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ROOT SYSTEMS

 
Acknowledgements
 
Introduction
      Reflections
      Definition of a Root System
      Simple and Positive Roots
      The Coxeter Group
      Nonreduced Root Systems
 
Constructing Root Systems
 
Operators on Root Systems
 
Properties of Root Systems
 
Roots and Coroots
      Accessing Roots and Coroots
      Reflections
      Operations and Properties for Roots and Coroot Indices
 
Building Root Systems
 
Related Structures
 
Bibliography







DETAILS

 
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 (H96E1)
      RootSystem(M) : AlgMatElt -> RootSys
      RootSystem(C) : AlgMatElt -> RootSys
      RootSystem(D) : GrphDir -> RootSys
      Example RootSys_CreatingRootSystemsMatrix (H96E2)
      RootSystem(A, B) : Mtrx, Mtrx -> RootSys
      Example RootSys_G2RootSystem (H96E3)
      IrreducibleRootSystem(X, n) : MonStgElt, RngIntElt -> RootSys
      StandardRootSystem(X, n) : MonStgElt, RngIntElt -> RootSys
      Example RootSys_IrreducibleRootSystem (H96E4)
      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 (H96E5)
      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 (H96E6)
      BaseField(R) : RootSys -> Fld
      RealInjection(R) : RootSys -> .
      Rank(R) : RootSys -> RngIntElt
      Dimension(R) : RootSys -> RngIntElt
      CoxeterGroupOrder(R) : RootSys -> RngIntElt
      Example RootSys_BasicOperations (H96E7)

 
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 (H96E8)

 
Roots and Coroots

      Accessing Roots and Coroots
            RootSpace(R) : RootSys -> ModTupFld
            SimpleRoots(R) : RootSys -> Mtrx
            Example RootSys_RootSpace (H96E9)
            NumberOfPositiveRoots(R) : RootSys -> RngIntElt
            Roots(R) : RootSys -> (@@)
            PositiveRoots(R) : RootSys -> (@@)
            Root(R, r) : RootSys, RngIntElt -> (@@)
            RootPosition(R, v) : RootSys, . -> (@@)
            Example RootSys_RootsCoroots (H96E10)
            HighestRoot(R) : RootSys -> .
            HighestLongRoot(R) : RootSys -> .
            HighestShortRoot(R) : RootSys -> .
            Example RootSys_HeighestRoots (H96E11)
            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 (H96E12)

      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 (H96E13)
            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 (H96E14)
            AdditiveOrder(R) : RootSys -> SeqEnum
            IsAdditiveOrder(R, Q) : RootSys, [RngIntElt] -> BoolElt
            Example RootSys_AdditiveOrder (H96E15)

 
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 (H96E16)
      DirectSumDecomposition(R) : RootSys -> []
      Dual(R) : RootSys -> RootSys
      IndivisibleSubsystem(R) : RootSys -> RootSys
      Example RootSys_DirectSumDual (H96E17)

 
Related Structures
      RootDatum(R) : RootSys -> RootDtm
      CoxeterGroup(GrpFPCox, R) : Cat, RootSys -> RngIntElt
      CoxeterGroup(R) : RootSys -> RngIntElt
      ReflectionGroup(R) : RootSys -> GrpMat
      LieAlgebra(R, k) : RootSys -> GrpMat
      MatrixLieAlgebra(R, k) : RootSys -> GrpMat
      Example RootSys_Related (H96E18)

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012