On certain submodules of Weyl modules for SO(2n+1,F) with char(F) = 2

For $k = 1, 2,...,n-1$ let $V_k = V(\lambda_k)$ be the Weyl module for the special orthogonal group G = \mathrm{SO}(2n+1,\F) with respect to the $k$-th fundamental dominant weight $\lambda_k$ of the root system of type $B_n$ and put $V_n = V(2\lambda_n)$. 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 $V_k$ admits a chain of submodules $V_k = M_k \supset M_{k-1}\supset ... \supset M_1\supset M_0 \supset M_{-1} = 0$ where $M_i \cong V_i$ for $1,..., k-1$ and $M_0$ is the trivial 1-dimensional module. We also show that for $i = 1, 2,..., k$ the quotient $M_i/M_{i-2}$ is isomorphic to the so called $i$-th Grassmann module for $G$. Resting on this fact we can give a geometric description of $M_{i-1}/M_{i-2}$ as a submodule of the $i$-th Grassmann module. When \F is perfect G\cong \mathrm{Sp}(2n,\F) and $M_i/M_{i-1}$ is isomorphic to the Weyl module for \mathrm{Sp}(2n,\F) relative to the $i$-th fundamental dominant weight of the root system of type $C_n$. All irreducible sections of the latter modules are known. Thus, when \F is perfect, all irreducible sections of $V_k$ are known as well

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

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

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 $2$ and $3$ 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