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 \({\mathbb{Z}}^n\), where \(n\) is \(\dim(R) - {\rm rk}(R)\). 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 \({\mathbb{Z}}^n\), where \(n\) is \(\dim(R) - {\rm rk}(R)\). Note that if \(R\) is semisimple then this function is identical to IsSimplyConnected.

Example: Testing Properties of a Root Datum (ex-ece71b)

Run in calculator (opens in a new tab)

> 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)

Run in calculator (opens in a new tab)

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 \(\Gamma\)-action is trivial.

IsTwisted(R): RootDtm -> BoolElt: Returns true if, and only if, the root datum \(R\) is twisted, i.e. the \(\Gamma\)-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 \(X=X_0\).