Properties of Root Systems

IsIrreducible(R) : RootSys -> BoolElt
Returns true if, and only if, the root system R is irreducible.
IsProjectivelyIrreducible(R) : RootSys -> BoolElt
Returns true if, and only if, the root system R is a direct sum of a simple system and a toral system. This is equivalent to R having a connected Coxeter diagram.
IsReduced(R) : RootSys -> BoolElt
Returns true if, and only if, the root system R is reduced.
IsSemisimple(R) : RootSys-> BoolElt
Returns true if, and only if, the root system R is semisimple, i.e. its rank is equal to its dimension.
IsCrystallographic(R) : RootSys -> BoolElt
Returns true if, and only if, the root system R is crystallographic, i.e. its Cartan matrix is integral.
IsSimplyLaced(R) : RootSys-> BoolElt
Returns true if, and only if, the root system R is simply laced, i.e. its Coxeter graph contains no labelled edges.

Example RootSys_Properties (H103E8)

> R := RootSystem("A5 B2");
> IsIrreducible(R);
false
> IsSemisimple(R);
true
> IsCrystallographic(R);
true
> IsSimplyLaced(R);
false
V2.28, 13 July 2023