Construction of Coxeter Groups
CoxeterGroup( RD ) : RootDtm -> GrpCox
CoxeterGroup( t ) : MonStgElt -> GrpCox
Example GrpCox_Create (H34E1)
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 (H34E2)
Transversal( W, H ) : GrpCox, GrpCox -> @ @
TransversalElt( W, H, x ) : GrpCox, GrpPermElt-> GrpPermElt
Example GrpCox_Transversals (H34E3)
Accessing the Root Datum
RootDatum( W ) : GrpCox -> RootDtm
RootDatum( F ) : GrpCox -> RootDtm
Example GrpCox_RDFromCox (H34E4)
Operations
RootSpace( W ) : GrpCox -> .
SimpleRoots( W ) : GrpCox -> Mtrx
CartanMatrix( W ) : GrpCox -> AlgMatElt
Rank( W ) : GrpCox -> RngIntElt
Dimension( W ) : GrpCox -> RngIntElt
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 (H34E5)
LeftDescentSet( W, w ) : GrpCox, GrpPermElt -> { }
RightDescentSet( W, w ) : GrpCox, GrpPermElt -> { }
Example GrpCox_DescentSets (H34E6)
AdditiveOrder( W ) : GrpCox -> SeqEnum
Example GrpCox_AdditiveOrder (H34E7)
BasicDegrees( W ) : GrpCox -> RngIntElt
Example GrpCox_BasicDegrees (H34E8)
Actions on Roots and Coroots
RootGSet( W ) : GrpCox -> GSet
Example GrpCox_GSets (H34E9)
RootAction( W ) : GrpCox -> Map
Example GrpCox_CorootAction (H34E10)
ReflectionGroup( W ) : GrpCox -> GrpMat, Map
Example GrpCox_ReflectionGroups (H34E11)
Finitely Presented Coxeter Groups
Creation
CoxeterGroup( GrpFP, t ) : Cat, MonStgElt -> GrpFP
Example GrpCox_F4 (H34E12)
Conversion
CoxeterGroup( GrpFP, W ) : Cat, GrpCox -> GrpFP, Map
CoxeterGroup( F ) : GrpFP -> GrpCox, Map
Example GrpCox_FPCoxeterGroups (H34E13)
Operations on FP Coxeter Groups
LongestElement( F ) : GrpFP -> SeqEnum
CoxeterElement( F ) : GrpFP -> SeqEnum
AdditiveOrder( F ) : GrpFP -> SeqEnum
Example GrpCox_WordOperations (H34E14)
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 (H34E15)
Actions
Action( F ) : GrpFP -> Map
RootAction( F ) : GrpFP -> Map
Example GrpCox_Actions (H34E16)
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 (H34E17)
The Standard Action
StandardAction( W ) : GrpCox -> Map
StandardActionGroup( W ) : GrpCox -> GrpPerm, Map
Example GrpCox_StandardAction (H34E18)
Weights
DominantWeight( W, v ) : GrpCox, . -> ModTupFldElt, []
WeightOrbit( W, v ) : GrpCox, . -> @ @
Example GrpCox_DominantWeights (H34E19)
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