73 research outputs found
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
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
Implementing Line-Hermitian Grassmann codes
In [I. Cardinali and L. Giuzzi. Line Hermitian Grassmann codes and their
parameters. Finite Fields Appl., 51: 407-432, 2018] we introduced line
Hermitian Grassmann codes and determined their parameters. The aim of this
paper is to present (in the spirit of [I. Cardinali and L. Giuzzi. Enumerative
coding for line polar Grassmannians with applications to codes. Finite Fields
Appl., 46:107-138, 2017]) an algorithm for the point enumerator of a line
Hermitian Grassmannian which can be usefully applied to get efficient encoders,
decoders and error correction algorithms for the aforementioned codes.Comment: 26 page
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
Regular partitions of half-spin geometries
We describe several families of regular partitions of half-spin geometries and determine their associated parameters and eigenvalues. We also give a general method for computing the eigenvalues of regular partitions of half-spin geometries
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
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
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