383 research outputs found
Veronesean embeddings of dual polar spaces of orthogonal type
Given a point-line geometry P and a pappian projective space S,a veronesean
embedding of P in S is an injective map e from the point-set of P to the set of
points of S mapping the lines of P onto non-singular conics of S and such that
e(P) spans S. In this paper we study veronesean embeddings of the dual polar
space \Delta_n associated to a non-singular quadratic form q of Witt index n >=
2 in V = V(2n + 1; F). Three such embeddings are considered,namely the
Grassmann embedding gr_n,the composition vs_n of the spin (projective)
embedding of \Delta_n in PG(2n-1; F) with the quadric veronesean map of V(2n;
F) and a third embedding w_n defined algebraically in the Weyl module V
(2\lambda_n),where \lambda_n is the fundamental dominant weight associated to
the n-th simple root of the root system of type Bn. We shall prove that w_n and
vs_n are isomorphic. If char(F) is different from 2 then V (2\lambda_n) is
irreducible and w_n is isomorphic to gr_n while if char(F) = 2 then gr_n is a
proper quotient of w_n. In this paper we shall study some of these submodules.
Finally we turn to universality,focusing on the case of n = 2. We prove that if
F is a finite field of odd order q > 3 then sv_2 is relatively universal. On
the contrary,if char(F) = 2 then vs_2 is not universal. We also prove that if F
is a perfect field of characteristic 2 then vs_n is not universal,for any n>=2
On certain submodules of Weyl modules for SO(2n+1,F) with char(F) = 2
For let be the Weyl module for the
special orthogonal group G = \mathrm{SO}(2n+1,\F) with respect to the -th
fundamental dominant weight of the root system of type and
put . It is well known that all of these modules are
irreducible when \mathrm{char}(\F) \neq 2 while when \mathrm{char}(\F) = 2
they admit many proper submodules. In this paper, assuming that
\mathrm{char}(\F) = 2, we prove that admits a chain of submodules
where for and is the trivial 1-dimensional
module. We also show that for the quotient is
isomorphic to the so called -th Grassmann module for . Resting on this
fact we can give a geometric description of as a submodule of
the -th Grassmann module. When \F is perfect G\cong \mathrm{Sp}(2n,\F)
and is isomorphic to the Weyl module for \mathrm{Sp}(2n,\F)
relative to the -th fundamental dominant weight of the root system of type
. All irreducible sections of the latter modules are known. Thus, when
\F is perfect, all irreducible sections of are known as well
A geometric approach to alternating -linear forms
Given an -dimensional vector space over a field , let
. There is a natural correspondence between the alternating
-linear forms of and the linear functionals of
. Let be the Plucker embedding of the -Grassmannian
of . Then
is a
hyperplane of the point-line geometry . All hyperplanes of
can be obtained in this way. For a hyperplane of
, let be the subspace of formed by the -subspaces such that
contains all -subspaces that contain . In other words, if is
the (unique modulo a scalar) alternating -linear form defining , then the
elements of are the -subspaces of such that for all
. When is even it might be that . When
is odd, then , since every -subspace
of is contained in at least one member of . If every
-subspace of is contained in precisely one member of
we say that is spread-like. In this paper we obtain some
results on which answer some open questions from the literature
and suggest the conjecture that, if is even and at least , then
but for one exception with and , while if is odd and at least
then is never spread-like.Comment: 29 Page
Grassmann embeddings of polar Grassmannians
In this paper we compute the dimension of the Grassmann embeddings of the
polar Grassmannians associated to a possibly degenerate Hermitian, alternating
or quadratic form with possibly non-maximal Witt index. Moreover, in the
characteristic case, when the form is quadratic and non-degenerate with
bilinearization of minimal Witt index, we define a generalization of the
so-called Weyl embedding (see [I. Cardinali and A. Pasini, Grassmann and Weyl
embeddings of orthogonal Grassmannians. J. Algebr. Combin. 38 (2013), 863-888])
and prove that the Grassmann embedding is a quotient of this generalized
"Weyl-like" embedding. We also estimate the dimension of the latter.Comment: 25 pages/revised version after revie
Line Polar Grassmann Codes of Orthogonal Type
Polar Grassmann codes of orthogonal type have been introduced in I. Cardinali
and L. Giuzzi, \emph{Codes and caps from orthogonal Grassmannians}, {Finite
Fields Appl.} {\bf 24} (2013), 148-169. They are subcodes of the Grassmann code
arising from the projective system defined by the Pl\"ucker embedding of a
polar Grassmannian of orthogonal type. In the present paper we fully determine
the minimum distance of line polar Grassmann Codes of orthogonal type for
odd
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