MAGMA Computational Algebra System

REFLECTION GROUPS

DETAILS

 
Introduction

 
Construction of Reflections and Pseudoreflections
      Reflection(r, c) : ModTupRngElt, ModTupRngElt -> AlgMatElt
      IsReflection(R) : Mtrx -> BoolElt, ModTupRngElt, ModTupRngElt
      Example GrpRfl_Reflections (H75E1)
      Pseudoreflection(r, c, o) : ModTupRngElt, ModTupRngElt, RngIntElt -> AlgMatElt
      IsPseudoreflection(R) : Mtrx -> BoolElt, ModTupRngElt, ModTupRngElt, RngIntElt
      Example GrpRfl_Pseudoreflections (H75E2)

 
Construction of Reflection Groups
      IsReflectionGroup(G) : GrpMat -> BoolElt, [RngIntElt], Mtrx, Mtrx
      ReflectionGroup(A, B, m) : Mtrx, Mtrx, [RngIntElt] -> GrpMat
      ReflectionGroup(A, B, m) : [ModTupRngElt], [ModTupRngElt], [RngIntElt] -> GrpMat
      ReflectionGroup(A, B) : Mtrx, Mtrx -> GrpMat
      ReflectionGroup(A, B) : [ModTupRngElt], [ModTupRngElt] -> GrpMat
      Example GrpRfl_ReflectionGroups (H75E3)

 
Construction of Real Reflection Groups
      ReflectionGroup(M) : AlgMatElt -> GrpMat
      ReflectionGroup(N) : MonStgElt -> GrpMat
      IrreducibleReflectionGroup(X, n) : MonStgElt, RngIntElt -> GrpMat
      Example GrpRfl_RealReflectionGroupByCartan (H75E4)
      ReflectionGroup(R) : RootSys -> GrpMat
      Example GrpRfl_RealReflectionGroupByRootDatum (H75E5)
      ReflectionGroup(W) : GrpFPCox -> GrpMat, Map
      ReflectionGroup(W) : Cat, GrpPermCox -> GrpMat, Map
      Example GrpRfl_ReflectionGroupConversion (H75E6)

 
Construction of Finite Complex Reflection Groups
      ImprimitiveReflectionGroup(m, p, n) : RngIntElt, RngIntElt, RngIntElt -> GrpMat, Fld
      Example GrpRfl_ImprimitiveReflectionGroup (H75E7)
      RootSystemMatrix(X, n) : MonStgElt, RngIntElt -> AlgMatElt
      ReflectionGroup(M) : AlgMatElt -> GrpMat, Fld
      Example GrpRfl_ComplexReflectionGroupByMatrix (H75E8)
      ComplexReflectionGroup(X, n) : MonStgElt, RngIntElt -> AlgMatElt
      ShephardTodd(n) : RngIntElt -> GrpMat, Fld
      Example GrpRfl_ComplexReflectionGroups (H75E9)

 
Operations on Reflection Groups
      IsCoxeterIsomorphic(W1, W2) : GrpMat, GrpMat -> BoolElt
      IsCartanEquivalent(W1, W2) : GrpMat, GrpMat -> BoolElt
      Example GrpRfl_Isomorphism (H75E10)
      CartanName(W) : GrpMat -> List
      CoxeterDiagram(W) : GrpMat ->
      DynkinDiagram(W) : GrpMat ->
      Example GrpRfl_NameAndDiagram (H75E11)
      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 (H75E12)
      FundamentalGroup(W) : GrpMat -> GrpAb
      IsogenyGroup(W) : GrpMat -> GrpAb, Map
      CoisogenyGroup(W) : GrpMat -> GrpAb, Map
      BasicDegrees(W) : GrpMat -> RngIntElt
      Example GrpRfl_BasicDegrees (H75E13)
      LongestElement(W) : GrpMat -> SeqEnum
      CoxeterElement(W) : GrpMat -> SeqEnum
      CoxeterNumber(W) : GrpMat -> SeqEnum
      Example GrpRfl_Operations (H75E14)
      LeftDescentSet(W, w) : GrpMat, GrpMatElt ->
      RightDescentSet(W, w) : GrpMat, GrpMatElt ->
      Example GrpRfl_DescentSets (H75E15)

 
Properties of Reflection Groups
      IsReflectionGroup(G) : GrpMat -> BoolElt, [RngIntElt], [ModTupRngElt], [ModTupRngElt]
      IsRealReflectionGroup(G) : GrpMat -> BoolElt, [], []
      Example GrpRfl_IsReflectionGroup (H75E16)
      IsCrystallographic(W) : GrpMat -> BoolElt
      IsSimplyLaced(W) : GrpMat -> BoolElt
      Example GrpRfl_Properties (H75E17)
      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 (H75E18)
            SimpleOrders(W) : GrpMat -> [RngIntElt]
            SimpleRoots(W) : GrpMat -> Mtrx
            Example GrpRfl_RootSpace (H75E19)
            NumberOfPositiveRoots(W) : GrpMat -> RngIntElt
            Roots(W) : GrpMat -> {@@}
            PositiveRoots(W) : GrpMat -> {@@}
            Root(W, r) : GrpMat, RngIntElt -> {@@}
            RootPosition(W, v) : GrpMat, . -> {@@}
            Example GrpRfl_RootsCoroots (H75E20)

      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 (H75E21)
            Length(w) : GrpMatElt -> RngIntElt

      Weights
            WeightLattice(W) : GrpMat -> Lat
            FundamentalWeights(W) : GrpMat -> Mtrx
            Example GrpRfl_Weights (H75E22)
            IsDominant(R, v) : RootDtm, . -> ModTupFldElt, GrpFPCoxElt
            DominantWeight(W, v) : GrpMat, . -> ModTupFldElt, GrpFPCoxElt
            WeightOrbit(W, v) : GrpMat, . -> @ ModTupFldElt @, [GrpFPCoxElt]
            Example GrpRfl_DominantWeights (H75E23)

 
Related Structures
      CoxeterGroup(GrpFPCox, W) : Cat, GrpMat -> GrpPermCox
      CoxeterGroup(GrpPermCox, W) : Cat, GrpMat -> GrpPermCox
      LieAlgebra(W, R) : GrpMat, Rng -> AlgLie

 
Bibliography

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