Operations on Finite Coxeter Groups

IsIsomorphic( W1, W2 ) : GrpFPCox, GrpFPCox -> BoolElt

Returns true if, and only if, W_1 and W_2 are isomorphic as abstract groups.

IsCoxeterIsomorphic( W1, W2 ) : GrpFPCox, GrpFPCox -> BoolElt

Returns true if, and only if, W_1 and W_2 are isomorphic as Coxeter groups.

IsCartanEquivalent( W1, W2 ) : GrpFPCox, GrpFPCox -> BoolElt

Returns true if, and only if, the crystallographic Coxeter groups W_1 and W_2 have Cartan equivalent Cartan matrices.

Example GrpPermCox_Isomorphism (H84E4)

> W1 := CoxeterGroup( "G2" );
> W2 := CoxeterGroup( "A1A2" );
> IsIsomorphic( W1, W2 );
true
> IsCoxeterIsomorphic( W1, W2 );
false

> W1 := CoxeterGroup( "B3" );
> W2 := CoxeterGroup( "C3" );
> IsIsomorphic( W1, W2 );
true
> IsCoxeterIsomorphic( W1, W2 );
true [ 1, 2, 3 ]
> IsCartanEquivalent( W1, W2 );
false

RootSystem( W ) : GrpPermCox -> RootDtm

The underlying root system of the Coxeter group W.

RootDatum( W ) : GrpPermCox -> RootDtm

The root datum of the Coxeter group W. If W does not have a root datum, an error is flagged.

Example GrpPermCox_GroupToRoot (H84E5)

> W := CoxeterGroup( "C5" );
> RootSystem( W );
Root system of type C5
> RootDatum( W );
Root datum of type C5
> 
> W := CoxeterGroup( "H4" );
> RootSystem( W );
Root system of type H4

> RootDatum( W );
Error: This group does not have a root datum

CartanName( W ) : GrpPermCox -> MonStgElt

The Cartan name of the Coxeter group W.

CoxeterDiagram( W ) : GrpPermCox ->

Print the Coxeter diagram of the finite Coxeter group W.

DynkinDiagram( W ) : GrpPermCox ->

Print the Dynkin diagram of the Coxeter group W. If W is not crystallographic, an error is flagged.

Example GrpPermCox_NamesDiagrams (H84E6)

> W := CoxeterGroup( "F4" );
> CartanName( W );
F4
> DynkinDiagram( W );

F4    1 - 2 =>= 3 - 4
> CoxeterDiagram( W );

F4    1 - 2 === 3 - 4

CoxeterMatrix( W ) : GrpFPCox -> AlgMatElt

The Coxeter matrix of the Coxeter group W.

CoxeterGraph( W ) : GrpFPCox -> GrphUnd

The Coxeter graph of the Coxeter group W.

CartanMatrix( W ) : GrpFPCox -> AlgMatElt

The Cartan matrix of the Coxeter group W.

DynkinDigraph( W ) : GrpFPCox -> GrphDir

The Dynkin digraph of the Coxeter group W.

Rank( W ) : GrpPermCox -> RngIntElt

NumberOfGenerators( W ) : GrpPermCox -> RngIntElt

The rank of the Coxeter group W, ie. the number of simple (co)roots.

Dimension( W ) : GrpPermCox -> RngIntElt

The dimension of the Coxeter group W, ie. the dimension of the root space.

Example GrpPermCox_RankDimension (H84E7)

> R := StandardRootSystem( "A", 4 );
> W := CoxeterGroup( R );
> Rank( W );
4
> Dimension( W );
5

FundamentalGroup( W ) : GrpPermCox -> GrpAb

The fundamental group of the Coxeter group W. The roots and coroots of W must have integral components.

IsogenyGroup( W ) : GrpPermCox -> GrpAb

The isogeny group of the Coxeter group W. The roots and coroots of W must have integral components.

CoisogenyGroup( W ) : GrpPermCox -> GrpAb

The coisogeny group of the Coxeter group W. The roots and coroots of W must have integral components.

BasicDegrees( W ) : GrpPermCox -> RngIntElt

The degrees of the basic invariant polynomials of the Coxeter group W. These are computed using the table in [Car72, page 155].

Example GrpPermCox_BasicDegrees (H84E8)

> W := CoxeterGroup( "E6" );
> degs := BasicDegrees( W );
> degs;
[ 2, 5, 6, 8, 9, 12 ]
> &*degs eq #W;
true
> &+degs eq NumPosRoots(W) + Rank(W);
true