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: Isomorphism

Note that the root systems \(B_n\) and \(C_n\) are isomorphic but not Cartan equivalent. Hence Cartan equivalence is not an invariant of a root system since it depends on the particular representation of the (co)roots within the (co)root space.

> R := RootSystem("B4");  S := RootSystem("C4");
> IsIsomorphic(R, S);
true
%%a> assert $1;
> IsCartanEquivalent(R, S);
false
%%a> assert not $1;

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 root system \(R\) (Section Dynkin Digraphs). If \(R\) is not crystallographic, an error is flagged.

Example: Diagrams

> 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.

RealInjection(R): RootSys → .

The real injection of the root system \(R\) (Section Constructing Root Systems).

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: Basic Operations

> 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
%%a> assert $1;
> CoxeterGroupOrder(R);
14
%%a> assert $1 eq 14;