Constructing Finite Coxeter Groups
CoxeterGroup(N) : MonStgElt -> GrpPermCox
IrreducibleCoxeterGroup(X, n) : MonStgElt, RngIntElt -> GrpPermCox
Example GrpPermCox_ConstructByName (H87E1)
CoxeterGroup(R) : RootSys -> GrpPermCox
Example GrpPermCox_ConstructByRoot (H87E2)
CoxeterGroup(M) : AlgMatElt -> GrpPermCox
CoxeterGroup(A, B) : Mtrx, Mtrx -> GrpPermCox
CoxeterGroup(GrpPermCox, W) : Cat, GrpFPCox -> GrpPermCox, Map
CoxeterGroup(GrpPermCox, W) : Cat, GrpMat -> GrpPermCox, Map
Example GrpPermCox_ConstructByGroup (H87E3)
Operations on Finite Coxeter Groups
IsIsomorphic(W1, W2) : GrpFPCox, GrpFPCox -> BoolElt
IsCoxeterIsomorphic(W1, W2) : GrpFPCox, GrpFPCox -> BoolElt
IsCartanEquivalent(W1, W2) : GrpFPCox, GrpFPCox -> BoolElt
Example GrpPermCox_Isomorphism (H87E4)
RootSystem(W) : GrpPermCox -> RootDtm
RootDatum(W) : GrpPermCox -> RootDtm
Example GrpPermCox_GroupToRoot (H87E5)
CartanName(W) : GrpPermCox -> MonStgElt
CoxeterDiagram(W) : GrpPermCox ->
DynkinDiagram(W) : GrpPermCox ->
Example GrpPermCox_NamesDiagrams (H87E6)
CoxeterMatrix(W) : GrpFPCox -> AlgMatElt
CoxeterGraph(W) : GrpFPCox -> GrphUnd
CartanMatrix(W) : GrpFPCox -> AlgMatElt
DynkinDigraph(W) : GrpFPCox -> GrphDir
Rank(W) : GrpPermCox -> RngIntElt
Dimension(W) : GrpPermCox -> RngIntElt
Example GrpPermCox_RankDimension (H87E7)
FundamentalGroup(W) : GrpPermCox -> GrpAb
IsogenyGroup(W) : GrpPermCox -> GrpAb
CoisogenyGroup(W) : GrpPermCox -> GrpAb
BasicDegrees(W) : GrpPermCox -> RngIntElt
Example GrpPermCox_BasicDegrees (H87E8)
Properties of Coxeter Groups
IsIrreducible(W) : GrpPermCox -> BoolElt
IsSemisimple(W) : GrpPermCox -> BoolElt
IsCrystallographic(W) : GrpPermCox -> BoolElt
IsSimplyLaced(W) : GrpPermCox-> BoolElt
Example GrpPermCox_Properties (H87E9)
Operations on Elements
Length(w) : GrpPermCox, GrpPermElt -> RngIntElt
LongestElement(W) : GrpPermCox -> GrpPermElt
CoxeterElement(W) : GrpPermCox -> GrpPermElt
CoxeterNumber(W) : GrpPermCox -> GrpPermElt
Example GrpPermCox_LongestCoxeterElements (H87E10)
LeftDescentSet(W, w) : GrpPermCox, GrpPermElt -> {}
RightDescentSet(W, w) : GrpPermCox, GrpPermElt -> {}
Example GrpPermCox_DescentSets (H87E11)
Roots, Coroots and Reflections
Accessing Roots and Coroots
RootSpace(W) : GrpPermCox -> .
SimpleRoots(W) : GrpPermCox -> Mtrx
Example GrpPermCox_RootSpace (H87E12)
NumberOfPositiveRoots(W) : GrpPermCox -> RngIntElt
Roots(W) : GrpPermCox -> {@@}
PositiveRoots(W) : GrpPermCox -> {@@}
Root(W, r) : GrpPermCox, RngIntElt -> {@@}
RootPosition(W, v) : GrpPermCox, . -> {@@}
Example GrpPermCox_RootsCoroots (H87E13)
HighestRoot(W) : GrpPermCox -> .
HighestShortRoot(W) : GrpPermCox -> .
Example GrpPermCox_HeighestRoots (H87E14)
CoxeterForm(W) : GrpPermCox -> AlgMatElt
Reflections
IsReflection(w) : GrpPermCoxElt -> BoolElt, ., ., RngInt
Reflections(W) : GrpPermCox -> GrpFPCoxElt
Reflection(W, r) : GrpPermCox, RngIntElt -> GrpFPCoxElt
SimpleReflectionMatrices(W) : GrpPermCox -> []
ReflectionMatrices(W) : GrpPermCox -> []
ReflectionMatrix(W, r) : GrpPermCox, RngIntElt -> []
ReflectionWords(R) : GrpPermCox -> []
ReflectionWord(R, r) : GrpPermCox, RngIntElt -> []
Example GrpPermCox_Action (H87E15)
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 GrpPermCox_RootArithmetic (H87E16)
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 GrpPermCox_RootOperations (H87E17)
Weights
WeightLattice(W) : RootDtm -> Lat
FundamentalWeights(W) : GrpPermCox -> SeqEnum
DominantWeight(W, v) : GrpPermCox, . -> ModTupFldElt, []
WeightOrbit(W, v) : GrpPermCox, . -> @ @
Example GrpPermCox_DominantWeights (H87E18)
Constructing Coxeter Groups from Existing Coxeter Groups
ReflectionSubgroup(W, a) : GrpPermCox, {} -> GrpPermCox
ReflectionSubgroup(W, s) : GrpPermCox, [] -> GrpPermCox
StandardParabolicSubgroup(W, s) : GrpPermCox, {} -> GrpPermCox
IsReflectionSubgroup(W, H) : GrpPermCox -> GrpPermCox
IsStandardParabolicSubgroup(W, H) : GrpPermCox -> GrpPermCox
Overgroup(H) : GrpPermCox -> GrpPermCox
Overdatum(H) : GrpPermCox -> GrpPermCox
LocalCoxeterGroup(H) : GrpPermCox -> GrpPermCox, Map
Example GrpPermCox_ReflectionSubgroups (H87E19)
Transversal(W, H) : GrpPermCox, GrpPermCox -> @ @
TransversalElt(W, H, x) : GrpPermCox, GrpPermElt-> GrpPermElt
Example GrpPermCox_Transversals (H87E20)
DirectSum(W1, W2) : GrpPermCox, GrpPermCox -> GrpPermCox
Dual(W) : GrpPermCox -> GrpPermCox
Example GrpPermCox_SumDual (H87E21)
Actions
RootGSet(W) : GrpPermCox -> GSet
Example GrpPermCox_GSets (H87E22)
RootAction(W) : GrpPermCox -> Map
Example GrpPermCox_CorootAction (H87E23)
ReflectionGroup(W) : GrpPermCox -> GrpMat, Map
Example GrpPermCox_ReflectionGroups (H87E24)
StandardAction(W) : GrpPermCox -> Map
StandardActionGroup(W) : GrpPermCox -> GrpPerm, Map
Example GrpPermCox_StandardAction (H87E25)
Related Structures
CoxeterGroup(GrpFPCox, W) : Cat, GrpPermCox -> GrpFPCox
ReflectionGroup(W) : GrpPermCox -> GrpMat
LieAlgebra(W, R) : GrpPermCox, Rng -> AlgLie