# 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: Properties (ex-a19f70)`

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

```