Construction of Pseudo- reflections
PseudoReflection(a, b) : ModTupRngElt, ModTupRngElt -> AlgMatElt
Transvection(a, b) : ModTupRngElt, ModTupRngElt -> AlgMatElt
Reflection(a, b) : ModTupRngElt, ModTupRngElt -> AlgMatElt
IsPseudoReflection(r) : Mtrx -> BoolElt, ModTupRngElt, ModTupRngElt
IsTransvection(r) : Mtrx -> BoolElt, ModTupRngElt, ModTupRngElt
IsReflection(r) : Mtrx -> BoolElt, ModTupRngElt, ModTupRngElt
IsReflectionGroup(G) : GrpMat -> BoolElt
Example GrpRfl_pseudoreflection (H99E1)
Example GrpRfl_ref-group (H99E2)
Example GrpRfl_transvections (H99E3)
Pseudo-reflections Preserving Reflexive Forms
SymplecticTransvection(a, alpha) : ModTupRngElt, FldElt -> AlgMatElt
UnitaryTransvection(a, alpha) : ModTupRngElt, FldElt -> AlgMatElt
UnitaryReflection(a, zeta) : ModTupRngElt, FldElt -> AlgMatElt
OrthogonalReflection(a) : ModTupFldElt -> AlgMatElt
Example GrpRfl_unitary-transvection (H99E4)
Construction of Reflection Groups
PseudoReflectionGroup(A, B) : Mtrx, Mtrx -> GrpMat, Map
Example GrpRfl_ReflectionGroups (H99E5)
Construction of Real Reflection Groups
ReflectionGroup(M) : AlgMatElt -> GrpMat
ReflectionGroup(N) : MonStgElt -> GrpMat
IrreducibleReflectionGroup(X, n) : MonStgElt, RngIntElt -> GrpMat
Example GrpRfl_RealReflectionGroupByCartan (H99E6)
ReflectionGroup(R) : RootSys -> GrpMat
Example GrpRfl_RealReflectionGroupByRootDatum (H99E7)
ReflectionGroup(W) : GrpFPCox -> GrpMat, Map
ReflectionGroup(W) : Cat, GrpPermCox -> GrpMat, Map
Example GrpRfl_ReflectionGroupConversion (H99E8)
Construction of Finite Complex Reflection Groups
ShephardTodd(n) : RngIntElt -> GrpMat, Fld
Example GrpRfl_ComplexReflectionGroups (H99E9)
ComplexReflectionGroup(C) : Mtrx -> GrpMat, Map
ComplexReflectionGroup(X, n) : MonStgElt, RngIntElt -> GrpMat, Map
Example GrpRfl_reflection-subgroups (H99E10)
ShephardTodd(m, p, n) : RngIntElt, RngIntElt, RngIntElt -> GrpMat, Fld
Example GrpRfl_ImprimitiveReflectionGroup (H99E11)
ComplexRootMatrices(k) : RngIntElt -> AlgMatElt, AlgMatElt, AlgMatElt, RngElt, RngIntElt
Example GrpRfl_ComplexReflectionGroupByMatrix (H99E12)
ComplexCartanMatrix(k) : RngIntElt -> AlgMatElt
BasicRootMatrices(C) : Mtrx -> AlgMatElt, AlgMatElt
CohenCoxeterName(k) : RngIntElt -> MonStgElt, RngIntElt
ShephardToddNumber(X, n) : MonStgElt, RngIntElt -> RngIntElt
Example GrpRfl_NameConversion (H99E13)
Example GrpRfl_ReflectionGroupNames (H99E14)
ComplexRootDatum(k) : RngIntElt -> SeqEnum, SeqEnum, Map, GrpMat, AlgMatElt
Operations on Reflection Groups
IsCoxeterIsomorphic(W1, W2) : GrpMat, GrpMat -> BoolElt
IsCartanEquivalent(W1, W2) : GrpMat, GrpMat -> BoolElt
Example GrpRfl_Isomorphism (H99E15)
CartanName(W) : GrpMat -> List
CoxeterDiagram(W) : GrpMat ->
DynkinDiagram(W) : GrpMat ->
Example GrpRfl_NameAndDiagram (H99E16)
RootSystem(W) : GrpMat -> RootDtm
RootDatum(W) : GrpMat -> RootDtm
CoxeterMatrix(W) : GrpMat -> AlgMatElt
CoxeterGraph(W) : GrpMat -> GrphUnd
CartanMatrix(W) : GrpMat -> AlgMatElt
DynkinDigraph(W) : GrpMat -> GrphDir
Rank(W) : GrpMat -> RngIntElt
Example GrpRfl_RankDimension (H99E17)
FundamentalGroup(W) : GrpMat -> GrpAb
IsogenyGroup(W) : GrpMat -> GrpAb, Map
CoisogenyGroup(W) : GrpMat -> GrpAb, Map
BasicDegrees(W) : GrpMat -> RngIntElt
BasicCodegrees(W) : GrpMat -> RngIntElt
Example GrpRfl_BasicDegrees (H99E18)
LongestElement(W) : GrpMat -> SeqEnum
CoxeterElement(W) : GrpMat -> SeqEnum
CoxeterNumber(W) : GrpMat -> SeqEnum
Example GrpRfl_Operations (H99E19)
LeftDescentSet(W, w) : GrpMat, GrpMatElt ->()
RightDescentSet(W, w) : GrpMat, GrpMatElt ->()
Example GrpRfl_DescentSets (H99E20)
Properties of Reflection Groups
IsReflectionGroup(G) : GrpMat -> BoolElt
RootsAndCoroots(G) : GrpMat -> [RngIntElt], [ModTupRngElt], [ModTupRngElt]
IsRealReflectionGroup(G) : GrpMat -> BoolElt, [], []
Example GrpRfl_IsReflectionGroup (H99E21)
IsCrystallographic(W) : GrpMat -> BoolElt
IsSimplyLaced(W) : GrpMat -> BoolElt
Example GrpRfl_Properties (H99E22)
Dual(G) : GrpMat -> BoolElt
Overgroup(H) : GrpMat -> GrpMat
Overdatum(H) : GrpMat -> RootDtm
StandardAction(W) : GrpMat -> Map
StandardActionGroup(W) : GrpMat -> GrpPerm, Map
Roots, Coroots and Reflections
Accessing Roots and Coroots
RootSpace(W) : GrpMat -> Lat
Example GrpRfl_RootSpace (H99E23)
SimpleOrders(W) : GrpMat -> [RngIntElt]
SimpleRoots(W) : GrpMat -> Mtrx
NumberOfPositiveRoots(W) : GrpMat -> RngIntElt
Roots(W) : GrpMat -> (@@)
PositiveRoots(W) : GrpMat -> (@@)
Root(W, r) : GrpMat, RngIntElt -> (@@)
RootPosition(W, v) : GrpMat, . -> (@@)
Example GrpRfl_RootsCoroots (H99E24)
Reflections
ReflectionMatrices(W) : GrpMat -> [AlgMatElt]
SimpleReflectionMatrices(W) : GrpMat -> [AlgMatElt]
ReflectionMatrix(W, r) : GrpMat, RngIntElt -> AlgMatElt
SimpleReflectionPermutations(W) : GrpMat -> []
ReflectionPermutations(W) : GrpMat -> []
ReflectionPermutation(W, r) : GrpMat, RngIntElt -> []
ReflectionWords(W) : GrpMat -> []
ReflectionWord(W, r) : GrpMat, RngIntElt -> []
Example GrpRfl_Action (H99E25)
Length(w) : GrpMatElt -> RngIntElt
Weights
WeightLattice(W) : GrpMat -> Lat
FundamentalWeights(W) : GrpMat -> Mtrx
Example GrpRfl_Weights (H99E26)
IsDominant(R, v) : RootDtm, . -> ModTupFldElt, GrpFPCoxElt
DominantWeight(W, v) : GrpMat, . -> ModTupFldElt, GrpFPCoxElt
WeightOrbit(W, v) : GrpMat, . -> @ ModTupFldElt @, [GrpFPCoxElt]
Example GrpRfl_DominantWeights (H99E27)
Related Structures
CoxeterGroup(GrpFPCox, W) : Cat, GrpMat -> GrpPermCox
CoxeterGroup(GrpPermCox, W) : Cat, GrpMat -> GrpPermCox
LieAlgebra(W, R) : GrpMat, Rng -> AlgLie
GroupOfLieType(W, k) : GrpMat, Rng -> GrpLie
Bibliography
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Version: V2.19 of
Wed Apr 24 15:09:57 EST 2013