Constructing Coxeter Groups
CoxeterGroup(GrpFPCox, N) : Cat, MonStgElt -> GrpFPCox
IrreducibleCoxeterGroup(GrpFPCox, X, n) : Cat, MonStgElt, RngIntElt -> GrpFPCox
Example GrpCox_ConstructByName (H98E1)
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 (H98E2)
CoxeterGroup(GrpFPCox, R) : Cat, RootSys -> GrpFPCox
CoxeterGroup(A, B) : Mtrx, Mtrx -> GrpPermCox
Example GrpCox_ConstructByRoot (H98E3)
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 (H98E4)
ReflectionGroup(W) : GrpFPCox -> GrpMat, Map
ReflectionGroup(W) : GrpPermCox -> GrpMat, Map
Example GrpCox_ReflectionGroupConversion (H98E5)
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 (H98E6)
RootSystem(W) : GrpPermCox -> RootDtm
RootDatum(W) : GrpPermCox -> RootDtm
Example GrpCox_GroupToRoot (H98E7)
CartanName(W) : GrpFPCox -> List
CoxeterDiagram(W) : GrpFPCox ->
DynkinDiagram(W) : GrpPermCox ->
Example GrpCox_NamesDiagrams (H98E8)
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 (H98E9)
ConjugacyClasses(W) : GrpFPCox -> [GrpFPCoxElt]
FundamentalGroup(W) : GrpPermCox -> GrpAb
IsogenyGroup(W) : GrpPermCox -> GrpAb
CoisogenyGroup(W) : GrpPermCox -> GrpAb
BasicDegrees(W) : GrpFPCox -> RngIntElt
BasicCodegrees(W) : GrpFPCox -> RngIntElt
Example GrpCox_BasicDegrees (H98E10)
BruhatLessOrEqual(x, y) : GrpPermElt, GrpPermElt -> BoolElt
BruhatDescendants(x) : GrpPermElt -> SetEnum
BruhatDescendants(X) : SetEnum -> SetEnum
Example GrpCox_BruhatDescendants (H98E11)
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 (H98E12)
Operations on Elements
Example GrpCox_WordArithmetic (H98E13)
# w : GrpFPCoxElt -> RngIntElt
LongestElement(W) : GrpFPCox -> SeqEnum
CoxeterElement(W) : GrpFPCox -> SeqEnum
CoxeterNumber(W) : GrpFPCox -> SeqEnum
Example GrpCox_LongestCoxeterElements (H98E14)
LeftDescentSet(W, w) : GrpFPCox, GrpFPCoxElt -> ()
RightDescentSet(W, w) : GrpFPCox, GrpFPCoxElt -> ()
Example GrpCox_DescentSets (H98E15)
Roots, Coroots and Reflections
Accessing Roots and Coroots
RootSpace(W) : GrpPermCox -> .
SimpleRoots(W) : GrpPermCox -> Mtrx
Example GrpCox_RootSpace (H98E16)
NumberOfPositiveRoots(W) : GrpPermCox -> RngIntElt
Roots(W) : GrpPermCox -> (@@)
PositiveRoots(W) : GrpPermCox -> (@@)
Root(W, r) : GrpPermCox, RngIntElt -> (@@)
RootPosition(W, v) : GrpPermCox, . -> (@@)
Example GrpCox_RootsCoroots (H98E17)
HighestRoot(W) : GrpPermCox -> .
HighestShortRoot(W) : GrpPermCox -> .
Example GrpCox_HeighestRoots (H98E18)
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 (H98E19)
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 (H98E20)
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 (H98E21)
Reflections
IsReflection(w) : GrpFPElt -> BoolElt
Reflections(W) : GrpFPCox -> [GrpFPCoxElt]
Example GrpCox_Reflections (H98E22)
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 (H98E23)
Reflection Subgroups
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 (H98E24)
Transversal(W, H) : GrpPermCox, GrpPermCox -> @ @
TransversalWords(W, H) : GrpPermCox, GrpPermCox -> @ @
TransversalElt(W, H, x) : GrpPermCox, GrpPermCox, GrpPermElt -> GrpPermElt
Example GrpCox_Transversals (H98E25)
TransversalElt(W, x, H) : GrpPermCox, GrpPermElt, GrpPermCox -> GrpPermElt
TransversalElt(W, H, x, J) : GrpPermCox, GrpPermCox, GrpPermElt, GrpPermCox -> GrpPermElt
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 (H98E26)
Root Actions
RootGSet(W) : GrpPermCox -> GSet
Example GrpCox_GSets (H98E27)
RootAction(W) : GrpPermCox -> Map
Example GrpCox_CorootAction (H98E28)
ReflectionGroup(W) : GrpPermCox -> GrpMat, Map
Example GrpCox_ReflectionGroups (H98E29)
Standard Action
StandardAction(W) : GrpFPCox -> Map
StandardActionGroup(W) : GrpFPCox -> GrpPerm, Map
Example GrpCox_StandardAction (H98E30)
Braid Groups
BraidGroup(W) : GrpFPCox -> GrpFP, Map
PureBraidGroup(W) : GrpFPCox -> GrpFP, Map
Example GrpCox_BraidGroups (H98E31)
W-graphs
SetVerbose("WGraph", v) : MonStgElt, RngIntElt ->
Mij2EltRootTable(seq) : SeqEnum -> SeqEnum[SeqEnum[RngIntElt]]
Name2Mij(name) : MonStgElt -> SeqEnum
Example GrpCox_mijseq (H98E32)
Partition2WGtable(pi) : SeqEnum -> SeqEnum, GrpFPCox
WGtable2WG(table) : SeqEnum -> GrphUnd
TestWG(W,wg) : GrpFPCox, GrphUnd -> .
Example GrpCox_SpechtWgraph (H98E33)
WGelement2WGtable(g,K) : GrpFPCoxElt, SetEnum -> SeqEnum, SeqEnum
Example GrpCox_B5Wgraph (H98E34)
GetCells(wg) : GrphUnd -> SeqEnum
InduceWG(W,wg,seq) : GrpFPCox, GrphUnd, SeqEnum -> GrphUnd
InduceWGtable(J, table, W) : SeqEnum, SeqEnum, GrpFPCox -> SeqEnum[SeqEnum[RngIntElt]]
IsWGsymmetric(dwg) : GrphDir -> BoolElt, GrphDir
MakeDirected(uwg) : GrphUnd -> GrphDir
TestHeckeRep(W,r) : GrpFPCox, SeqEnum -> .
WG2GroupRep(wg) : GrphUnd -> SeqEnum
WG2HeckeRep(W,wg) : GrpFPCox, GrphUnd -> SeqEnum
WGidealgens2WGtable(dgens,K) : SeqEnum, SetEnum -> SeqEnum[SeqEnum[RngIntElt]], SetIndx
Example GrpCox_WgraphIdeal (H98E35)
WriteWG(file,uwg) : MonStgElt, GrphUnd -> .
Related Structures
CoxeterGroup(GrpFP, W) : Cat, GrpPermCox -> GrpFPCox
ReflectionGroup(W) : GrpPermCox -> GrpMat
LieAlgebra(W, R) : GrpPermCox, Rng -> AlgLie
GroupOfLieType(W, R) : GrpPermCox, Rng -> GrpLie
Bibliography
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Wed Apr 24 15:09:57 EST 2013