Returns true for any root datum R.
Returns true if, and only if, the root datum R is irreducible.
Returns true if, and only if, the split version of the root datum R is irreducible.
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.
Returns true if, and only if, the root datum R is reduced.
Returns true if, and only if, the root datum R is semisimple, i.e. its rank is equal to its dimension.
Returns true for any root datum R.
Returns true if, and only if, the root datum R is simply laced, i.e. its Dynkin diagram contains no multiple bonds.
Returns true if, and only if, the root datum R is adjoint, i.e. its isogeny group is trivial.
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.
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.
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.
> R := RootDatum("A5 B2" : Isogeny := "SC"); > IsIrreducible(R); false > IsSimplyLaced(R); false > IsSemisimple(R); true > IsAdjoint(R); falseFor 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); trueThere exist root data that are neither adjoint nor simply connected.
> R := RootDatum("A3" : Isogeny := 2); > IsAdjoint(R), IsSimplyConnected(R); false falseFinally, 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)
Returns true if, and only if, the root datum R is reduced.
Returns true if, and only if, the root datum R is split, i.e. the Γ-action is trivial.
Returns true if, and only if, the root datum R is twisted, i.e. the Γ-action is not trivial.
Returns true if, and only if, the root datum R is quasisplit, i.e. the anisotropic subdatum is trivial.
Returns true if, and only if, the root datum R is inner (resp. outer).
Returns true if, and only if, the root datum R is anisotropic, i.e. when X=X0.[Next][Prev] [Right] [Left] [Up] [Index] [Root]