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.
- 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.