Properties of Coxeter Groups

IsFinite(W): GrpFPCox → BoolElt

Returns true if, and only if, the Coxeter group \(W\) is finite.

IsAffine(W): GrpFPCox → BoolElt

Returns true if, and only if, the Coxeter group \(W\) is affine (Section Finite and Affine Coxeter Groups).

IsHyperbolic(W): GrpFPCox → BoolElt

Returns true if, and only if, the Coxeter group \(W\) is hyperbolic (Section SectCartanHyperbolic).

IsCompactHyperbolic(W): GrpFPCox → BoolElt

Returns true if, and only if, the Coxeter group \(W\) is compact hyperbolic (Section SectCartanHyperbolic).

IsIrreducible(W): GrpFPCox → BoolElt

IsIrreducible(W): GrpPermCox → BoolElt

Returns true if, and only if, the Coxeter group \(W\) is irreducible.

IsSemisimple(W): GrpPermCox → BoolElt

Returns true if, and only if, the permutation Coxeter group \(W\) is semisimple, i.e. its rank is equal to its dimension.

IsCrystallographic(W): GrpPermCox → BoolElt

Returns true if, and only if, the permutation Coxeter group \(W\) is crystallographic, i.e. if the corresponding reflection representation is defined over the integers.

IsSimplyLaced(W): GrpPermCox → BoolElt

IsSimplyLaced(W): GrpFPCox → BoolElt

Returns true if, and only if, the Coxeter group \(W\) is simply laced, i.e. its Coxeter graph has no labels.

Example: Properties

> W := CoxeterGroup(GrpFPCox, HyperbolicCoxeterMatrix(22));
> IsFinite(W);
false
%%a> assert not $1;
> IsAffine(W);
false
%%a> assert not $1;
> IsHyperbolic(W);
true
%%a> assert $1;
> IsCompactHyperbolic(W);
false
%%a> assert not $1;
> IsIrreducible(W);
true
%%a> assert $1;
> IsSimplyLaced(W);
true
%%a> assert $1;
> W := CoxeterGroup("A2 D4");
> IsIrreducible(W);
false
%%a> assert not $1;
> IsSemisimple(W);
true
%%a> assert $1;
> IsCrystallographic(W);
true
%%a> assert $1;
> IsSimplyLaced(W);
true
%%a> assert $1;