Coxeter Systems
- Introduction
- Coxeter Matrices
- Coxeter Graphs
- Cartan Matrices
- Dynkin Digraphs
- Finite and Affine Coxeter Groups
IsCoxeterFinite(M): AlgMatElt → BoolEltIsCoxeterFinite(G): GrphUnd → BoolEltIsCoxeterFinite(C): AlgMatElt → BoolEltIsCoxeterFinite(D): GrphDir → BoolEltIsCoxeterFinite(N): MonStgElt → BoolEltIsCoxeterAffine(M): AlgMatElt → BoolEltIsCoxeterAffine(G): GrphUnd → BoolEltIsCoxeterAffine(C): AlgMatElt → BoolEltIsCoxeterAffine(D): GrphDir → BoolEltIsCoxeterAffine(N): MonStgElt → BoolElt- Example: Testing Finite and Affine Types
CoxeterMatrix(N): MonStgElt → AlgMatEltCoxeterGraph(N): MonStgElt → GrpUndCartanMatrix(N): MonStgElt → AlgMatEltDynkinDigraph(N): MonStgElt → GrphDir- Example: Matrices and Graphs from Cartan Names
IrreducibleCoxeterMatrix(X, n): MonStgElt, RngIntElt → AlgMatEltIrreducibleCoxeterGraph(X, n): MonStgElt, RngIntElt → GrpUndIrreducibleCartanMatrix(X, n): MonStgElt, RngIntElt → AlgMatEltIrreducibleDynkinDigraph(X, n): MonStgElt, RngIntElt → GrphDir- Example: Irreducible Coxeter Graphs
IsCoxeterIsomorphic(N1, N2): MonStgElt, MonStgElt → BoolEltIsCartanEquivalent(N1, N2): MonStgElt, MonStgElt → BoolElt- Example: Isomorphism and Equivalence
IsSimplyLaced(N): MonStgElt → BoolEltCoxeterGroupOrder(N): MonStgElt → .CoxeterGroupFactoredOrder(N): MonStgElt → .NumberOfPositiveRoots(N): MonStgElt → .NumPosRoots(N): MonStgElt → .FundamentalGroup(N): MonStgElt → GrpAb- Example: Coxeter Group Orders
CartanName(M): AlgMatElt → MonStgEltCartanName(G): GrphUnd → MonStgEltCartanName(C): AlgMatElt → MonStgEltCartanName(D): GrphDir → MonStgElt- Example: Cartan Name of a Coxeter Matrix
DynkinDiagram(M): AlgMatEltDynkinDiagram(G): GrphUndDynkinDiagram(C): AlgMatEltDynkinDiagram(D): GrphDirDynkinDiagram(N): MonStgElt- Example: Dynkin Diagram of a Reducible Type
CoxeterDiagram(M): AlgMatEltCoxeterDiagram(G): GrphUndCoxeterDiagram(C): AlgMatEltCoxeterDiagram(D): GrphDirCoxeterDiagram(N): MonStgElt- Example: Coxeter Diagram of a Reducible Type
- Hyperbolic Groups
- Related Structures
RootSystem(M): AlgMatElt → RootSysRootSystem(G): GrphUnd → RootSysRootSystem(C): AlgMatElt → RootSysRootSystem(D): GrphDir → RootSysRootSystem(N): MonStgElt → RootSysRootDatum(C): AlgMatElt → RootDtmRootDatum(M): AlgMatElt → RootDtmRootDatum(G): GrphUnd → RootDtmRootDatum(D): GrphDir → RootDtmRootDatum(N): MonStgElt → RootDtmCoxeterGroup(grpcat, M): Cat, AlgMatElt → grpcatCoxeterGroup(grpcat, G): Cat, GrphUnd → grpcatCoxeterGroup(grpcat, D): Cat, GrphDir → grpcatCoxeterGroup(grpcat, N): Cat, MonStgElt → grpcatCoxeterGroup(M): AlgMatElt → GrpCoxeterGroup(C): AlgMatElt → GrpCoxeterGroup(G): GrphUnd → GrpCoxeterGroup(D): GrphDir → GrpCoxeterGroup(N): MonStgElt → GrpReflectionGroup(M): AlgMatElt → GrpMatReflectionGroup(G): GrphUnd → GrpMatReflectionGroup(C): AlgMatElt → GrpMatReflectionGroup(D): GrphDir → GrpMatReflectionGroup(N): MonStgElt → GrpMatLieAlgebra(C, k): AlgMatElt, Rng → AlgLieLieAlgebra(D, k): GrphDir, Rng → AlgLieLieAlgebra(N, k): MonStgElt, Rng → AlgLieMatrixLieAlgebra(C, k): AlgMatElt, Rng → AlgLieMatrixLieAlgebra(D, k): GrphDir, Rng → AlgLieMatrixLieAlgebra(N, k): MonStgElt, Rng → AlgLieGroupOfLieType(C, k): AlgMatElt, Rng → GrpLieGroupOfLieType(D, k): GrphDir, Rng → GrpLieGroupOfLieType(N, k): MonStgElt, Rng → GrpLie
- Bibliography