[Next][Prev] [Right] [Left] [Up] [Index] [Root]

COXETER GROUPS

 
Acknowledgements
 
Introduction
      The Normal Form for Words
 
Constructing Coxeter Groups
 
Converting Between Types of Coxeter Group
 
Operations on Coxeter Groups
 
Properties of Coxeter Groups
 
Operations on Elements
 
Roots, Coroots and Reflections
      Accessing Roots and Coroots
      Operations and Properties for Root and Coroot Indices
      Weights
 
Reflections
 
Reflection Subgroups
 
Root Actions
 
Standard Action
 
Braid Groups
 
W-graphs
 
Related Structures
 
Bibliography







DETAILS

 
Introduction

      The Normal Form for Words

 
Constructing Coxeter Groups
      CoxeterGroup(GrpFPCox, N) : Cat, MonStgElt -> GrpFPCox
      IrreducibleCoxeterGroup(GrpFPCox, X, n) : Cat, MonStgElt, RngIntElt -> GrpFPCox
      Example GrpCox_ConstructByName (H98E1)
      CoxeterGroup(GrpFPCox, M) : Cat, AlgMatElt -> GrpFPCox
      CoxeterGroup(GrpFPCox, G) : Cat, GrphUnd -> GrpFPCox
      CoxeterGroup(GrpFPCox, C) : Cat, AlgMatElt -> GrpFPCox
      CoxeterGroup(GrpFPCox, D) : Cat, GrphDir -> GrpFPCox
      Example GrpCox_ConstructFromMatrix (H98E2)
      CoxeterGroup(GrpFPCox, R) : Cat, RootSys -> GrpFPCox
      CoxeterGroup(A, B) : Mtrx, Mtrx -> GrpPermCox
      Example GrpCox_ConstructByRoot (H98E3)

 
Converting Between Types of Coxeter Group
      CoxeterGroup(GrpFPCox, W) : Cat, GrpPermCox -> GrpFPCox, Map
      CoxeterGroup(GrpFPCox, W) : Cat, GrpMat -> GrpFPCox
      CoxeterGroup(GrpPermCox, W) : Cat, GrpFPCox -> GrpPermCox, Map
      CoxeterGroup(GrpPermCox, W) : Cat, GrpMat -> GrpPermCox, Map
      Example GrpCox_ConstructByGroup (H98E4)
      ReflectionGroup(W) : GrpFPCox -> GrpMat, Map
      ReflectionGroup(W) : GrpPermCox -> GrpMat, Map
      Example GrpCox_ReflectionGroupConversion (H98E5)
      CoxeterGroup(GrpFP, W) : Cat, GrpFPCox -> GrpFP, Map
      CoxeterGroup(GrpFP, W) : Cat, GrpPermCox -> GrpFP, Map
      CoxeterGroup(GrpFP, W) : Cat, GrpMat -> GrpPermCox, Map
      CoxeterGroup(GrpPerm, W) : Cat, GrpFPCox -> GrpPerm, Map
      CoxeterGroup(GrpPerm, W) : Cat, GrpPermCox -> GrpPerm, Map
      CoxeterGroup(GrpPerm, W) : Cat, GrpMat -> GrpPermCox, Map

 
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 (H98E6)
      RootSystem(W) : GrpPermCox -> RootDtm
      RootDatum(W) : GrpPermCox -> RootDtm
      Example GrpCox_GroupToRoot (H98E7)
      CartanName(W) : GrpFPCox -> List
      CoxeterDiagram(W) : GrpFPCox ->
      DynkinDiagram(W) : GrpPermCox ->
      Example GrpCox_NamesDiagrams (H98E8)
      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 (H98E9)
      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 (H98E10)
      BruhatLessOrEqual(x, y) : GrpPermElt, GrpPermElt -> BoolElt
      BruhatDescendants(x) : GrpPermElt -> SetEnum
      BruhatDescendants(X) : SetEnum -> SetEnum
      Example GrpCox_BruhatDescendants (H98E11)

 
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 (H98E12)

 
Operations on Elements
      Example GrpCox_WordArithmetic (H98E13)
      # w : GrpFPCoxElt -> RngIntElt
      LongestElement(W) : GrpFPCox -> SeqEnum
      CoxeterElement(W) : GrpFPCox -> SeqEnum
      CoxeterNumber(W) : GrpFPCox -> SeqEnum
      Example GrpCox_LongestCoxeterElements (H98E14)
      LeftDescentSet(W, w) : GrpFPCox, GrpFPCoxElt -> ()
      RightDescentSet(W, w) : GrpFPCox, GrpFPCoxElt -> ()
      Example GrpCox_DescentSets (H98E15)

 
Roots, Coroots and Reflections

      Accessing Roots and Coroots
            RootSpace(W) : GrpPermCox -> .
            SimpleRoots(W) : GrpPermCox -> Mtrx
            Example GrpCox_RootSpace (H98E16)
            NumberOfPositiveRoots(W) : GrpPermCox -> RngIntElt
            Roots(W) : GrpPermCox -> (@@)
            PositiveRoots(W) : GrpPermCox -> (@@)
            Root(W, r) : GrpPermCox, RngIntElt -> (@@)
            RootPosition(W, v) : GrpPermCox, . -> (@@)
            Example GrpCox_RootsCoroots (H98E17)
            HighestRoot(W) : GrpPermCox -> .
            HighestShortRoot(W) : GrpPermCox -> .
            Example GrpCox_HeighestRoots (H98E18)
            CoxeterForm(W) : GrpPermCox -> AlgMatElt
            AdditiveOrder(W) : GrpPermCox -> 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 (H98E19)
            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 (H98E20)

      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 (H98E21)

 
Reflections
      IsReflection(w) : GrpFPElt -> BoolElt
      Reflections(W) : GrpFPCox -> [GrpFPCoxElt]
      Example GrpCox_Reflections (H98E22)
      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 (H98E23)

 
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 (H98E24)
      Transversal(W, H) : GrpPermCox, GrpPermCox -> @ @
      TransversalWords(W, H) : GrpPermCox, GrpPermCox -> @ @
      TransversalElt(W, H, x) : GrpPermCox, GrpPermCox, GrpPermElt -> GrpPermElt
      Example GrpCox_Transversals (H98E25)
      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 (H98E26)

 
Root Actions
      RootGSet(W) : GrpPermCox -> GSet
      Example GrpCox_GSets (H98E27)
      RootAction(W) : GrpPermCox -> Map
      Example GrpCox_CorootAction (H98E28)
      ReflectionGroup(W) : GrpPermCox -> GrpMat, Map
      Example GrpCox_ReflectionGroups (H98E29)

 
Standard Action
      StandardAction(W) : GrpFPCox -> Map
      StandardActionGroup(W) : GrpFPCox -> GrpPerm, Map
      Example GrpCox_StandardAction (H98E30)

 
Braid Groups
      BraidGroup(W) : GrpFPCox -> GrpFP, Map
      PureBraidGroup(W) : GrpFPCox -> GrpFP, Map
      Example GrpCox_BraidGroups (H98E31)

 
W-graphs
      SetVerbose("WGraph", v) : MonStgElt, RngIntElt ->
      Mij2EltRootTable(seq) : SeqEnum -> SeqEnum[SeqEnum[RngIntElt]]
      Name2Mij(name) : MonStgElt -> SeqEnum
      Example GrpCox_mijseq (H98E32)
      Partition2WGtable(pi) : SeqEnum -> SeqEnum, GrpFPCox
      WGtable2WG(table) : SeqEnum -> GrphUnd
      TestWG(W,wg) : GrpFPCox, GrphUnd -> .
      Example GrpCox_SpechtWgraph (H98E33)
      WGelement2WGtable(g,K) : GrpFPCoxElt, SetEnum -> SeqEnum, SeqEnum
      Example GrpCox_B5Wgraph (H98E34)
      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 (H98E35)
      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

[Next][Prev] [Right] [____] [Up] [Index] [Root]
Version: V2.19 of Mon Dec 17 14:40:36 EST 2012