Polar Spaces More Generally

In V2.29 (September 2025), in order to support more functionality for polar spaces, a category SpcPlr has been added, and one can create a polar spaces also in this category.

Creation of PolarSpaces

AmbientPolarSpace(J) : AlgMatElt -> SpcPlr

AmbientPolarSpace(J, a) : AlgMatElt, FldAut -> SpcPlr

Builds a polar space with respect to the matrix J and the field automorphism a.

PolarSpace(L) : LatNF -> SpcPlr

PolarSpace(L) : Lat -> SpcPlr

AmbientSpace(L) : LatNF -> SpcPlr

The polar space associated to the lattice L.

ChangeRing(V, R) : SpcPlr, Rng -> SpcPlr

The polar space obtained from V by base change to R.

Properties of Polar Spaces

BaseField(V) : SpcPlr -> Fld

BaseRing(V) : SpcPlr -> Fld

CoefficientField(V) : SpcPlr -> Fld

CoefficientRing(V) : SpcPlr -> Fld

The field over which V is defined.

VectorSpace(V) : SpcPlr -> ModTupFld

KSpace(V) : SpcPlr -> ModTupFld

The underlying vector space.

Dimension(V) : SpcPlr -> RngIntElt

The dimension of V.

InnerForm(V) : SpcPlr -> AlgMatElt

The inner form associated to V.

Involution(V) : SpcPlr -> FldAut

The involution with respect to which V is hermitian or skew-hermitian.

SpaceType(V) : SpcPlr -> MonStgElt

The type of the polar space V, returned as a string.

Diagonal(V) : SpcPlr -> AlgMatElt

The coefficients of the diagonalized form.

Predicates on Polar Spaces

IsDefinite(V) : SpcPlr -> BoolElt

Whether V space is totally positive definite, or totally negative definite.