Properties of Reductive Groups

ConnectedComponent(G): GrpRed -> GrpRed

The connected component of the identity, \(G^{\circ}\).

ComponentGroup(G): GrpRed -> GrpRed

The component group of \(G\).

SplittingField(G): GrpRed -> Fld

The field over which \(G\) splits.

FieldOfDefinition(G): GrpRed -> Fld

BaseRing(G): GrpRed -> Fld

The field of definition of \(G\).

CartanName(G): GrpRed -> MonStgElt

A string concatenating the Cartan types of all simple factors and tori of the connected component of \(G\).

InnerForm(G, i): GrpRed, RngIntElt -> SpcPlr

The inner form corresponding to the \(i\)-th simple factor.

InnerForms(G): GrpRed -> [ SpcPlr ]

The inner forms defining the simple factors of \(G\).

Dimension(G): GrpRed -> RngIntElt

Degree(G): GrpRed -> RngIntElt

Returns \(n\) such that \(G\) embeds into \({\rm GL}_n\) through the representation via its inner forms.

Rank(G): GrpRed -> RngIntElt

The rank of \(G\).

Predicates

IsConnected(G): GrpRed -> BoolElt

Returns true iff \(G\) is connected.

IsOrthogonal(G): GrpRed -> BoolElt

Returns true iff \(G\) is an orthogonal group.

IsSpecialOrthogonal(G): GrpRed -> BoolElt

Returns true iff \(G\) is a special orthogonal group.

IsSymplectic(G): GrpRed -> BoolElt

Returns true iff \(G\) is a symplectic group.

IsUnitary(G): GrpRed -> BoolElt

Returns true iff \(G\) is a unitary group.

IsCompact(G): GrpRed -> BoolElt

Returns true iff \(G\) is a compact form. Only relevant for groups defined over number fields.