MAGMA Computational Algebra System

Properties of Root Data

IsFinite(R) : RootStr -> BoolElt

Returns true for any root datum R.

IsIrreducible(R) : RootStr -> BoolElt

Returns true if, and only if, the root datum R is irreducible.

IsAbsolutelyIrreducible(R) : RootStr -> BoolElt

Returns true if, and only if, the split version of the root datum R is irreducible.

IsProjectivelyIrreducible(R) : RootStr -> BoolElt

Returns true if, and only if, the quotient of the root datum R modulo its radical is irreducible. This is equivalent for R to have a connected Coxeter diagram.

IsReduced(R) : RootDtm -> BoolElt

Returns true if, and only if, the root datum R is reduced.

IsSemisimple(R) : RootStr -> BoolElt

Returns true if, and only if, the root datum R is semisimple, i.e. its rank is equal to its dimension.

IsCrystallographic(R) : RootStr -> BoolElt

Returns true for any root datum R.

IsSimplyLaced(R) : RootStr -> BoolElt

Returns true if, and only if, the root datum R is simply laced, i.e. its Dynkin diagram contains no multiple bonds.

IsAdjoint(R) : RootDtm -> BoolElt

Returns true if, and only if, the root datum R is adjoint, i.e. its isogeny group is trivial.

IsSimplyConnected(R) : RootDtm -> BoolElt

Returns true if, and only if, the root datum R is simply connected, i.e. its isogeny group is equal to the fundamental group, i.e. its coisogeny group is trivial.

Example RootDtm_Properties (H78E14)

> R := RootDatum("A5 B2" : Isogeny := "SC");
> IsIrreducible(R);
false
> IsSimplyLaced(R);
false
> IsSemisimple(R);
true
> IsAdjoint(R);
false

> R := RootDatum("G2");
> IsAdjoint(R);
true
> IsSimplyConnected(R);
true

IsReduced(R) : RootStr -> BoolElt

Returns true if, and only if, the root datum R is reduced.

IsSplit(R) : RootDtm -> BoolElt

Returns true if, and only if, the root datum R is split, i.e. the Γ-action is trivial.

IsTwisted(R) : RootDtm -> BoolElt

Returns true if, and only if, the root datum R is twisted, i.e. the Γ-action is not trivial.

IsQuasisplit(R) : RootDtm -> BoolElt

Returns true if, and only if, the root datum R is quasisplit, i.e. the anisotropic subdatum is trivial.

IsInner(R) : RootDtm -> BoolElt

IsOuter(R) : RootDtm -> BoolElt

Returns true if, and only if, the root datum R is inner (resp. outer).

IsAnisotropic(R) : RootDtm -> BoolElt

Returns true if, and only if, the root datum R is anisotropic, i.e. when X=X₀.