Properties of Reductive Groups

Contents

ConnectedComponent(G) : GrpRed -> GrpRed
The connected component of the identity, G.
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 GLn 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.
V2.29, 21 October 2025