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: 

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: Maximal Subgroups from the Database (ex-23802f)

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Using the subgroup database:

> MaximalSubgroups("E7");
[ A2, A1, A1, A1F4, G2C3, A1G2, A1A1, D6A1, A7, A5A2 ]
> 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

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