Constructing Groups of Lie Type
GroupOfLieType(N, k) : MonStgElt, Rng -> GrpLie
GroupOfLieType(N, q) : MonStgElt, RngIntElt -> GrpLie
GroupOfLieType(W, k) : GrpFPCox, Rng -> GrpLie
GroupOfLieType(W, q) : GrpFPCox, RngIntElt -> GrpLie
GroupOfLieType(R, k) : RootDtm, Rng -> GrpLie
GroupOfLieType(R, q) : RootDtm, RngIntElt -> GrpLie
GroupOfLieType(C, k) : Mtrx, Rng -> GrpLie
GroupOfLieType(C, q) : Mtrx, RngIntElt -> GrpLie
SimpleGroupOfLieType(X, n, k) : MonStgElt, RngIntElt, Rng -> GrpLie
SimpleGroupOfLieType(X, n, q) : MonStgElt, RngIntElt, RngIntElt -> GrpLie
IsNormalising(G) : GrpLie -> BoolElt
SetNormalising( G, Normalising) : GrpLie, BoolElt -> .
Example GrpLie_Create (H89E1)
Operations on Groups of Lie Type
G eq H : GrpLie, GrpLie -> BoolElt
IsAlgebraicallyIsomorphic(G, H) : GrpLie, GrpLie -> BoolElt
IsIsogenous(G, H) : GrpLie, GrpLie -> BoolElt
BaseRing(G) : GrpLie -> Rng
ExtendRing(G, K) : GrpLie -> GrpLie, Rng
Generators(G) : GrpLie ->
AlgebraicGenerators(G) : GrpLie ->
Example GrpLie_Generators (H89E2)
Order(G) : GrpLie -> RngIntElt
FactoredOrder(G) : GrpLie -> RngIntElt
Example GrpLie_Orders (H89E3)
CartanName(G) : GrpLie -> Mtrx
RootDatum(G) : GrpLie -> RootDtm
DynkinDiagram(G) : GrpLie ->
CoxeterDiagram(G) : GrpLie ->
CoxeterMatrix(G) : AlgMatElt -> AlgMatElt
CoxeterGraph(G) : AlgMatElt -> GrphUnd
CartanMatrix(G) : AlgMatElt -> GrphUnd
DynkinDigraph(G) : AlgMatElt -> GrphUnd
Rank(G) : GrpLie -> RngIntElt
SemisimpleRank(G) : GrpLie -> RngIntElt
WeylGroup(G) : GrpLie -> GrpPermCox
WeylGroup(GrpFPCox, G) : Cat, GrpLie -> GrpFPCox
WeylGroup(GrpMat, G) : Cat, GrpLie -> GrpCox
FundamentalGroup(G) : GrpLie -> GrpAb, Map
IsogenyGroup(G) : GrpLie -> GrpAb, Map
CoisogenyGroup(G) : GrpLie -> GrpAb, Map
Properties of Groups of Lie Type
IsSimple(G) : GrpLie -> BoolElt
IsSimplyLaced(G) : GrpLie-> BoolElt
IsSemisimple(G) : GrpLie-> BoolElt
IsAdjoint(G) : GrpLie-> BoolElt
IsSimplyConnected(G) : GrpLie-> BoolElt
Constructing Elements
elt<G | L> : GrpLie, List -> GrpMatElt
Identity(G) : GrpLie -> GrpLieElt
Example GrpLie_ElementCreate (H89E4)
TorusTerm(G, r, t) : GrpLie, RngIntElt, . -> GrpLieElt
CoxeterElement(G) : GrpCox -> GrpPermElt
Random(G) : GrpLie -> GrpLieElt
Eltlist(g) : GrpLieElt -> List
Operations on Elements
g * h : GrpLieElt, GrpLieElt -> GrpLieElt
Example GrpLie_GrpLieEltProduct (H89E5)
g ^ n : GrpLieElt, RngIntElt -> GrpLieElt
g ^ h : GrpLieElt, GrpLieELt -> GrpLieElt
(g, h) : GrpLieElt, GrpLieELt -> GrpLieElt
Normalise( g) : GrpLieElt ->
Example GrpLie_GrpLieEltArith (H89E6)
Bruhat(g) : GrpLieElte -> GrpLieElt, GrpLieElt, GrpLieElt, GrpLieElt
Example GrpLie_Bruhat (H89E7)
CentrePolynomials(G) : GrpLie ->
IsCentral(x) : GrpLieElt -> BoolElt
Example GrpLie_Centre (H89E8)
Accessing Roots and Coroots
RootSpace(G) : GrpLie -> Lat
SimpleRoots(G) : GrpLie -> Mtrx
NumberOfPositiveRoots(G) : GrpLie -> RngIntElt
Roots(G) : GrpLie -> {@@}
PositiveRoots(G) : GrpLie -> {@@}
Root(G, r) : GrpLie, RngIntElt -> {@@}
RootPosition(G, v) : GrpLie, . -> {@@}
Example GrpLie_RootsCoroots (H89E9)
HighestRoot(G) : GrpLie -> LatElt
HighestShortRoot(G) : GrpLie -> LatElt
Example GrpLie_HeighestRoots (H89E10)
Reflections
Reflections(G) : GrpLie -> GrpLieElt
Reflection(G, r) : GrpLie, RngIntElt -> GrpLieElt
Example GrpLie_Reflections (H89E11)
Operations and Properties for Root and Coroot indices
RootHeight(G, r) : GrpLie, RngIntElt -> RngIntElt
RootNorms(G) : GrpLie -> [RngIntElt]
RootNorm(G, r) : GrpLie, RngIntElt -> RngIntElt
IsLongRoot(G, r) : GrpLie, RngIntElt -> BoolElt
IsShortRoot(G, r) : GrpLie, RngIntElt -> BoolElt
AdditiveOrder(G) : GrpLie -> SeqEnum
Example GrpLie_AdditiveOrder (H89E12)
Weights
WeightLattice(G) : RootDtm -> Lat
FundamentalWeights(G) : GrpLie -> SeqEnum
Automorphisms
InnerAutomorphism(G, x) : GrpLie, GrpLieElt -> Map
DiagonalAutomorphism(G, v) : GrpLie, ModTupRngElt -> Map
GraphAutomorphism(G, p) : GrpLie, GrpPermElt -> Map
FieldAutomorphism(G, sigma) : GrpLie, Map -> Map
DecomposeAutomorphism(h) : Map -> Map, Map, Map, Rec
Example GrpLie_Automorphism (H89E13)
Constructing Representations
StandardRepresentation(G) : GrpLie -> Map
AdjointRepresentation(G) : GrpLie -> Map
LieAlgebra(G) : GrpLie -> AlgLie, Map
HighestWeightRepresentation(G, v) : GrpLie, . -> Map
Example GrpLie_StandardRepresentation (H89E14)
Operations on Representations
Weight(rho, v) : Map, ModTupRngElt -> LatElt
HighestWeightVectors(rho) : Map -> [ModTupRngElt]
HighestWeights(rho) : Map -> [LatElt], [ModTupRngElt]
WeightVectors(rho) : Map -> [ModTupRngElt]
Weights(rho) : Map -> [LatElt], [ModTupRngElt]