# Operations on Groups of Lie Type Many of the basic operations for Coxeter groups are shortcuts for obtaining information about the underlying root datum (Chapter [Root Data](../RootData/index-root-data.md#chaprootdtm)). Such functions are listed here; see Sections [Operations on Root Data](../RootData/operators-root-dtm.md#sectrdop), [Properties of Root Data](../RootData/properties-root-datum.md#sectrdprop), [Roots, Coroots and Weights](../RootData/roots-coroots-weights.md#sectrdroot), and [Operations on Coxeter Groups](../CoxeterGroups/group-op.md#sectgrppermcoxop) for more details and examples of their use. ## `G eq H: GrpLie, GrpLie -> BoolElt` Returns `true` iff the groups of Lie type $G$ and $H$ are equal. ## `G subset H: GrpLie, GrpLie -> BoolElt` Returns `true` iff the group of Lie type $G$ is a subset of $H$. ## `IsAlgebraicallyIsomorphic(G, H): GrpLie, GrpLie -> BoolElt, Map` Returns `true` if the semisimple groups $G$ and $H$ are isomorphic as algebraic groups (i.e. they have the same base rings and isomorphic root data). If `true`, then the second value returned is an isomorphism. ## `IsIsogenous(G, H): GrpLie, GrpLie -> BoolElt` Returns `true` if $G$ and $H$ are isogenous. The groups must be semisimple and defined over the same field. If `true`, the subsequent values returned are: the corresponding adjoint group $G_{ad}$, the homomorphisms $G_{ad}\to G$ and $G_{ad}\to H$, the corresponding simply connected root datum $G_{sc}$, and the homomorphisms $G\to G_{sc}$ and $H\to G_{sc}$. ## `IsCartanEquivalent(G, H): GrpLie, GrpLie -> BoolElt` Returns `true` if, and only if, the groups of Lie type $G$ and $H$ are Cartan equivalent, i.e. they have isomorphic Dynkin diagrams and defined over the same ring. ## `BaseRing(G): GrpLie -> Rng` ## `CoefficientRing(G): GrpLie -> Rng` The base ring $k$ of the group of Lie type $G$. ## `BaseExtend(G, K): GrpLie, Rng -> GrpLie, Map` Given a group of Lie type $G$ with base ring $k$ and a larger ring $K$, return the group $G(K)$ gotten by extending the base ring and the injection $G\to G(K)$. ## `ChangeRing(G, K): GrpLie, Rng -> GrpLie` Given a group of Lie type $G$ and a ring $K$, return the group with the same root datum, but defined over a different ring. ## `Generators(G): GrpLie -> {}` Generators for the group of Lie type $G$ as an abstract group. This is currently only implemented when the base ring is a finite field. ## `NumberOfGenerators(G): GrpLie -> RngIntElt` ## `Ngens(G): GrpLie -> RngIntElt` The number of generators for the group of Lie type $G$ as an abstract group. This is currently only implemented when the base ring is a finite field. ## `AlgebraicGenerators(G): GrpLie -> {}` A set of generators for the group of Lie type $G$ as an algebraic group. ## `NumberOfAlgebraicGenerators(G): GrpLie -> RngIntElt` ## `Nalggens(G): GrpLie -> RngIntElt` The number of generators for the group of Lie type $G$ as an algebraic group. ## `Example: Generators (ex-a929c8)` ```magma > k := GF(4); > G := GroupOfLieType("A2", k : Normalising:=false); > Generators(G); [ x1(1) , x4(1) , x1(z) , x4(z) , x2(1) , x5(1) , x2(z) , x5(z) , ( z 1) , (1 z) ] > AlgebraicGenerators(G); [ x1(1) , x2(1) , x4(1) , x5(1) , ( z 1) , ( 1 z) ] ``` ## `Order(G): GrpLie -> RngIntElt` ## `# G: GrpLie -> RngIntElt` The order of the group of Lie type $G$. ## `FactoredOrder(G): GrpLie -> RngIntElt` The factored order of the group of Lie type $G$. ## `Dimension(G): GrpLie -> RngIntElt` The dimension of the group of Lie type $G$, considered as an algebraic variety. ## `Example: Orders (ex-1128fd)` ```magma > G := GroupOfLieType("G2", 3); > Order(G); 4245696 > FactoredOrder(G); [ <2, 8>, <13, 1>, <3, 6>, <7, 1> ] > G := GroupOfLieType("G2", Rationals()); > Order(G); Infinity > Dimension(G); 14 ``` ## `CartanName(G): GrpLie -> Mtrx` The Cartan name of the group of Lie type $G$. ## `RootDatum(G): GrpLie -> RootDtm` The root datum of the group of Lie type $G$. ## `DynkinDiagram(G): GrpLie` Print the Dynkin diagram of the group of Lie type $G$. ## `CoxeterDiagram(G): GrpLie` Print the Coxeter diagram of the group of Lie type $G$. ## `CoxeterMatrix(G): GrpLie -> AlgMatElt` The Coxeter matrix of the group of Lie type $G$. ## `CoxeterGraph(G): GrpLie -> GrphUnd` The Coxeter graph of the group of Lie type $G$. ## `CartanMatrix(G): GrpLie -> GrphUnd` The Cartan matrix of the group of Lie type $G$. ## `DynkinDigraph(G): GrpLie -> GrphUnd` The Dynkin digraph of the group of Lie type $G$. ## `Rank(G): GrpLie -> RngIntElt` ## `ReductiveRank(G): GrpLie -> RngIntElt` The reductive rank of the group of Lie type $G$, i.e. the dimension of the underlying root datum. ## `SemisimpleRank(G): GrpLie -> RngIntElt` The semisimple rank of the group of Lie type $G$, i.e. the rank of the underlying root datum. ## `CoxeterNumber(G): GrpLie -> RngIntElt` The Coxeter number of the group of Lie type $G$, i.e. the order of the Coxeter element in the Weyl group of $G$. ## `WeylGroup(G): GrpLie -> GrpPermCox` ## `WeylGroup(GrpPermCox, G): Cat, GrpLie -> GrpPermCox` The Weyl group of the group of Lie type $G$ as a permutation Coxeter group. This is a crystallographic Coxeter group, see Chapter ChapGrpPermCox. ## `WeylGroup(GrpFPCox, G): Cat, GrpLie -> GrpFPCox` The Weyl group of the group of Lie type $G$ as a finitely presented Coxeter group. This is a crystallographic Coxeter group, see Chapter ChapGrpFPCox. ## `WeylGroup(GrpMat, G): Cat, GrpLie -> GrpMat` The Weyl group of the group of Lie type $G$ as a reflection group. This is a crystallographic Coxeter group, see Chapter [Reflection Groups](../ReflectionGroups/index-reflection-groups.md#chapgrprfl). ## `FundamentalGroup(G): GrpLie -> GrpAb, Map` The fundamental group of the group of Lie type $G$, together with the projection of the weight lattice onto the fundamental group. ## `IsogenyGroup(G): GrpLie -> GrpAb, Map` The isogeny group of the group of Lie type $G$, together with its injection into the fundamental group. ## `CoisogenyGroup(G): GrpLie -> GrpAb, Map` The coisogeny group of the group of Lie type $G$, together with its projection onto the fundamental group.