64 research outputs found
The generating rank of the unitary and symplectic Grassmannians
We prove that the Grassmannian of totally isotropic -spaces of the polar
space associated to the unitary group () has generating rank when . We also reprove the main result of Blok [Blok2007], namely that
the Grassmannian of totally isotropic -spaces associated to the symplectic
group has generating rank , when
On the Generation of Dual Polar Spaces of Symplectic Type over Finite Fields
AbstractIt is demonstrated that the dual polar space of typeSp2n(q),q>2, can be generated as a geometry by[formula]points
Highest weight modules and polarized embeddings of shadow spaces
Let Gamma be the K-shadow space of a spherical building Delta. An embedding V
of Gamma is called polarized if it affords all "singular" hyperplanes of Gamma.
Suppose that Delta is associated to a Chevalley group G. Then Gamma can be
embedded into what we call the Weyl module for G of highest weight lambda_K. It
is proved that this module is polarized and that the associated minimal
polarized embedding is precisely the irreducible G-module of highest weight
lambda_K. In addition a number of general results on polarized embeddings of
shadow spaces are proved. The last few sections are devoted to the study of
specific shadow spaces, notably minuscule weight geometries, polar
grassmannians, and projective flag-grassmannians. The paper is in part
expository in nature so as to make this material accessible to a wide audience.Comment: Improvement in exposition of Sections 1-3 and . Notation improved.
References added. Main results unchange
The hyperplanes of finite symplectic dual polar spaces which arise from projective embeddings
AbstractWe characterize the hyperplanes of the dual polar space DW(2n−1,q) which arise from projective embeddings as those hyperplanes H of DW(2n−1,q) which satisfy the following property: if Q is an ovoidal quad, then Q∩H is a classical ovoid of Q. A consequence of this is that all hyperplanes of the dual polar spaces DW(2n−1,4), DW(2n−1,16) and DW(2n−1,p) (p prime) arise from projective embeddings
The generating rank of the unitary and symplectic Grassmannians
AbstractWe prove that the Grassmannian of totally isotropic k-spaces of the polar space associated to the unitary group SU2n(F) (n∈N) has generating rank (2nk) when F≠F4. We also reprove the main result of Blok (2007) [3], namely that the Grassmannian of totally isotropic k-spaces associated to the symplectic group Sp2n(F) has generating rank (2nk)−(2nk−2), when Char(F)≠2
Minimum distance of Symplectic Grassmann codes
We introduce the Symplectic Grassmann codes as projective codes defined by
symplectic Grassmannians, in analogy with the orthogonal Grassmann codes
introduced in [4]. Note that the Lagrangian-Grassmannian codes are a special
class of Symplectic Grassmann codes. We describe the weight enumerator of the
Lagrangian--Grassmannian codes of rank and and we determine the minimum
distance of the line Symplectic Grassmann codes.Comment: Revised contents and biblograph
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