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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
Building Coxeter Groups
Root actions
Standard action
Braid Groups
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 (H74E1)
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 (H74E2)
CoxeterGroup(GrpFPCox, R) : Cat, RootSys -> GrpFPCox
CoxeterGroup(A, B) : Mtrx, Mtrx -> GrpPermCox
Example GrpCox_ConstructByRoot (H74E3)
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 (H74E4)
ReflectionGroup(W) : GrpFPCox -> GrpMat, Map
ReflectionGroup(W) : GrpPermCox -> GrpMat, Map
Example GrpCox_ReflectionGroupConversion (H74E5)
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 (H74E6)
RootSystem(W) : GrpPermCox -> RootDtm
RootDatum(W) : GrpPermCox -> RootDtm
Example GrpCox_GroupToRoot (H74E7)
CartanName(W) : GrpFPCox -> List
CoxeterDiagram(W) : GrpFPCox ->
DynkinDiagram(W) : GrpPermCox ->
Example GrpCox_NamesDiagrams (H74E8)
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 (H74E9)
ConjugacyClasses(W) : GrpFPCox -> [GrpFPCoxElt]
FundamentalGroup(W) : GrpPermCox -> GrpAb
IsogenyGroup(W) : GrpPermCox -> GrpAb
CoisogenyGroup(W) : GrpPermCox -> GrpAb
BasicDegrees(W) : GrpFPCox -> RngIntElt
Example GrpCox_BasicDegrees (H74E10)
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 (H74E11)
Operations on Elements
Example GrpCox_WordArithmetic (H74E12)
# w : GrpFPCoxElt -> RngIntElt
LongestElement(W) : GrpFPCox -> SeqEnum
CoxeterElement(W) : GrpFPCox -> SeqEnum
CoxeterNumber(W) : GrpFPCox -> SeqEnum
Example GrpCox_LongestCoxeterElements (H74E13)
LeftDescentSet(W, w) : GrpFPCox, GrpFPCoxElt ->
RightDescentSet(W, w) : GrpFPCox, GrpFPCoxElt ->
Example GrpCox_DescentSets (H74E14)
Roots, Coroots and Reflections
Accessing Roots and Coroots
RootSpace(W) : GrpPermCox -> .
SimpleRoots(W) : GrpPermCox -> Mtrx
Example GrpCox_RootSpace (H74E15)
NumberOfPositiveRoots(W) : GrpPermCox -> RngIntElt
Roots(W) : GrpPermCox -> {@@}
PositiveRoots(W) : GrpPermCox -> {@@}
Root(W, r) : GrpPermCox, RngIntElt -> {@@}
RootPosition(W, v) : GrpPermCox, . -> {@@}
Example GrpCox_RootsCoroots (H74E16)
HighestRoot(W) : GrpPermCox -> .
HighestShortRoot(W) : GrpPermCox -> .
Example GrpCox_HeighestRoots (H74E17)
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 (H74E18)
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 (H74E19)
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 (H74E20)
Reflections
IsReflection(w) : GrpFPElt -> BoolElt
Reflections(W) : GrpFPCox -> [GrpFPCoxElt]
Example GrpCox_Reflections (H74E21)
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 (H74E22)
Building Coxeter Groups
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 (H74E23)
Transversal(W, H) : GrpPermCox, GrpPermCox -> @ @
TransversalWds(W, H) : GrpPermCox, GrpPermCox -> @ @
TransversalElt(W, H, x) : GrpPermCox, GrpPermCox, GrpPermElt -> GrpPermElt
Example GrpCox_Transversals (H74E24)
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 (H74E25)
Root actions
RootGSet(W) : GrpPermCox -> GSet
Example GrpCox_GSets (H74E26)
RootAction(W) : GrpPermCox -> Map
Example GrpCox_CorootAction (H74E27)
ReflectionGroup(W) : GrpPermCox -> GrpMat, Map
Example GrpCox_ReflectionGroups (H74E28)
Standard action
StandardAction(W) : GrpFPCox -> Map
StandardActionGroup(W) : GrpFPCox -> GrpPerm, Map
Example GrpCox_StandardAction (H74E29)
Braid Groups
BraidGroup(W) : GrpFPCox -> GrpFP, Map
PureBraidGroup(W) : GrpFPCox -> GrpFP, Map
Example GrpCox_BraidGroups (H74E30)
Related Structures
CoxeterGroup(GrpFP, W) : Cat, GrpPermCox -> GrpFPCox
ReflectionGroup(W) : GrpPermCox -> GrpMat
LieAlgebra(W, R) : GrpPermCox, Rng -> AlgLie
Bibliography
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