Creation of Reductive Groups

In the current implementation only certain reductive groups are supported. In particular, the component group is assumed to give rise to a split extension, and if \(G^{\circ} \simeq G_1 \times \cdots \times G_n\) is a product of almost-simple groups, the only inner forms \(G_1, \ldots G_n\) that are supported are either orthogonal or hermitian, over a number field.

In future releases it will be possible to create more reductive groups.

ReductiveGroup(G0, Comp): GrpLie, Grp -> GrpRed

InnerForms: [ AlgMatElt ]                    Default: []

Creates the reductive group with connected component of the identity G0 and component group Comp, such that the extension is split. InnerForms is a list of inner forms, listing for each non-toroidal factor the inner form over the base field.

ReductiveGroup(group_data): List -> GrpRed

Creates the reductive group described by group_data, where group_data is formed as a list of pairs, each pair consisting of a field name and field value. The fields are "INNER_FORMS", "ROOT_DATUM", "BASE_FIELD" and "COMP_GROUP".

The value of the field "BASE_FIELD" should be of type Fld and consist of the field of definition of \(G\). The value of the field "ROOT_DATUM" should be a list of \(4\) pairs, consisting of a field name and field value, where the field names are "SIMPLE_ROOTS", "SIMPLE_COROOTS", "SIGNS" and "TYPES", and their values should be the same as in the constructor for RootDatum, two matrices representing the simple roots and coroots, signs for each extraspecial pair, and Cartan type.

SymplecticGroup(V): SpcPlr -> GrpRed

SymplecticGroup(n, F): RngIntElt, Fld -> GrpRed

SymplecticGroup(Q): AlgMatElt[Fld] -> GrpRed

SymplecticGroup(Q): AlgMatElt[Rng] -> GrpRed

The symplectic group \({\operatorname{Sp}}(V)\) associated to the symplectic space \(V\), the standard symplectic form on \(F^n\) (where \(n\) must be even), or the alternating form \(Q\).

OrthogonalGroup(V): SpcPlr -> GrpRed

OrthogonalGroup(n, F): RngIntElt, Fld -> GrpRed

OrthogonalGroup(Q): AlgMatElt[Fld] -> GrpRed

OrthogonalGroup(Q): AlgMatElt[Rng] -> GrpRed

The orthogonal group \({\rm O}(V)\) associated to the quadratic space \(V\), the standard split quadratic form on \(F^n\), or the symmetric form \(Q\).

SpecialOrthogonalGroup(V): SpcPlr -> GrpRed

SpecialOrthogonalGroup(n, F): RngIntElt, Fld -> GrpRed

SpecialOrthogonalGroup(Q): AlgMatElt[Fld] -> GrpRed

SpecialOrthogonalGroup(Q): AlgMatElt[Rng] -> GrpRed

The special orthogonal group \({\operatorname{SO}}(V)\) associated to the quadratic space \(V\), the standard split quadratic form on \(F^n\), or the symmetric form \(Q\).

UnitaryGroup(V): SpcPlr -> GrpRed

UnitaryGroup(n, F): RngIntElt, Fld -> GrpRed

UnitaryGroup(Q): AlgMatElt[Fld] -> GrpRed

UnitaryGroup(Q): AlgMatElt[Rng] -> GrpRed

The unitary group \(U(V)\) associated to the hermitian space \(V\), the standard hermitian form on \(F^n\), or the hermitian form \(Q\).