MAGMA Computational Algebra System

Subgroups of Small Rank

LiE contains a small database with the types of the maximal proper subgroups of complex reductive simply connected Lie groups g, where g is simple and of rank at most 8. We copied this list into Magma, and it can be accessed using the following functions.

LiEMaximalSubgroups() : -> SeqEnum

All maximal subgroups as described above, as a sequence of pairs. Each pair consists of a string denoting the simple group at hand, and a sequence of strings denoting its maximal subgroups.

MaximalSubgroups(G) : MonStgElt -> SeqEnum[MonStgElt]

The maximal subgroups of the complex reductive simply connected simple Lie group whose Cartan type is the string G, represented as a sequence of strings.

RestrictionMatrix(G, H) : MonStgElt, MonStgElt -> AlgMatElt

    Index: RngIntElt                    Default: -1

The restriction matrix for the maximal proper subgroup of type H of G. If more than one maximal subgroup of G is of type H, the parameter Index must be set to indicate which one is required.

Example LieReps_LieRepDecDBEx (H97E18)

> MaximalSubgroups("E7");
[ A2, A1, A1, A1F4, G2C3, A1G2, A1A1, D6A1, A7, A5A2 ]
> M := RestrictionMatrix("E7", "A1");
More than one subgroup of E7 of type A1 exists; Please specify which
one you want using the Index parameter
> M := RestrictionMatrix("E7", "A1" : Index := 2); M;
[26]
[37]
[50]
[72]
[57]
[40]
[21]
> R := RootDatum("E7" : Isogeny := "SC");
> S := RootDatum("A1" : Isogeny := "SC");
> D := AdjointRepresentationDecomposition(R);
> RepresentationDimension(D);
133
> E := Branch(S, D, M); #E;
8
> RepresentationDimension(E);
133