Properties of Root Systems

Contents

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#

> R := RootSystem("A5 B2");
> IsIrreducible(R);
false
%%a> assert not $1;
> IsSemisimple(R);
true
%%a> assert $1;
> IsCrystallographic(R);
true
%%a> assert $1;
> IsSimplyLaced(R);
false
%%a> assert not $1;