Construction of Coxeter groups
CoxeterGroup( RD ) : RootDtm -> GrpCox
CoxeterGroup( t ) : MonStgElt -> GrpCox
Example GrpCox_Create (H36E1)
Subgroups and transversals
ReflectionSubgroup( W, a ) : GrpCox, {} -> GrpCox
ReflectionSubgroup( W, s ) : GrpCox, [] -> GrpCox
StandardParabolicSubgroup( W, s ) : GrpCox, {} -> GrpCox
IsReflectionSubgroup( W, H ) : GrpCox -> GrpCox
IsStandardParabolicSubgroup( W, H ) : GrpCox -> GrpCox
Overgroup( H ) : GrpCox -> GrpCox
Overdatum( H ) : GrpCox -> GrpCox
LocalCoxeterGroup( H ) : GrpCox -> GrpCox, Map
Example GrpCox_ReflectionSubgroups (H36E2)
Transversal( W, H ) : GrpCox, GrpCox -> @ @
TransversalElt( W, H, x ) : GrpCox, GrpPermElt-> GrpPermElt
Example GrpCox_Transversals (H36E3)
Accessing the root datum
RootDatum( W ) : GrpCox -> RootDtm
RootDatum( F ) : GrpCox -> RootDtm
Example GrpCox_RDFromCox (H36E4)
Operations
RootSpace( W ) : GrpCox -> .
SimpleRoots( W ) : GrpCox -> Mtrx
CartanMatrix( W ) : GrpCox -> AlgMatElt
Rank( W ) : GrpCox -> RngIntElt
Dimension( W ) : GrpCox -> RngIntElt
IsogenyType( W ) : GrpCox -> List
DynkinDiagram( W ) : GrpCox ->
Properties
IsIrreducible( W ) : GrpCox -> BoolElt
IsCrystallographic( W ) : GrpCox -> BoolElt
Roots, coroots and weights
NumberOfPositiveRoots( W ) : GrpCox -> RngIntElt
Roots( W ) : GrpCox -> {@@}
PositiveRoots( W ) : GrpCox -> {@@}
Root( W, r ) : GrpCox, RngIntElt -> {@@}
RootPosition( W, v ) : GrpCox, . -> {@@}
WeightLattice( W ) : RootDtm -> Lat
FundamentalWeights( W ) : GrpCox -> SeqEnum
Operations on Coxeter groups and elements
Length( W, w ) : GrpCox, GrpPermElt -> RngIntElt
LongestElement( W ) : GrpCox -> GrpPermElt
CoxeterElement( W ) : GrpCox -> GrpPermElt
CoxeterNumber( W ) : GrpCox -> GrpPermElt
Example GrpCox_LongestCoxeterElements (H36E5)
LeftDescentSet( W, w ) : GrpCox, GrpPermElt -> {}
RightDescentSet( W, w ) : GrpCox, GrpPermElt -> {}
Example GrpCox_DescentSets (H36E6)
AdditiveOrder( W ) : GrpCox -> SeqEnum
Example GrpCox_AdditiveOrder (H36E7)
BasicDegrees( W ) : GrpCox -> RngIntElt
Example GrpCox_BasicDegrees (H36E8)
Actions on roots and coroots
RootGSet( W ) : GrpCox -> GSet
Example GrpCox_GSets (H36E9)
RootAction( W ) : GrpCox -> Map
Example GrpCox_CorootAction (H36E10)
ReflectionGroup( W ) : GrpCox -> GrpMat, Map
Example GrpCox_ReflectionGroups (H36E11)
Finitely presented Coxeter groups
Creation
CoxeterGroup( GrpFP, t ) : Cat, MonStgElt -> GrpFP
Example GrpCox_F4 (H36E12)
Conversion
CoxeterGroup( GrpFP, W ) : Cat, GrpCox -> GrpFP, Map
CoxeterGroup( F ) : GrpFP -> GrpCox, Map
Example GrpCox_FPCoxeterGroups (H36E13)
Operations on FP Coxeter groups
LongestElement( F ) : GrpFP -> SeqEnum
CoxeterElement( F ) : GrpFP -> SeqEnum
AdditiveOrder( F ) : GrpFP -> SeqEnum
Example GrpCox_WordOperations (H36E14)
Operations on words
WordProduct( F, w1, w2 ) : GrpFP, GrpFPElt, GrpFPElt -> GrpFPElt
Reduce( F, w ) : GrpFP, GrpFPElt -> GrpFPElt
Inverse( F, w ) : GrpFP, GrpFPElt -> GrpFPElt
IsEqual( F, w1, w2 ) : GrpFP, GrpFPElt, GrpFPElt -> BoolElt
Example GrpCox_WordArithmetic (H36E15)
Actions
Action( F ) : GrpFP -> Map
RootAction( F ) : GrpFP -> Map
Example GrpCox_Actions (H36E16)
Braid groups
BraidGroup( W ) : GrpCox -> GrpFP, Map
BraidGroup( F ) : GrpFP -> GrpFP, Map
PureBraidGroup( W ) : GrpCox -> GrpFP, Map
PureBraidGroup( F ) : GrpFP -> GrpFP, Map
Example GrpCox_BraidGroups (H36E17)
The standard action
StandardAction( W ) : GrpCox -> Map
StandardActionGroup( W ) : GrpCox -> GrpPerm, Map
Example GrpCox_StandardAction (H36E18)
Weights
DominantWeight( W, v ) : GrpCox, . -> ModTupFldElt, []
WeightOrbit( W, v ) : GrpCox, . -> @ @
Example GrpCox_DominantWeights (H36E19)
Element conversion functions
WordToPerm( W, w ) : GrpCox, [] -> GrpPermElt
WordToMatrix( W, w ) : GrpCox, [] -> AlgMatElt
PermToWord( W, p ) : GrpCox, GrpPermElt -> SeqEnum
MatrixToWord( W, M ) : GrpCox, AlgMatElt -> SeqEnum
MatrixToPerm( W, M ) : GrpCox, AlgMatElt -> GrpPermElt
PermToMatrix( W, M ) : GrpCox, GrpPermElt -> AlgMatElt
WordOnRoot( W, r, w ) : GrpCox, RngIntElt, . -> RngIntElt
WordOnRootSpace( W, v, w ) : GrpCox, ., . -> .
Related structures
LieAlgebra( W, R ) : GrpCox, Rng -> AlgLie