Introduction

      Reflections

      Definition of a Root System

      Simple and Positive Roots

      The Coxeter Group

      Nonreduced Root Systems

 
Constructing Root Systems
      RootSystem(N) : MonStgElt -> RootSys
      Example RootSys_CreatingRootSystemsName (H106E1)
      RootSystem(M) : AlgMatElt -> RootSys
      RootSystem(C) : AlgMatElt -> RootSys
      RootSystem(D) : GrphDir -> RootSys
      Example RootSys_CreatingRootSystemsMatrix (H106E2)
      RootSystem(A, B) : Mtrx, Mtrx -> RootSys
      Example RootSys_G2RootSystem (H106E3)
      IrreducibleRootSystem(X, n) : MonStgElt, RngIntElt -> RootSys
      StandardRootSystem(X, n) : MonStgElt, RngIntElt -> RootSys
      Example RootSys_IrreducibleRootSystem (H106E4)
      ToralRootSystem(n) : RngIntElt -> RootSys
      TrivialRootSystem() : -> RootSys

 
Operators on Root Systems
      R1 eq R2 : RootSys, RootSys -> BoolElt
      IsIsomorphic(R1, R2) : RootSys, RootSys -> BoolElt
      IsCartanEquivalent(R1, R2) : RootSys, RootSys -> BoolElt
      Example RootSys_Isomorphism (H106E5)
      CartanName(R) : RootSys -> List
      CoxeterDiagram(R) : RootSys ->
      DynkinDiagram(R) : RootSys ->
      CoxeterMatrix(R) : RootSys -> AlgMatElt
      CoxeterGraph(R) : RootSys -> GrphUnd
      CartanMatrix(R) : RootSys -> AlgMatElt
      DynkinDigraph(R) : RootSys -> GrphDir
      Example RootSys_Diagrams (H106E6)
      BaseField(R) : RootSys -> Fld
      RealInjection(R) : RootSys -> .
      Rank(R) : RootSys -> RngIntElt
      Dimension(R) : RootSys -> RngIntElt
      CoxeterGroupOrder(R) : RootSys -> RngIntElt
      Example RootSys_BasicOperations (H106E7)

 
Properties of Root Systems
      IsIrreducible(R) : RootSys -> BoolElt
      IsProjectivelyIrreducible(R) : RootSys -> BoolElt
      IsReduced(R) : RootSys -> BoolElt
      IsSemisimple(R) : RootSys-> BoolElt
      IsCrystallographic(R) : RootSys -> BoolElt
      IsSimplyLaced(R) : RootSys-> BoolElt
      Example RootSys_Properties (H106E8)

 
Roots and Coroots

      Accessing Roots and Coroots
            RootSpace(R) : RootSys -> ModTupFld
            SimpleRoots(R) : RootSys -> Mtrx
            Example RootSys_RootSpace (H106E9)
            NumberOfPositiveRoots(R) : RootSys -> RngIntElt
            Roots(R) : RootSys -> (@@)
            PositiveRoots(R) : RootSys -> (@@)
            Root(R, r) : RootSys, RngIntElt -> (@@)
            RootPosition(R, v) : RootSys, . -> (@@)
            Example RootSys_RootsCoroots (H106E10)
            HighestRoot(R) : RootSys -> .
            HighestLongRoot(R) : RootSys -> .
            HighestShortRoot(R) : RootSys -> .
            Example RootSys_HeighestRoots (H106E11)
            CoxeterForm(R) : RootSys -> AlgMatElt

      Reflections
            SimpleReflectionMatrices(R) : RootSys -> []
            ReflectionMatrices(R) : RootSys -> []
            ReflectionMatrix(R, r) : RootSys, RngIntElt -> []
            SimpleReflectionPermutations(R) : RootSys -> []
            ReflectionPermutations(R) : RootSys -> []
            ReflectionPermutation(R, r) : RootSys, RngIntElt -> []
            ReflectionWords(R) : RootSys -> []
            ReflectionWord(R, r) : RootSys, RngIntElt -> []
            Example RootSys_Action (H106E12)

      Operations and Properties for Roots and Coroot Indices
            Sum(R, r, s) : RootSys, RngIntElt, RngIntElt -> RngIntElt
            IsPositive(R, r) : RootSys, RngIntElt -> BoolElt
            IsNegative(R, r) : RootSys, RngIntElt -> BoolElt
            Negative(R, r) : RootSys, RngIntElt -> RngIntElt
            Example RootSys_RootArithmetic (H106E13)
            RootHeight(R, r) : RootSys, RngIntElt -> RngIntElt
            RootNorms(R) : RootSys -> [RngIntElt]
            RootNorm(R, r) : RootSys, RngIntElt -> RngIntElt
            IsLongRoot(R, r) : RootSys, RngIntElt -> BoolElt
            IsShortRoot(R, r) : RootSys, RngIntElt -> BoolElt
            IsIndivisibleRoot(R, r) : RootSys, RngIntElt -> BoolElt
            LeftString(R, r, s) : RootSys, RngIntElt, RngIntElt -> RngIntElt
            RightString(R, r, s) : RootSys, RngIntElt, RngIntElt -> RngIntElt
            LeftStringLength(R, r, s) : RootSys, RngIntElt, RngIntElt -> RngIntElt
            RightStringLength(R, r, s) : RootSys, RngIntElt, RngIntElt -> RngIntElt
            Example RootSys_RootOperations (H106E14)
            AdditiveOrder(R) : RootSys -> SeqEnum
            IsAdditiveOrder(R, Q) : RootSys, [RngIntElt] -> BoolElt
            Example RootSys_AdditiveOrder (H106E15)

 
Building Root Systems
      sub<R | a> : RootSys, SetEnum -> RootSys
      sub<R | s> : RootSys, SetEnum -> RootSys
      R1 subset R2 : RootSys, RootSys -> BoolElt, .
      R1 + R2 : RootSys, RootSys -> RootSys
      R1 join R2 : RootSys, RootSys -> RootSys
      Example RootSys_RootSysSums (H106E16)
      DirectSumDecomposition(R) : RootSys -> []
      Dual(R) : RootSys -> RootSys
      IndivisibleSubsystem(R) : RootSys -> RootSys
      Example RootSys_DirectSumDual (H106E17)

 
Related Structures
      RootDatum(R) : RootSys -> RootDtm
      CoxeterGroup(grpcat,R) : Cat, RootSys -> grpcat
      CoxeterGroup(R) : RootSys -> GrpPermCox
      CoxeterGroup(GrpPermCox, R) : Cat, RootSys -> RngIntElt
      CoxeterGroup(GrpPermCox, W) : Cat, RootSys -> GrpPermCox
      MatrixLieAlgebra(R, k) : RootSys -> GrpMat
      Example RootSys_Related (H106E18)

 
Bibliography

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