Given a finite dimensional vector space V over a field and a non-trivial but possibly degenerate reflexive sesquilinear form , let P(f) be the polar space associated to f, namely the collection of all totally f-isotropic subspaces of V, and R(f) the collection of all subspaces of V that meet non-trivially. Let r be the Witt index of f. Given a positive integer , let be the k-Grassmannian of V, namely the collection of all k-subspaces of V, and put and . Let be the Plücker embedding, mapping a subspace onto the point . The image of via is a well-known projective variety, called a Grassmann variety. Put and . In this paper, we find a tensor equation that, jointly with the equations of , describe as a subset of . The last two sections of the paper are devoted to the following two special cases: (1) f is bilinear and ; (2) f is alternating.
[1]
I. Cardinali,et al.
Line polar Grassmann codes of orthogonal type
,
2014,
1407.6149.
[2]
Jesús Carrillo-Pacheco,et al.
On Lagrangian–Grassmannian codes
,
2011,
Des. Codes Cryptogr..
[3]
Ilaria Cardinali,et al.
Codes and caps from orthogonal Grassmannians
,
2013,
Finite Fields Their Appl..
[4]
Ilaria Cardinali,et al.
Minimum distance of Symplectic Grassmann codes
,
2015,
ArXiv.
[5]
Antonio Pasini.
Diagram Geometry
,
1994
.
[6]
P. M. Cohn.
GROUPES ET ALGÉBRES DE LIE
,
1977
.
[7]
Antonio Pasini,et al.
Embeddings of Line-Grassmannians of Polar Spaces in Grassmann Varieties
,
2013,
1303.5609.
[8]
P. Dienes,et al.
On tensor geometry
,
1926
.
[9]
Jacques Tits,et al.
Buildings of Spherical Type and Finite BN-Pairs
,
1974
.