Introduction

 
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 (H109E1)
      Example GrpRfl_ref-group (H109E2)
      Example GrpRfl_transvections (H109E3)

      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 (H109E4)

 
Construction of Reflection Groups
      PseudoReflectionGroup(A, B) : Mtrx, Mtrx -> GrpMat, Map
      Example GrpRfl_ReflectionGroups (H109E5)

 
Construction of Real Reflection Groups
      ReflectionGroup(M) : AlgMatElt -> GrpMat
      ReflectionGroup(N) : MonStgElt -> GrpMat
      IrreducibleReflectionGroup(X, n) : MonStgElt, RngIntElt -> GrpMat
      Example GrpRfl_RealReflectionGroupByCartan (H109E6)
      ReflectionGroup(R) : RootSys -> GrpMat
      Example GrpRfl_RealReflectionGroupByRootDatum (H109E7)
      ReflectionGroup(W) : GrpFPCox -> GrpMat, Map
      ReflectionGroup(W) : GrpPermCox -> GrpMat, Map
      Example GrpRfl_ReflectionGroupConversion (H109E8)

 
Construction of Finite Complex Reflection Groups
      ShephardTodd(n) : RngIntElt -> GrpMat, Fld
      Example GrpRfl_ComplexReflectionGroups (H109E9)
      ComplexReflectionGroup(C) : Mtrx -> GrpMat, Map
      ComplexReflectionGroup(X, n) : MonStgElt, RngIntElt -> GrpMat, Map
      Example GrpRfl_reflection-subgroups (H109E10)
      ShephardTodd(m, p, n) : RngIntElt, RngIntElt, RngIntElt -> GrpMat, Fld
      Example GrpRfl_ImprimitiveReflectionGroup (H109E11)
      ComplexRootMatrices(k) : RngIntElt -> AlgMatElt, AlgMatElt, AlgMatElt, RngElt, RngIntElt
      Example GrpRfl_ComplexReflectionGroupByMatrix (H109E12)
      ComplexCartanMatrix(k) : RngIntElt -> AlgMatElt
      BasicRootMatrices(C) : Mtrx -> AlgMatElt, AlgMatElt
      CohenCoxeterName(k) : RngIntElt -> MonStgElt, RngIntElt
      ShephardToddNumber(X, n) : MonStgElt, RngIntElt -> RngIntElt
      Example GrpRfl_NameConversion (H109E13)
      Example GrpRfl_ReflectionGroupNames (H109E14)
      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 (H109E15)
      CartanName(W) : GrpMat -> List
      CoxeterDiagram(W) : GrpMat ->
      DynkinDiagram(W) : GrpMat ->
      Example GrpRfl_NameAndDiagram (H109E16)
      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 (H109E17)
      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 (H109E18)
      LongestElement(W) : GrpMat -> SeqEnum
      CoxeterElement(W) : GrpMat -> SeqEnum
      CoxeterNumber(W) : GrpMat -> SeqEnum
      Example GrpRfl_Operations (H109E19)
      LeftDescentSet(W, w) : GrpMat, GrpMatElt ->()
      RightDescentSet(W, w) : GrpMat, GrpMatElt ->()
      Example GrpRfl_DescentSets (H109E20)

 
Properties of Reflection Groups
      IsReflectionGroup(G) : GrpMat -> BoolElt
      RootsAndCoroots(G) : GrpMat -> [RngIntElt], [ModTupRngElt], [ModTupRngElt]
      IsRealReflectionGroup(G) : GrpMat -> BoolElt, [], []
      Example GrpRfl_IsReflectionGroup (H109E21)
      IsCrystallographic(W) : GrpMat -> BoolElt
      IsSimplyLaced(W) : GrpMat -> BoolElt
      Example GrpRfl_Properties (H109E22)
      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 (H109E23)
            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 (H109E24)

      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 (H109E25)
            Length(w) : GrpMatElt -> RngIntElt

      Weights
            WeightLattice(W) : GrpMat -> Lat
            FundamentalWeights(W) : GrpMat -> Mtrx
            Example GrpRfl_Weights (H109E26)
            IsDominant(R, v) : RootDtm, . -> ModTupFldElt, GrpFPCoxElt
            DominantWeight(W, v) : GrpMat, . -> ModTupFldElt, GrpFPCoxElt
            WeightOrbit(W, v) : GrpMat, . -> @ ModTupFldElt @, [GrpFPCoxElt]
            Example GrpRfl_DominantWeights (H109E27)

 
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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