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.

\name{IsAdjointR}{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 IsAdjointR.

\name{IsSimplyConnectedR}{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 .

Example: Properties

For some of the exceptional isogeny classes, there is only one isomorphism class of root data, which is both adjoint and simply connected. There exist root data that are neither adjoint nor simply connected. Finally, we demonstrate a case where the root datum is not adjoint, but is weakly adjoint.

> R := RootDatum("A5 B2" : Isogeny := "SC");
> IsIrreducible(R);
false
%%a> assert not $1;
> IsSimplyLaced(R);
false
%%a> assert not $1;
> IsSemisimple(R);
true
%%a> assert $1;
> IsAdjoint(R);
false
%%a> assert not $1;
> R := RootDatum("G2");
> IsAdjoint(R);
true
%%a> assert $1;
> IsSimplyConnected(R);
true
%%a> assert $1;
> R := RootDatum("A3" : Isogeny := 2);
> IsAdjoint(R), IsSimplyConnected(R);
false false
> 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 \(\Gamma\)-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 \(X=X_0\).