arXiv subjects

Antonio Pasini

Publications and source records attributed to Antonio Pasini.

25 records · Page 2Linked to original sources

arXiv2014

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 Plücker 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 $q$ odd.

math.CO↗

arXiv2014

Embedded polar spaces revisited

In this paper we introduce generalized pseudo-quadratic forms and develope some theory for them. Recall that the codomain of a $(σ,\varepsilon)$-quadratic form is the group $\overline{K} := K/K_{σ,\varepsilon}$, where $K$ is the underlying division ring of the vector space on which the form is defined and $K_{σ,\varepsilon} := \{t-t^σ\varepsilon\}_{t\in K}$. Generalized pseudo-quadratic forms are defined in the same way as $(σ,\varepsilon)$-quadratic forms but for replacing $\overline{K}$ with a quotient $\overline{K}/\overline{R}$ for a subgroup $\overline{R}$ of $\overline{K}$ such that $λ^σ\overline{R}λ= \overline{R}$ for any $λ\in K$. In particular, every non-trivial generalized pseudo-quadratic form admits a unique sesquilinearization, characterized by the same property as the sesquilinearization of a pseudo-quadratic form. Moreover, if $q:V\rightarrow \overline{K}/\overline{R}$ is a non-trivial generalized pseudo-quadratic form and $f:V\times V\rightarrow K$ is its sesquilinarization, the points and the lines of $\mathrm{PG}(V)$ where $q$ vanishes form a subspace $S_q$ of the polar space $S_f$ associated to $f$. After a discussion of quotients and covers of generalized pseudo-quadratic forms we prove the following: let $e:S\rightarrow \mathrm{PG}(V)$ be a projective embedding of a non-degenerate polar space $S$ of rank at least $2$; then $e(S)$ is either the polar space $S_q$ associated to a generalized pseudo-quadratic form $q$ or the polar space $S_f$ associated to an alternating form $f$. By exploiting this theorem we also obtain an elementary proof of the following well known fact: an embedding $e$ as above is dominant if and only if either $e(S) = S_q$ for a pseudo-quadratic form $q$ or $\mathrm{char}(K)\neq 2$ and $e(S) = S_f$ for an alternating form $f$.

math.RT↗

arXiv2013

Unipotent representations of Lie incidence geometries

If a geometry $Γ$ is isomorphic to the residue of a point $A$ of a shadow geometry of a spherical building $Δ$, a representation $\varepsilon_Δ^A$ of $Γ$ can be given in the unipotent radical $U_{A^*}$ of the stabilizer in $\mathrm{Aut}(Δ)$ of a flag $A^*$ of $Δ$ opposite to $A$, every element of $Γ$ being mapped onto a suitable subgroup of $U_{A^*}$. We call such a representation a unipotent representation. We develope some theory for unipotent representations and we examine a number of interesting cases, where a projective embedding of a Lie incidence geometry $Γ$ can be obtained as a quotient of a suitable unipotent representation $\varepsilon_Δ^A$ by factorizing over the derived subgroup of $U_{A^*}$, while $\varepsilon^A_Δ$ itself is not a proper quotient of any other representation of $Γ$.

math.GR↗

arXiv2013

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

For $k = 1, 2,...,n-1$ let $V_k = V(λ_k)$ be the Weyl module for the special orthogonal group $G = \mathrm{SO}(2n+1,\F)$ with respect to the $k$-th fundamental dominant weight $λ_k$ of the root system of type $B_n$ and put $V_n = V(2λ_n)$. 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.

math.RT↗

arXiv2013

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 Δ_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 Δ_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 (2λ_n),where λ_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 (2λ_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.

math.RT↗

arXiv2013

Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties

An embedding of a point-line geometry Γis usually defined as an injective mapping εfrom the point-set of Γto the set of points of a projective space such that ε(l) is a projective line for every line l of Γ, but different situations have lately been considered in the literature, where ε(l) is allowed to be a subline of a projective line or a curve. In this paper we propose a more general definition of embedding which includes all the above situations and we focus on a class of embeddings, which we call Grassmman embeddings, where the points of Γare firstly associated to lines of a projective geometry PG(V), next they are mapped onto points of PG(V\wedge V) via the usual projective embedding of the line-grassmannian of PG(V) in PG(V\wedge V). In the central part of our paper we study sets of points of PG(V\wedge V) corresponding to lines of PG(V) totally singular for a given pseudoquadratic form of V. Finally, we apply the results obtained in that part to the investigation of Grassmann embeddings of several generalized quadrangles.

math.AG↗

arXiv2007

Intransitive geometries and fused amalgams

We study geometries that arise from the natural $G_2(K)$ action on the geometry of one-dimensional subspaces, of nonsingular two-dimensional subspaces, and of nonsingular three-dimensional subspaces of the building geometry of type $C_3(K)$ where $K$ is a perfect field of characteristic 2. One of these geometries is intransitive in such a way that the non-standard geometric covering theory by the first and the last author is not applicable. In this paper we introduce the concept of fused amalgams in order to extend the geometric covering theory so that it applies to that geometry. This yields an interesting new amalgamation result for the group $G_2(K)$.

math.GR↗