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Returns true if, and only if, R_1 and R_2 are identical root data.
Returns true if, and only if, R_1 and R_2 are isomorphic root data.
If true, we also return a sequence
giving the simple root of R_2 corresponding to each simple root of R_1.
Returns true if, and only if, the root data R_1 and R_2 Cartan equivalent,
i.e. they have isomorphic Dynkin diagrams.
If true, we also return a sequence
giving the simple root of R_2 corresponding to each simple root of R_1.
Returns true if, and only if, R_1 and R_2 are
isogenous root data.
If true, we also return a sequence
giving the simple root of R_2 corresponding to each simple root of R_1.
An example of isogenous root data:
> R1 := RootDatum( "A3" );
> R2 := RootDatum( "A3" : Isogeny := "SC" );
> R1 eq R2;
false
> IsIsomorphic( R1, R2 );
false
> IsCartanEquivalent( R1, R2 );
true [ 1, 2, 3 ]
> IsIsogenous( R1, R2 );
true [ 1, 2, 3 ]
An example of distinct isomorphic root data:
> C := CartanMatrix( "B2" );
> R1 := RootDatum( C );
> R2 := RootDatum( Transpose( C ) );
> R1; R2;
Adjoint root datum of type B2
Adjoint root datum of type C2
> R1 eq R2;
false
> IsIsomorphic( R1, R2 );
true [ 2, 1 ]
The Cartan name of the root datum R (Section Finite and Affine Coxeter Groups).
Print the Coxeter diagram of the root datum R
(Section Finite and Affine Coxeter Groups).
Print the Dynkin diagram of the root datum R
(Section Finite and Affine Coxeter Groups).
The Coxeter matrix of the root datum R (Section Coxeter Matrices).
The Coxeter graph of the root datum R (Section Coxeter Graphs).
The Cartan matrix of the root datum R (Section Cartan Matrices).
The Dynkin digraph of the root datum R
(Section Dynkin Digraphs).
> R := RootDatum( "F4" );
> DynkinDiagram( R );
F4 1 - 2 =>= 3 - 4
> CoxeterDiagram( R );
F4 1 - 2 === 3 - 4
The ring of integers.
The rank of the root datum R, i.e. the
number of simple (co)roots.
The dimension of the root datum R, i.e. the dimension of the (co)root space.
This is at least as large as the rank, with equality when R is
semisimple.
The order of the Coxeter group of the root datum R.
> R := RootDatum( "G2" );
> RootSpace( R );
Standard Lattice of rank 2 and degree 2
> CorootSpace( R );
Standard Lattice of rank 2 and degree 2
> SimpleRoots( R );
[1 0]
[0 1]
> SimpleCoroots( R );
[ 2 -3]
[-1 2]
> CartanMatrix( R );
[ 2 -1]
[-3 2]
> Rank( R ) eq Dimension( R );
true
The order of the group of Lie type with root datum R over
the field of cardinality q.
The factored order of the group of Lie type with root datum R over the field
of order q.
As well as accepting a specific prime power, these functions also take an
indeterminate so that the generic order formula can be computed.
> P<q> := PolynomialRing( Integers() );
> R := RootDatum( "F4" );
> GroupOfLieTypeFactoredOrder( R, q );
[
<q^4 + 1, 1>,
<q, 24>,
<q^2 - q + 1, 2>,
<q^4 - q^2 + 1, 1>,
<q - 1, 8>,
<q^2 + 1, 2>,
<q + 1, 4>,
<q^2 + q + 1, 2>
]
>
> R := RootDatum( "B2" );
> ord := GroupOfLieTypeOrder( R, q );
> for r in [2..20] do
> if IsPrimePower( r ) then
> print r, Evaluate( ord, r ) eq GroupOfLieTypeOrder( R, r );
> end if;
> end for;
2 true
3 true
4 true
5 true
7 true
8 true
9 true
11 true
13 true
16 true
17 true
19 true
The fundamental group Lambda/Z Phi of
the root datum R together with the projection
Lambda -> Lambda/Z Phi.
See Subsection Isogeny.
The isogeny group X/Z Phi of the root datum R
together with the injection X/Z Phi -> Lambda/Z Phi.
See Subsection Isogeny.
The coisogeny group Y/Z Phi^star of the root datum
R together with the projection Y/Z Phi^star -> Lambda/Z Phi.
See Subsection Isogeny.
In the semisimple case, the fundamental group is a direct sum of the
isogeny group and the coisogeny group.
> R := RootDatum( "A5" : Isogeny := 3 );
> F := FundamentalGroup( R );
> G := IsogenyGroup( R );
> H := CoisogenyGroup( R );
> #G * #H eq #F;
true
Nonsemisimple root data have infinite isogeny groups.
> R := StandardRootDatum( "A", 5 );
> IsogenyGroup( R );
Abelian Group isomorphic to Z
Defined on 1 generator (free)
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