Reflection Groups
- Introduction
- Construction of Pseudo-reflections
PseudoReflection(a, b): ModTupRngElt, ModTupRngElt → AlgMatEltTransvection(a, b): ModTupRngElt, ModTupRngElt → AlgMatEltReflection(a, b): ModTupRngElt, ModTupRngElt → AlgMatEltIsPseudoReflection(r): Mtrx → BoolElt, ModTupRngElt, ModTupRngEltIsTransvection(r): Mtrx → BoolElt, ModTupRngElt, ModTupRngEltIsReflection(r): Mtrx → BoolElt, ModTupRngElt, ModTupRngEltIsReflectionGroup(G): GrpMat → BoolEltExample: pseudoreflectionExample: Ref GroupExample: transvections- Pseudo-reflections Preserving Reflexive Forms
- Construction of Reflection Groups
- Construction of Real Reflection Groups
- Construction of Finite Complex Reflection Groups
ShephardTodd(n): RngIntElt → GrpMat, FldExample: Complex Reflection GroupsComplexReflectionGroup(C): Mtrx → GrpMat, MapComplexReflectionGroup(X, n): MonStgElt, RngIntElt → GrpMat, MapExample: Reflection SubgroupsShephardTodd(m, p, n): RngIntElt, RngIntElt, RngIntElt → GrpMat, FldImprimitiveReflectionGroup(m, p, n): RngIntElt, RngIntElt, RngIntElt → GrpMat, FldExample: Imprimitive Reflection GroupComplexRootMatrices(k): RngIntElt → AlgMatElt, AlgMatElt, AlgMatElt, RngElt, RngIntEltComplexRootMatrices(m, p, n): RngIntElt, RngIntElt, RngIntElt → AlgMatElt, AlgMatElt, AlgMatElt, RngElt, RngIntEltExample: Complex Reflection Group By MatrixComplexCartanMatrix(k): RngIntElt → AlgMatEltComplexCartanMatrix(m, p, n): RngIntElt, RngIntElt, RngIntElt → AlgMatEltBasicRootMatrices(C): Mtrx → AlgMatElt, AlgMatEltCohenCoxeterName(k): RngIntElt → MonStgElt, RngIntEltShephardToddNumber(X, n): MonStgElt, RngIntElt → RngIntEltExample: Name ConversionExample: Reflection Group NamesComplexRootDatum(k): RngIntElt → SeqEnum, SeqEnum, Map, GrpMat, AlgMatEltComplexRootDatum(m, p, n): RngIntElt, RngIntElt, RngIntElt → SeqEnum, SeqEnum, Map, GrpMat, AlgMatElt
- Operations on Reflection Groups
IsCoxeterIsomorphic(W1, W2): GrpMat, GrpMat → BoolEltIsCartanEquivalent(W1, W2): GrpMat, GrpMat → BoolEltExample: IsomorphismCartanName(W): GrpMat → ListCoxeterDiagram(W): GrpMatDynkinDiagram(W): GrpMatExample: Name And DiagramRootSystem(W): GrpMat → RootDtmRootDatum(W): GrpMat → RootDtmCoxeterMatrix(W): GrpMat → AlgMatEltCoxeterGraph(W): GrpMat → GrphUndCartanMatrix(W): GrpMat → AlgMatEltDynkinDigraph(W): GrpMat → GrphDirRank(W): GrpMat → RngIntEltNumberOfGenerators(W): GrpMat → RngIntEltExample: Rank DimensionFundamentalGroup(W): GrpMat → GrpAbIsogenyGroup(W): GrpMat → GrpAb, MapCoisogenyGroup(W): GrpMat → GrpAb, MapBasicDegrees(W): GrpMat → RngIntEltBasicCodegrees(W): GrpMat → RngIntEltExample: Basic DegreesLongestElement(W): GrpMat → SeqEnumCoxeterElement(W): GrpMat → SeqEnumCoxeterNumber(W): GrpMat → SeqEnumExample: OperationsLeftDescentSet(W, w): GrpMat, GrpMatElt → SetEnumRightDescentSet(W, w): GrpMat, GrpMatElt → SetEnumExample: Descent Sets
- Properties of Reflection Groups
- Roots, Coroots and Reflections
- Related Structures
