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 → BoolElt- Example: Creating a Pseudo-reflection
- Example: Group Generated by Reflections
- Example: Special Linear Groups Generated by 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, Fld- Example: Identifying the Complex Reflection Group G24
ComplexReflectionGroup(C): Mtrx → GrpMat, MapComplexReflectionGroup(X, n): MonStgElt, RngIntElt → GrpMat, Map- Example: Reflection Subgroups
ShephardTodd(m, p, n): RngIntElt, RngIntElt, RngIntElt → GrpMat, FldImprimitiveReflectionGroup(m, p, n): RngIntElt, RngIntElt, RngIntElt → GrpMat, Fld- Example: Imprimitive Reflection Groups
ComplexRootMatrices(k): RngIntElt → AlgMatElt, AlgMatElt, AlgMatElt, RngElt, RngIntEltComplexRootMatrices(m, p, n): RngIntElt, RngIntElt, RngIntElt → AlgMatElt, AlgMatElt, AlgMatElt, RngElt, RngIntElt- Example: Complex Reflection Group from Root Matrices
ComplexCartanMatrix(k): RngIntElt → AlgMatEltComplexCartanMatrix(m, p, n): RngIntElt, RngIntElt, RngIntElt → AlgMatEltBasicRootMatrices(C): Mtrx → AlgMatElt, AlgMatEltCohenCoxeterName(k): RngIntElt → MonStgElt, RngIntEltShephardToddNumber(X, n): MonStgElt, RngIntElt → RngIntElt- Example: Converting Reflection Group Names
- Example: Constructing a Reflection Group from Its Name
ComplexRootDatum(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 → BoolElt- Example: Coxeter Isomorphism and Cartan Equivalence
CartanName(W): GrpMat → ListCoxeterDiagram(W): GrpMatDynkinDiagram(W): GrpMat- Example: Cartan Name and Coxeter Diagram
RootSystem(W): GrpMat → RootDtmRootDatum(W): GrpMat → RootDtmCoxeterMatrix(W): GrpMat → AlgMatEltCoxeterGraph(W): GrpMat → GrphUndCartanMatrix(W): GrpMat → AlgMatEltDynkinDigraph(W): GrpMat → GrphDirRank(W): GrpMat → RngIntEltNumberOfGenerators(W): GrpMat → RngIntElt- Example: Rank and Dimension
FundamentalGroup(W): GrpMat → GrpAbIsogenyGroup(W): GrpMat → GrpAb, MapCoisogenyGroup(W): GrpMat → GrpAb, MapBasicDegrees(W): GrpMat → RngIntEltBasicCodegrees(W): GrpMat → RngIntElt- Example: Basic Degrees
LongestElement(W): GrpMat → SeqEnumCoxeterElement(W): GrpMat → SeqEnumCoxeterNumber(W): GrpMat → SeqEnum- Example: Longest and Coxeter Elements
LeftDescentSet(W, w): GrpMat, GrpMatElt → SetEnumRightDescentSet(W, w): GrpMat, GrpMatElt → SetEnum- Example: Left and Right Descent Sets
- Properties of Reflection Groups
- Roots, Coroots, and Reflections
- Related Structures
- Bibliography