Operators on Root Systems

R1 eq R2 : RootSys, RootSys -> BoolElt

Returns true if, and only if, the root systems R_1 and R_2 are identical.

IsIsomorphic(R1, R2) : RootSys, RootSys -> BoolElt

Returns true if, and only if, root systems R_1 and R_2 are isomorphic.

IsCartanEquivalent(R1, R2) : RootSys, RootSys -> BoolElt

Returns true if, and only if, the crystallographic root systems R_1 and R_2 are Cartan equivalent, i.e. their Cartan matrices are the same modulo a permutation of the underlying basis.

Example RootSys_Isomorphism (H79E5)

> R := RootSystem( "B4" );  S := RootSystem( "C4" );
> IsIsomorphic( R, S );
true
> IsCartanEquivalent( R, S );
false

CartanName(R) : RootSys -> List

The Cartan name of the root system R (Section Finite and Affine Coxeter Groups).

CoxeterDiagram(R) : RootSys ->

Print the Coxeter diagram of the root system R (Section Finite and Affine Coxeter Groups).

DynkinDiagram(R) : RootSys ->

Print the Dynkin diagram of the root system R (Section Finite and Affine Coxeter Groups). If R is not crystallographic, an error is flagged.

CoxeterMatrix(R) : RootSys -> AlgMatElt

The Coxeter matrix of the root system R (Section Coxeter Matrices).

CoxeterGraph(R) : RootSys -> GrphUnd

The Coxeter graph of the root system R (Section Coxeter Graphs).

CartanMatrix(R) : RootSys -> AlgMatElt

The Cartan matrix of the root system R (Section Cartan Matrices).

DynkinDigraph(R) : RootSys -> GrphDir

The Dynkin digraph of the crystallographic root system R (Section Dynkin Digraphs).

Example RootSys_Diagrams (H79E6)

> R := RootSystem( "F4" );
> DynkinDiagram( R );

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

F4    1 - 2 === 3 - 4

BaseField(R) : RootSys -> Fld

BaseRing(R) : RootSys -> Fld

The field over which the root system R is defined.

Rank(R) : RootSys -> RngIntElt

The rank of the root system R, i.e. the number of simple (co)roots.

Dimension(R) : RootSys -> RngIntElt

The dimension of the root system R, i.e. the dimension of the (co)root space. This is always at least as large as the rank, with equality when R is semisimple.

CoxeterGroupOrder(R) : RootSys -> RngIntElt

The order of the Coxeter group of the root system R.

Example RootSys_BasicOperations (H79E7)

> R := RootSystem( "I2(7)" );
> BaseField( R );
Number Field with defining polynomial x^3 - x^2 - 2*x + 1 over the
Rational Field
> Rank( R ) eq Dimension( R );
true
> CoxeterGroupOrder( R );
14