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.
IsWeaklyAdjoint(R) : RootDtm -> BoolElt
Returns true if, and only if, the root datum R is weakly adjoint, i.e. its isogeny group is isomorphic to Zn, where n is dim(R) - (rk)(R). Note that if R is semisimple then this function is identical to IsAdjoint.
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.
IsWeaklySimplyConnected(R) : RootDtm -> BoolElt
Returns true if, and only if, the root datum R is weakly simply connected, i.e. its coisogeny group is isomorphic to Zn, where n is dim(R) - (rk)(R). Note that if R is semisimple then this function is identical to IsSimplyConnected.

Example RootDtm_Properties (H104E13)

> R := RootDatum("A5 B2" : Isogeny := "SC");
> IsIrreducible(R);
false
> IsSimplyLaced(R);
false
> IsSemisimple(R);
true
> IsAdjoint(R);
false
For some of the exceptional isogeny classes, there is only one isomorphism class of root data, which is both adjoint and simply connected. 
> R := RootDatum("G2");
> IsAdjoint(R);
true
> IsSimplyConnected(R);
true
There exist root data that are neither adjoint nor simply connected. 
> R := RootDatum("A3" : Isogeny := 2);
> IsAdjoint(R), IsSimplyConnected(R);
false false
Finally, we demonstrate a case where the root datum is not adjoint, but is weakly adjoint. 
> R := RootDatum("A2T1");
> IsAdjoint(R), IsWeaklyAdjoint(R);
false true
> Dimension(R), Rank(R);
3 2
> G := IsogenyGroup(R); G;
Abelian Group isomorphic to Z
Defined on 1 generator (free)
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.
V2.28, 13 July 2023