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 Hyperbolic Groups).

IsCompactHyperbolic(W): GrpFPCox -> BoolElt

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

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 of a Hyperbolic Coxeter Group (ex-5b007e)

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> W := CoxeterGroup(GrpFPCox, HyperbolicCoxeterMatrix(22));
> IsFinite(W);
false
> IsAffine(W);
false
> IsHyperbolic(W);
true
> IsCompactHyperbolic(W);
false
> IsIrreducible(W);
true
> IsSimplyLaced(W);
true
> W := CoxeterGroup("A2 D4");
> IsIrreducible(W);
false
> IsSemisimple(W);
true
> IsCrystallographic(W);
true
> IsSimplyLaced(W);
true

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