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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 GrpCox_Properties (H98E12)

> W := CoxeterGroup(GrpFPCox, HyperbolicCoxeterMatrix(22));
> IsFinite(W);
false
> IsAffine(W);
false
> IsHyperbolic(W);
true
> IsCompactHyperbolic(W);
true
> 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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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013