GrpPermCox-> BoolElt
      Example GrpCox_Properties (H108E12)

 
Operations on Elements
      Example GrpCox_WordArithmetic (H108E13)
      # w : GrpFPCoxElt -> RngIntElt
      LongestElement(W) : GrpFPCox -> SeqEnum
      CoxeterElement(W) : GrpFPCox -> SeqEnum
      CoxeterNumber(W) : GrpFPCox -> SeqEnum
      Example GrpCox_LongestCoxeterElements (H108E14)
      LeftDescentSet(W, w) : GrpFPCox, GrpFPCoxElt -> ()
      RightDescentSet(W, w) : GrpFPCox, GrpFPCoxElt -> ()
      Example GrpCox_DescentSets (H108E15)

 
Roots, Coroots and Reflections

      Accessing Roots and Coroots
            RootSpace(W) : GrpPermCox -> .
            SimpleRoots(W) : GrpPermCox -> Mtrx
            Example GrpCox_RootSpace (H108E16)
            NumberOfPositiveRoots(W) : GrpPermCox -> RngIntElt
            Roots(W) : GrpPermCox -> (@@)
            PositiveRoots(W) : GrpPermCox -> (@@)
            Root(W, r) : GrpPermCox, RngIntElt -> (@@)
            RootPosition(W, v) : GrpPermCox, . -> (@@)
            Example GrpCox_RootsCoroots (H108E17)
            HighestRoot(W) : GrpPermCox -> .
            HighestShortRoot(W) : GrpPermCox -> .
            Example GrpCox_HeighestRoots (H108E18)
            CoxeterForm(W) : GrpPermCox -> AlgMatElt
            AdditiveOrder(W) : GrpPermCox -> SeqEnum
            PapiOrder(W,w) : GrpPermCox, GrpPermElt -> SeqEnum

      Operations and Properties for Root and Coroot Indices
            Sum(W, r, s) : GrpPermCox, RngIntElt, RngIntElt -> RngIntElt
            IsPositive(W, r) : GrpPermCox, RngIntElt -> BoolElt
            IsNegative(W, r) : GrpPermCox, RngIntElt -> BoolElt
            Negative(W, r) : GrpPermCox, RngIntElt -> RngIntElt
            LeftString(W, r, s) : GrpPermCox, RngIntElt, RngIntElt -> RngIntElt
            RightString(W, r, s) : GrpPermCox, RngIntElt, RngIntElt -> RngIntElt
            LeftStringLength(W, r, s) : GrpPermCox, RngIntElt, RngIntElt -> RngIntElt
            RightStringLength(W, r, s) : GrpPermCox, RngIntElt, RngIntElt -> RngIntElt
            Example GrpCox_RootArithmetic (H108E19)
            RootHeight(W, r) : GrpPermCox, RngIntElt -> RngIntElt
            RootNorms(W) : GrpPermCox -> [RngIntElt]
            RootNorm(W, r) : GrpPermCox, RngIntElt -> RngIntElt
            IsLongRoot(W, r) : GrpPermCox, RngIntElt -> BoolElt
            IsShortRoot(W, r) : GrpPermCox, RngIntElt -> BoolElt
            Example GrpCox_RootOperations (H108E20)

      Weights
            WeightLattice(W) : GrpPermCox -> Lat
            FundamentalWeights(W) : GrpPermCox -> SeqEnum
            IsDominant(R, v) : RootDtm, . -> ModTupFldElt, GrpFPCoxElt
            DominantWeight(W, v) : GrpPermCox, . -> ModTupFldElt, GrpFPCoxElt
            WeightOrbit(W, v) : GrpPermCox, . -> @ ModTupFldElt @, [GrpFPCoxElt]
            Example GrpCox_DominantWeights (H108E21)

 
Reflections
      IsReflection(w) : GrpFPElt -> BoolElt
      Reflections(W) : GrpFPCox -> [GrpFPCoxElt]
      Example GrpCox_Reflections (H108E22)
      SimpleReflections(W) : GrpFPCox -> [GrpFPCoxElt]
      SimpleReflectionPermutations(W) : GrpPermCox -> [GrpPermElt]
      Reflection(W, r) : GrpPermCox, RngIntElt -> GrpPermElt
      SimpleReflectionMatrices(W) : GrpPermCox -> []
      ReflectionMatrices(W) : GrpPermCox -> []
      ReflectionMatrix(W, r) : GrpPermCox, RngIntElt -> []
      ReflectionWords(W) : GrpPermCox -> []
      ReflectionWord(W, r) : GrpPermCox, RngIntElt -> []
      Example GrpCox_Action (H108E23)

 
Reflection Subgroups
      ReflectionSubgroup(W, a) : GrpPermCox, () -> GrpPermCox
      ReflectionSubgroup(W, s) : GrpPermCox, [] -> GrpPermCox
      StandardParabolicSubgroup(W, J) : GrpPermCox, () -> GrpPermCox
      IsReflectionSubgroup(W, H) : GrpPermCox, GrpPermCox -> BoolElt
      IsParabolicSubgroup(W, H) : GrpPermCox, GrpPermCox -> BoolElt
      IsStandardParabolicSubgroup(W, H) : GrpPermCox, GrpPermCox -> BoolElt
      Overgroup(H) : GrpPermCox -> GrpPermCox
      Overdatum(H) : GrpPermCox -> RootDtm
      LocalCoxeterGroup(H) : GrpPermCox -> GrpPermCox, Map
      Example GrpCox_ReflectionSubgroups (H108E24)
      Transversal(W, H) : GrpPermCox, GrpPermCox -> @ @
      TransversalWords(W, H) : GrpPermCox, GrpPermCox -> @ @
      TransversalElt(W, H, x) : GrpPermCox, GrpPermCox, GrpPermElt -> GrpPermElt
      Example GrpCox_Transversals (H108E25)
      TransversalElt(W, x, H) : GrpPermCox, GrpPermElt, GrpPermCox -> GrpPermElt
      TransversalElt(W, H, x, J) : GrpPermCox, GrpPermCox, GrpPermElt, GrpPermCox -> GrpPermElt
      Transversal(W, J) : GrpFPCox, (RngIntElt} -> (@ GrpFPCoxElt @)
      Transversal(W, J, K) : GrpFPCox, (RngIntElt}, (RngIntElt} -> [ GrpFPCoxElt ], [ ]
      DirectProduct(W1, W2) : GrpPermCox, GrpPermCox -> GrpPermCox
      Dual(W) : GrpPermCox -> GrpPermCox
      Example GrpCox_SumDual (H108E26)

 
Root Actions
      RootGSet(W) : GrpPermCox -> GSet
      Example GrpCox_GSets (H108E27)
      RootAction(W) : GrpPermCox -> Map
      Example GrpCox_CorootAction (H108E28)
      ReflectionGroup(W) : GrpPermCox -> GrpMat, Map
      Example GrpCox_ReflectionGroups (H108E29)

 
Standard Action
      StandardAction(W) : GrpFPCox -> Map
      StandardActionGroup(W) : GrpFPCox -> GrpPerm, Map
      Example GrpCox_StandardAction (H108E30)

 
Braid Groups
      BraidGroup(W) : GrpFPCox -> GrpFP, Map
      PureBraidGroup(W) : GrpFPCox -> GrpFP, Map
      Example GrpCox_BraidGroups (H108E31)

 
W-graphs
      SetVerbose("WGraph", v) : MonStgElt, RngIntElt ->
      Mij2EltRootTable(seq) : SeqEnum -> SeqEnum[SeqEnum[RngIntElt]]
      Name2Mij(name) : MonStgElt -> SeqEnum
      Example GrpCox_mijseq (H108E32)
      Partition2WGtable(pi) : SeqEnum -> SeqEnum, GrpFPCox
      WGtable2WG(table) : SeqEnum -> GrphUnd
      TestWG(W,wg) : GrpFPCox, GrphUnd -> .
      Example GrpCox_SpechtWgraph (H108E33)
      WGelement2WGtable(g,K) : GrpFPCoxElt, SetEnum -> SeqEnum, SeqEnum
      Example GrpCox_B5Wgraph (H108E34)
      GetCells(wg) : GrphUnd -> SeqEnum
      InduceWG(W,wg,seq) : GrpFPCox, GrphUnd, SeqEnum -> GrphUnd
      InduceWGtable(J, table, W) : SeqEnum, SeqEnum, GrpFPCox -> SeqEnum[SeqEnum[RngIntElt]]
      IsWGsymmetric(dwg) : GrphDir -> BoolElt, GrphDir
      MakeDirected(uwg) : GrphUnd -> GrphDir
      TestHeckeRep(W,r) : GrpFPCox, SeqEnum -> .
      WG2GroupRep(wg) : GrphUnd -> SeqEnum
      WG2HeckeRep(W,wg) : GrpFPCox, GrphUnd -> SeqEnum
      WGidealgens2WGtable(dgens,K) : SeqEnum, SetEnum -> SeqEnum[SeqEnum[RngIntElt]], SetIndx
      Example GrpCox_WgraphIdeal (H108E35)
      WriteWG(file,uwg) : MonStgElt, GrphUnd -> .

 
Related Structures
      CoxeterGroup(GrpFP, W) : Cat, GrpPermCox -> GrpFPCox
      ReflectionGroup(W) : GrpPermCox -> GrpMat
      LieAlgebra(W, R) : GrpPermCox, Rng -> AlgLie
      GroupOfLieType(W, R) : GrpPermCox, Rng -> GrpLie

 
Bibliography

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