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)
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