Constructing Finite Coxeter Groups
CoxeterGroup( N ) : MonStgElt -> GrpPermCox
Example GrpPermCox_ConstructByName (H84E1)
CoxeterGroup( R ) : RootSys -> GrpPermCox
Example GrpPermCox_ConstructByRoot (H84E2)
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 (H84E3)
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 (H84E4)
RootSystem( W ) : GrpPermCox -> RootDtm
RootDatum( W ) : GrpPermCox -> RootDtm
Example GrpPermCox_GroupToRoot (H84E5)
CartanName( W ) : GrpPermCox -> MonStgElt
CoxeterDiagram( W ) : GrpPermCox ->
DynkinDiagram( W ) : GrpPermCox ->
Example GrpPermCox_NamesDiagrams (H84E6)
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 (H84E7)
FundamentalGroup( W ) : GrpPermCox -> GrpAb
IsogenyGroup( W ) : GrpPermCox -> GrpAb
CoisogenyGroup( W ) : GrpPermCox -> GrpAb
BasicDegrees( W ) : GrpPermCox -> RngIntElt
Example GrpPermCox_BasicDegrees (H84E8)
Properties of Coxeter Groups
IsIrreducible( W ) : GrpPermCox -> BoolElt
IsSemisimple( W ) : GrpPermCox -> BoolElt
IsCrystallographic( W ) : GrpPermCox -> BoolElt
IsSimplyLaced( W ) : GrpPermCox-> BoolElt
Example GrpPermCox_Properties (H84E9)
Operations on Elements
Length( w ) : GrpPermCox, GrpPermElt -> RngIntElt
LongestElement( W ) : GrpPermCox -> GrpPermElt
CoxeterElement( W ) : GrpPermCox -> GrpPermElt
CoxeterNumber( W ) : GrpPermCox -> GrpPermElt
Example GrpPermCox_LongestCoxeterElements (H84E10)
LeftDescentSet( W, w ) : GrpPermCox, GrpPermElt -> {}
RightDescentSet( W, w ) : GrpPermCox, GrpPermElt -> {}
Example GrpPermCox_DescentSets (H84E11)
Roots, Coroots and Reflections
Accessing Roots and Coroots
RootSpace( W ) : GrpPermCox -> .
SimpleRoots( W ) : GrpPermCox -> Mtrx
Example GrpPermCox_RootSpace (H84E12)
NumberOfPositiveRoots( W ) : GrpPermCox -> RngIntElt
Roots( W ) : GrpPermCox -> {@@}
PositiveRoots( W ) : GrpPermCox -> {@@}
Root( W, r ) : GrpPermCox, RngIntElt -> {@@}
RootPosition( W, v ) : GrpPermCox, . -> {@@}
Example GrpPermCox_RootsCoroots (H84E13)
HighestRoot( W ) : GrpPermCox -> .
HighestShortRoot( W ) : GrpPermCox -> .
Example GrpPermCox_HeighestRoots (H84E14)
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 (H84E15)
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 (H84E16)
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 (H84E17)
Weights
WeightLattice( W ) : RootDtm -> Lat
FundamentalWeights( W ) : GrpPermCox -> SeqEnum
DominantWeight( W, v ) : GrpPermCox, . -> ModTupFldElt, []
WeightOrbit( W, v ) : GrpPermCox, . -> @ @
Example GrpPermCox_DominantWeights (H84E18)
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 (H84E19)
Transversal( W, H ) : GrpPermCox, GrpPermCox -> @ @
TransversalElt( W, H, x ) : GrpPermCox, GrpPermElt-> GrpPermElt
Example GrpPermCox_Transversals (H84E20)
W1 + W2 : GrpPermCox, GrpPermCox -> GrpPermCox
Dual( W ) : GrpPermCox -> GrpPermCox
Example GrpPermCox_SumDual (H84E21)
Actions
RootGSet( W ) : GrpPermCox -> GSet
Example GrpPermCox_GSets (H84E22)
RootAction( W ) : GrpPermCox -> Map
Example GrpPermCox_CorootAction (H84E23)
ReflectionGroup( W ) : GrpPermCox -> GrpMat, Map
Example GrpPermCox_ReflectionGroups (H84E24)
StandardAction( W ) : GrpPermCox -> Map
StandardActionGroup( W ) : GrpPermCox -> GrpPerm, Map
Example GrpPermCox_StandardAction (H84E25)
Related Structures
CoxeterGroup( GrpFPCox, W ) : Cat, GrpPermCox -> GrpFPCox
ReflectionGroup( W ) : GrpPermCox -> GrpMat
LieAlgebra( W, R ) : GrpPermCox, Rng -> AlgLie