Root Systems
- Introduction
- Constructing Root Systems
RootSystem(N): MonStgElt → RootSys- Example: Root Systems from a Name
RootSystem(M): AlgMatElt → RootSysRootSystem(G): GrphUnd → RootSysRootSystem(C): AlgMatElt → RootSysRootSystem(D): GrphDir → RootSys- Example: Root Systems from a Matrix
RootSystem(A, B): Mtrx, Mtrx → RootSys- Example: Nonsemisimple Root System of Type G2
IrreducibleRootSystem(X, n): MonStgElt, RngIntElt → RootSysStandardRootSystem(X, n): MonStgElt, RngIntElt → RootSys- Example: Irreducible Dihedral Root Systems
ToralRootSystem(n): RngIntElt → RootSysTrivialRootSystem() → RootSys
- Operators on Root Systems
R1 eq R2: RootSys, RootSys → BoolEltIsIsomorphic(R1, R2): RootSys, RootSys → BoolEltIsCartanEquivalent(R1, R2): RootSys, RootSys → BoolElt- Example: Isomorphism and Cartan Equivalence
CartanName(R): RootSys → ListCoxeterDiagram(R): RootSysDynkinDiagram(R): RootSysCoxeterMatrix(R): RootSys → AlgMatEltCoxeterGraph(R): RootSys → GrphUndCartanMatrix(R): RootSys → AlgMatEltDynkinDigraph(R): RootSys → GrphDir- Example: Dynkin and Coxeter Diagrams
BaseField(R): RootSys → FldBaseRing(R): RootSys → FldRealInjection(R): RootSys → .Rank(R): RootSys → RngIntEltDimension(R): RootSys → RngIntEltCoxeterGroupOrder(R): RootSys → RngIntElt- Example: Base Field, Rank, and Coxeter Group Order
- Properties of Root Systems
- Roots and Coroots
- Accessing Roots and Coroots
RootSpace(R): RootSys → ModTupFldCorootSpace(R): RootSys → ModTupFldSimpleRoots(R): RootSys → MtrxSimpleCoroots(R): RootSys → Mtrx- Example: Root and Coroot Spaces
NumberOfPositiveRoots(R): RootSys → RngIntEltNumPosRoots(R): RootSys → RngIntEltRoots(R): RootSys → SetIndxCoroots(R): RootSys → SetIndxPositiveRoots(R): RootSys → SetIndxPositiveCoroots(R): RootSys → SetIndxRoot(R, r): RootSys, RngIntElt → SetIndxCoroot(R, r): RootSys, RngIntElt → SetIndxRootPosition(R, v): RootSys, . → SetIndxCorootPosition(R, v): RootSys, . → SetIndx- Example: Roots and Coroots of a Root System
HighestRoot(R): RootSys → .HighestCoroot(R): RootSys → .HighestLongRoot(R): RootSys → .HighestLongCoroot(R): RootSys → .HighestShortRoot(R): RootSys → .HighestShortCoroot(R): RootSys → .- Example: Highest Roots
CoxeterForm(R): RootSys → AlgMatEltDualCoxeterForm(R): RootSys → AlgMatElt
- Reflections
SimpleReflectionMatrices(R): RootSys → []SimpleCoreflectionMatrices(R): RootSys → []ReflectionMatrices(R): RootSys → []CoreflectionMatrices(R): RootSys → []ReflectionMatrix(R, r): RootSys, RngIntElt → []CoreflectionMatrix(R, r): RootSys, RngIntElt → []SimpleReflectionPermutations(R): RootSys → []ReflectionPermutations(R): RootSys → []ReflectionPermutation(R, r): RootSys, RngIntElt → []ReflectionWords(R): RootSys → []ReflectionWord(R, r): RootSys, RngIntElt → []- Example: Reflection Matrices, Permutations, and Words
- Operations and Properties for Roots and Coroot Indices
Sum(R, r, s): RootSys, RngIntElt, RngIntElt → RngIntEltIsPositive(R, r): RootSys, RngIntElt → BoolEltIsNegative(R, r): RootSys, RngIntElt → BoolEltNegative(R, r): RootSys, RngIntElt → RngIntElt- Example: Arithmetic with Root Indices
RootHeight(R, r): RootSys, RngIntElt → RngIntEltCorootHeight(R, r): RootSys, RngIntElt → RngIntEltRootNorms(R): RootSys → [RngIntElt]CorootNorms(R): RootSys → [RngIntElt]RootNorm(R, r): RootSys, RngIntElt → RngIntEltCorootNorm(R, r): RootSys, RngIntElt → RngIntEltIsLongRoot(R, r): RootSys, RngIntElt → BoolEltIsShortRoot(R, r): RootSys, RngIntElt → BoolEltIsIndivisibleRoot(R, r): RootSys, RngIntElt → BoolEltLeftString(R, r, s): RootSys, RngIntElt, RngIntElt → RngIntEltRightString(R, r, s): RootSys, RngIntElt, RngIntElt → RngIntEltLeftStringLength(R, r, s): RootSys, RngIntElt, RngIntElt → RngIntEltRightStringLength(R, r, s): RootSys, RngIntElt, RngIntElt → RngIntElt- Example: Heights, Norms, and Root Strings
AdditiveOrder(R): RootSys → SeqEnumIsAdditiveOrder(R, Q): RootSys, [RngIntElt] → BoolElt- Example: Additive Order on Positive Roots
- Accessing Roots and Coroots
- Building Root Systems
sub<R | a>: RootSys, SetEnum → RootSyssub<R | s>: RootSys, SetEnum → RootSysR1 subset R2: RootSys, RootSys → BoolElt, .R1 + R2: RootSys, RootSys → RootSysDirectSum(R1, R2): RootSys, RootSys → RootSysR1 join R2: RootSys, RootSys → RootSys- Example: Sums of Root Systems
DirectSumDecomposition(R): RootSys → []IndecomposableSummands(R): RootDtm → [], RootDtm, MapDual(R): RootSys → RootSysIndivisibleSubsystem(R): RootSys → RootSys- Example: Direct Sums and Duals of Root Systems
- Related Structures
RootDatum(R): RootSys → RootDtmCoxeterGroup(grpcat,R): Cat, RootSys → grpcatCoxeterGroup(R): RootSys → GrpPermCoxWeylGroup(R): RootSys → GrpPermCoxCoxeterGroup(GrpPermCox, R): Cat, RootSys → RngIntEltReflectionGroup(R): RootSys → GrpMatCoxeterGroup(GrpPermCox, W): Cat, RootSys → GrpPermCoxLieAlgebra(R, k): RootSys, Rng → AlgLieMatrixLieAlgebra(R, k): RootSys → GrpMat- Example: Structures Related to a Root System
- Bibliography