Coxeter Groups
- Introduction
- Constructing Coxeter Groups
CoxeterGroup(grpcat, N): Cat, MonStgElt → grpcatCoxeterGroup(N): MonStgElt → GrpIrreducibleCoxeterGroup(grpcat, X, n): Cat, MonStgElt, RngIntElt → grpcatIrreducibleCoxeterGroup(X, n): MonStgElt, RngIntElt → .- Example: Constructing a Coxeter Group by Name
CoxeterGroup(grpcat, M): Cat, AlgMatElt → grpcatCoxeterGroup(grpcat, G): Cat, GrphUnd → grpcatCoxeterGroup(grpcat, D): Cat, GrphDir → grpcatCoxeterGroup(M): AlgMatElt → GrpCoxeterGroup(G): GrphUnd → GrpCoxeterGroup(D): GrphDir → Grp- Example: Coxeter Group from a Coxeter Matrix
CoxeterGroup(grpcat, R): Cat, RootStr → grpcatCoxeterGroup(R): RootStr → GrpPermCoxCoxeterGroup(A, B): Mtrx, Mtrx → GrpPermCoxCoxeterGroup(grpcat, A, B): Cat, Mtrx, Mtrx → grpcat- Example: Coxeter Group from a Root Datum
- Converting Between Types of Coxeter Group
- Operations on Coxeter Groups
IsIsomorphic(W1, W2): GrpPermCox, GrpPermCox → BoolEltIsCoxeterIsomorphic(W1, W2): GrpFPCox, GrpFPCox → BoolEltIsCoxeterIsomorphic(W1, W2): GrpPermCox, GrpPermCox → BoolEltIsCartanEquivalent(W1, W2): GrpPermCox, GrpPermCox → BoolElt- Example: Testing Coxeter Isomorphism
RootSystem(W): GrpPermCox → RootDtmRootDatum(W): GrpPermCox → RootDtm- Example: Root System of a Coxeter Group
CartanName(W): GrpFPCox → ListCartanName(W): GrpPermCox → MonStgEltCoxeterDiagram(W): GrpFPCoxCoxeterDiagram(W): GrpPermCoxDynkinDiagram(W): GrpPermCox- Example: Cartan Names and Diagrams
CoxeterMatrix(W): GrpFPCox → AlgMatEltCoxeterMatrix(W): GrpPermCox → AlgMatEltCoxeterGraph(W): GrpFPCox → GrphUndCoxeterGraph(W): GrpPermCox → GrphUndCartanMatrix(W): GrpPermCox → AlgMatEltDynkinDigraph(W): GrpPermCox → GrphDirRank(W): GrpFPCox → RngIntEltRank(W): GrpPermCox → RngIntEltNumberOfGenerators(W): GrpFPCox → RngIntEltNumberOfGenerators(W): GrpPermCox → RngIntEltNumberOfPositiveRoots(W): GrpFPCox → RngIntEltNumberOfPositiveRoots(W): GrpPermCox → RngIntEltNumPosRoots(W): GrpFPCox → RngIntEltNumPosRoots(W): GrpPermCox → RngIntEltDimension(W): GrpPermCox → RngIntElt- Example: Rank and Dimension
ConjugacyClasses(W): GrpFPCox → [GrpFPCoxElt]ConjugacyClasses(W): GrpPermCox → [GrpPermElt]FundamentalGroup(W): GrpPermCox → GrpAbIsogenyGroup(W): GrpPermCox → GrpAbCoisogenyGroup(W): GrpPermCox → GrpAbBasicDegrees(W): GrpFPCox → RngIntEltBasicDegrees(W): GrpPermCox → RngIntEltBasicCodegrees(W): GrpFPCox → RngIntEltBasicCodegrees(W): GrpPermCox → RngIntElt- Example: Basic Degrees of E6
BruhatLessOrEqual(x, y): GrpPermElt, GrpPermElt → BoolEltBruhatDescendants(x): GrpPermElt → SetEnumBruhatDescendants(X): SetEnum → SetEnum- Example: Bruhat Descendants
- Properties of Coxeter Groups
- Operations on Elements
- Example: Arithmetic with Coxeter Words
# w: GrpFPCoxElt → RngIntEltLength(w): GrpFPCoxElt → RngIntEltLength(W, w): GrpPermCox, GrpPermElt → RngIntEltCoxeterLength(w): GrpFPCoxElt → RngIntEltCoxeterLength(W, w): GrpPermCox, GrpPermElt → RngIntEltLongestElement(W): GrpFPCox → SeqEnumLongestElement(W): GrpPermCox → GrpPermEltCoxeterElement(W): GrpFPCox → SeqEnumCoxeterElement(W): GrpPermCox → GrpPermEltCoxeterNumber(W): GrpFPCox → SeqEnumCoxeterNumber(W): GrpPermCox → GrpPermElt- Example: Longest Elements and Coxeter Elements
LeftDescentSet(W, w): GrpFPCox, GrpFPCoxElt → SetEnumLeftDescentSet(W, w): GrpPermCox, GrpPermElt → SetEnumRightDescentSet(W, w): GrpFPCox, GrpFPCoxElt → SetEnumRightDescentSet(W, w): GrpPermCox, GrpPermElt → SetEnum- Example: Left and Right Descent Sets
- Roots, Coroots, and Reflections
- Accessing Roots and Coroots
RootSpace(W): GrpPermCox → .CorootSpace(W): GrpPermCox → .SimpleRoots(W): GrpPermCox → MtrxSimpleCoroots(W): GrpPermCox → Mtrx- Example: Root Space and Coroot Space
NumberOfPositiveRoots(W): GrpPermCox → RngIntEltNumPosRoots(W): GrpPermCox → RngIntEltRoots(W): GrpPermCox → SetIndxCoroots(W): GrpPermCox → SetIndxPositiveRoots(W): GrpPermCox → SetIndxPositiveCoroots(W): GrpPermCox → SetIndxRoot(W, r): GrpPermCox, RngIntElt → SetIndxCoroot(W, r): GrpPermCox, RngIntElt → SetIndxRootPosition(W, v): GrpPermCox, . → SetIndxCorootPosition(W, v): GrpPermCox, . → SetIndx- Example: Roots and Coroots from Matrices
HighestRoot(W): GrpPermCox → .HighestLongRoot(W): GrpPermCox → .HighestShortRoot(W): GrpPermCox → .- Example: Highest Roots of G2
CoxeterForm(W): GrpPermCox → AlgMatEltDualCoxeterForm(W): GrpPermCox → AlgMatEltAdditiveOrder(W): GrpPermCox → SeqEnumPapiOrder(W,w): GrpPermCox, GrpPermElt → SeqEnumPapiOrder(W, wd): GrpPermCox, SeqEnum[RngIntElt] → SeqEnum
- Operations and Properties for Root and Coroot Indices
Sum(W, r, s): GrpPermCox, RngIntElt, RngIntElt → RngIntEltIsPositive(W, r): GrpPermCox, RngIntElt → BoolEltIsNegative(W, r): GrpPermCox, RngIntElt → BoolEltNegative(W, r): GrpPermCox, RngIntElt → RngIntEltLeftString(W, r, s): GrpPermCox, RngIntElt, RngIntElt → RngIntEltRightString(W, r, s): GrpPermCox, RngIntElt, RngIntElt → RngIntEltLeftStringLength(W, r, s): GrpPermCox, RngIntElt, RngIntElt → RngIntEltRightStringLength(W, r, s): GrpPermCox, RngIntElt, RngIntElt → RngIntElt- Example: Arithmetic with Root Indices
RootHeight(W, r): GrpPermCox, RngIntElt → RngIntEltCorootHeight(W, r): GrpPermCox, RngIntElt → RngIntEltRootNorms(W): GrpPermCox → [RngIntElt]CorootNorms(W): GrpPermCox → [RngIntElt]RootNorm(W, r): GrpPermCox, RngIntElt → RngIntEltCorootNorm(W, r): GrpPermCox, RngIntElt → RngIntEltIsLongRoot(W, r): GrpPermCox, RngIntElt → BoolEltIsShortRoot(W, r): GrpPermCox, RngIntElt → BoolElt- Example: Root Heights, Norms, and Strings
- Weights
- Reflections
IsReflection(w): GrpFPElt → BoolEltIsReflection(w): GrpPermElt → BoolElt, ., ., RngIntReflections(W): GrpFPCox → [GrpFPCoxElt]Reflections(W): GrpPermCox → [GrpPermElt]- Example: Reflections in a Coxeter Group
SimpleReflections(W): GrpFPCox → [GrpFPCoxElt]SimpleReflections(W): GrpPermCox → [GrpPermElt]SimpleReflectionPermutations(W): GrpPermCox → [GrpPermElt]Reflection(W, r): GrpPermCox, RngIntElt → GrpPermEltReflectionPermutation(W, r): GrpPermCox, RngIntElt → GrpPermEltSimpleReflectionMatrices(W): GrpPermCox → []SimpleCoreflectionMatrices(W): GrpPermCox → []ReflectionMatrices(W): GrpPermCox → []CoreflectionMatrices(W): GrpPermCox → []ReflectionMatrix(W, r): GrpPermCox, RngIntElt → []CoreflectionMatrix(W, r): GrpPermCox, RngIntElt → []ReflectionWords(W): GrpPermCox → []ReflectionWord(W, r): GrpPermCox, RngIntElt → []- Example: Reflection Matrices, Permutations, and Words
- Reflection Subgroups
ReflectionSubgroup(W, a): GrpPermCox, SetEnum → GrpPermCoxReflectionSubgroup(W, s): GrpPermCox, [] → GrpPermCoxStandardParabolicSubgroup(W, J): GrpPermCox, SetEnum → GrpPermCoxIsReflectionSubgroup(W, H): GrpPermCox, GrpPermCox → BoolEltIsParabolicSubgroup(W, H): GrpPermCox, GrpPermCox → BoolEltIsStandardParabolicSubgroup(W, H): GrpPermCox, GrpPermCox → BoolEltOvergroup(H): GrpPermCox → GrpPermCoxOverdatum(H): GrpPermCox → RootDtmLocalCoxeterGroup(H): GrpPermCox → GrpPermCox, Map- Example: Local Coxeter Group of a Parabolic Subgroup
Transversal(W, H): GrpPermCox, GrpPermCox → SetIndxTransversalWords(W, H): GrpPermCox, GrpPermCox → SetIndxTransversalElt(W, H, x): GrpPermCox, GrpPermCox, GrpPermElt → GrpPermElt- Example: Transversals of a Parabolic Subgroup
TransversalElt(W, x, H): GrpPermCox, GrpPermElt, GrpPermCox → GrpPermEltTransversalElt(W, H, x, J): GrpPermCox, GrpPermCox, GrpPermElt, GrpPermCox → GrpPermEltTransversal(W, J): GrpFPCox, {RngIntElt} → {@ GrpFPCoxElt @}Transversal(W, J, L): GrpFPCox, {RngIntElt}, RngIntElt → {@ GrpFPCoxElt @}Transversal(W, J, K): GrpFPCox, {RngIntElt}, {RngIntElt} → [ GrpFPCoxElt ], [ ]DirectProduct(W1, W2): GrpPermCox, GrpPermCox → GrpPermCoxDual(W): GrpPermCox → GrpPermCox- Example: Direct Sums and Duals
- Root Actions
- Standard Action
- Braid Groups
- \(W\)-graphs
SetVerbose("WGraph", v): MonStgElt, RngIntEltMij2EltRootTable(seq): SeqEnum → SeqEnum[SeqEnum[RngIntElt]]Name2Mij(name): MonStgElt → SeqEnum- Example: Reflection Table from Coxeter Matrix Entries
Partition2WGtable(pi): SeqEnum → SeqEnum, GrpFPCoxWGtable2WG(table): SeqEnum → GrphUndTestWG(W,wg): GrpFPCox, GrphUnd → .TestWG(W,wg): GrpFPCox, GrphDir → .TestWG(tp,wg): MonStgElt, GrphDir → .TestWG(tp,wg): MonStgElt, GrphDir → .- Example: W-Graph of a Specht Module
WGelement2WGtable(g,K): GrpFPCoxElt, SetEnum → SeqEnum, SeqEnum- Example: W-Graph of Type B5
GetCells(wg): GrphUnd → SeqEnumGetCells(wg): GrphDir → SeqEnumInduceWG(W,wg,seq): GrpFPCox, GrphUnd, SeqEnum → GrphUndInduceWG(W,wg,seq): GrpFPCox, GrphDir, SeqEnum → GrphDirInduceWGtable(J, table, W): SeqEnum, SeqEnum, GrpFPCox → SeqEnum[SeqEnum[RngIntElt]]IsWGsymmetric(dwg): GrphDir → BoolElt, GrphDirMakeDirected(uwg): GrphUnd → GrphDirTestHeckeRep(W,r): GrpFPCox, SeqEnum → .WG2GroupRep(wg): GrphUnd → SeqEnumWG2GroupRep(wg): GrphDir → SeqEnumWG2HeckeRep(W,wg): GrpFPCox, GrphUnd → SeqEnumWG2HeckeRep(W,wg): GrpFPCox, GrphDir → SeqEnumWGidealgens2WGtable(dgens,K): SeqEnum, SetEnum → SeqEnum[SeqEnum[RngIntElt]], SetIndx- Example: W-Graph Ideal in Type E6
WriteWG(file,uwg): MonStgElt, GrphUnd → .WriteWG(file,dwg): MonStgElt, GrphDir → .
- Related Structures
- Bibliography