MAGMA Computational Algebra System

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 (H77E8)

> R := RootSystem("A5 B2");
> IsIrreducible(R);
false
> IsSemisimple(R);
true
> IsCrystallographic(R);
true
> IsSimplyLaced(R);
false