arXiv subjects

Antonio Pasini

Publications and source records attributed to Antonio Pasini.

At least 19 recordsLinked to original sources

arXiv2026

Bijection Between Point-Hyperplane Anti-Flags of $V(n, 2)$ and Non-Singular Points of $O^+(2n, 2)$

We give a bijection between the point-hyperplane antiflags of $V(n, 2)$ and the nonsingular points of $V(2n, \allowbreak 2)$ with respect to a hyperbolic quadric. With the help of this bijection, we give a description of the strongly regular graph $NO^+_{2n}(2)$ in $V(2n, 2)$. We also describe a graph with respect to a hyperbolic quadric in $V(2n, 2)$ that was recently defined by Stanley and Takeda in $V(n, 2)$. Similarly, we give a bijection between the point-hyperplane antiflags of $V(n, 3)$ and the nonsingular points of one type in $V(2n, 3)$ with respect to a hyperbolic quadric.

math.CO↗

arXiv2026

Four relations on the set of point-hyperplane anti-flags

There are precisely four arrangements of two point-hyperplane anti-flags. We consider the corresponding relations on the set of such anti-flags and show that each of them can be recovered from any other except in one special case. If the field consists of two elements, then one of the relations cannot be used to recover each of the remaining three. This is related to a bijection between anti-flags and exterior points of the hyperbolic polar space which exists in this case.

math.CO↗

arXiv2025

On the $1$-cohomology of $\mathrm{SL}(n,{\mathbb K})$ on the dual of its adjoint module

Given a field $\mathbb K$, for any $n\geq 3$ the first cohomology group $H^1(G_n,A^*_n)$ of the special linear group $G_n = \mathrm{SL}(n,{\mathbb K})$ over the dual $A^*_n$ of its adjoint module $A_n$ is isomorphic to the space $\mathrm{Der}({\mathbb K})$ of the derivations of $\mathbb K$, except possibly when $|{\mathbb K}| \in \{2, 4\}$ and $n$ is even. This fact is stated by S. Smith and H. Völklein in their paper "A geometric presentation for the adjont module of $\mathrm{SL}_3(k)$" (J. Algebra 127 (1989), 127--138). They claim that when $|{\mathbb K}| > 9$ this fact follows from the main result of Völklein's paper "The 1-cohomology of the adjoint module of a Chevalley group" (Forum Math. 1 (1989), 1--13), but say nothing that can help the reader to deduce it from that result. When $|{\mathbb K}| \leq 9$ they obtain the isomorphism $H^1(G_n,A^*_n) \cong \mathrm{Der}({\mathbb K})$ by means of other results from homological algebra, which however miss the case $|{\mathbb K}| \in\{2, 4\}$ with $n $ even. In the present paper we shall provide a straightforward proof of the isomorphism $H^1(G_n,A^*_n) \cong \mathrm{Der}({\mathbb K})$ under the hypothesis $n > 3$. Our proof also covers the above mentioned missing case.

math.GR↗

arXiv2023

Geometric hyperplanes of the Lie geometry $A_{n,\{1,n\}}(\mathbb{F})$

In this paper we investigate hyperplanes of the point-line geometry $\mathit{A}_{n,\{1,n\}}(\mathbb{F})$ of point-hyerplane flags of the projective geometry $\mathrm{PG}(n,\mathbb{F})$. Renouncing a complete classification, which is not yet within our reach, we describe the hyperplanes which arise from the natural embedding of $\mathit{A}_{n,\{1,n\}}(\mathbb{F})$, that is the embedding which yields the adjoint representation of $\mathrm{SL}(n+1,\mathbb{F})$. The information we shall collect on these hyperplanes will allow us to prove that all hyperplanes of $\mathit{A}_{n,\{1,n\}}(\mathbb{F})$ are maximal subspaces of $\mathit{A}_{n,\{1,n\}}(\mathbb{F})$. Hyperplanes of $\mathit{A}_{n,\{1,n\}}(\mathbb{F})$ can also be contructed starting from suitable line-spreads of $\mathrm{PG}(n,\mathbb{F})$ (provided that $\mathrm{PG}(n,\mathbb{F})$ admits line-spreads, of course). Explicitly, let $\mathfrak{S}$ be a line-spread of $\mathrm{PG}(n,\mathbb{K})$ satisfying certain conditions to be stated in this paper (which hold for all line-spreads obtained via the most popular constructions). The set of point-hyperplane flags $(p,\mathit{H})$ of $\mathrm{PG}(n,\mathbb{F})$ such that $\mathit{H}$ contains the member of $\mathfrak{S}$ through the point $p$ is a hyperplane of $\mathit{A}_{n,\{1,n\}}(\mathbb{F})$. We call these hyperplanes {\em hyperplanes of spread type}. Many of them arise from the natural embedding. We don't know if this is the case for all of them.

math.CO↗

arXiv2023

Embeddings and hyperplanes of the Lie incidence geometry $A_{n,\{1,n\}}(\mathbb{F})

In this paper we consider a family of projective embeddings of the geometry $Γ= A_{n,\{1,n\}}(F)$ of point-hyperplanes flags of the projective geometry $Σ= PG(n,F)$. The natural embedding $\varepsilon_{mathrm{nat}}$ is one of them. It maps every point-hyperplane flag $(p,H)$ of $Σ$ onto the vector-line $\langle x\otimesξ\rangle$, where $x$ is a representative vector of $p$ and $ξ$ is a linear functional describing $H$. The other embeddings have been discovered by Thas and Van Maldeghem (2000) for the case $n = 2$ and later generalized to any $n$ by De Schepper, Schillewaert and Van Maldeghem (2023). They are obtained as twistings of $\varepsilon_{\mathrm{nat}}$ by non-trivial automorphisms of $F$. Explicitly, for $σ\in Aut(F)\setminus\{\mathrm{id}_F\}$, the twisting $\varepsilon_σ$ of $\varepsilon_{\mathrm{nat}}$ by $σ$ maps $(p,H)$ onto $\langle xσ\otimes ξ\rangle$. We shall prove that, when $|Aut(F)| > 1$ a geometric hyperplane $\cal H$ of $Γ$ arises from $\varepsilon_{\mathrm{nat}}$ and one of its twistings or from two distinct twistings of $\varepsilon_{\mathrm{nat}}$ if and only if ${\cal H} = \{(p,H)\in Γ\mid p\in A \mbox{ or } a \in H\}$ for a possibly non-incident point-hyperplane pair $(a,A)$ of $Σ$. We call these hyperplanes quasi-singular hyperplanes. With the help of this result we shall prove that if $|Aut(F)| > 1$ then $Γ$ admits no absolutely universal embedding.

math.CO↗

arXiv2023

Regularity in polar spaces of infinite rank

In this paper we propose a definition of regularity suited for polar spaces of infinite rank and we investigate to which extent properties of regular polar spaces of finite rank can be generalized to polar spaces of infinite rank.

math.CO↗

arXiv2022

Characterizations of symplectic polar spaces

A polar space S is said to be symplectic if it admits an embedding e in a projective geometry PG(V) such that the e-image e(S) of S is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially when S admits different (non isomorphic) embeddings, as it is the case (precisely) when S is defined over a field of characteristic 2.

math.SG↗

arXiv2022

On the generation of some Lie-type geometries

Let $X_n(K)$ be a building of Coxeter type $X_n = A_n$ or $X_n = D_n$ defined over a given division ring $K$ (a field when $X_n = D_n$). For a non-connected set $J$ of nodes of the diagram $X_n$, let $Γ(K) = Gr_J(X_n(K))$ be the $J$-Grassmannian of $X_n(K)$. We prove that $Γ(K)$ cannot be generated over any proper sub-division ring $K_0$ of $K$. As a consequence, the generating rank of $Γ(K)$ is infinite when $K$ is not finitely generated. In particular, if $K$ is the algebraic closure of a finite field of prime order then the generating rank of $Gr_{1,n}(A_n(K))$ is infinite, although its embedding rank is either $(n+1)^2-1$ or $(n+1)^2$.

math.CO↗

arXiv2021

Computations regarding certain graphs associated to finite polar spaces

We consider various regular graphs defined on the set of elements of given rank of a finite polar space. It is likely that no two such graphs, of the same kind but defined for different ranks, can have the same degree. We shall prove this conjecture under the hypothesis that the considered rank are not too small.

math.CO↗

arXiv2020

Nearly all subspaces of a classical polar space arise from its universal embedding

Let $Γ$ be an embeddable non-degenerate polar space of finite rank $n \geq 2$. Assuming that $Γ$ admits the universal embedding (which is true for all embeddable polar spaces except grids of order at least $5$ and certain generalized quadrangles defined over quaternion division rings), let $\varepsilon:Γ\to\mathrm{PG}(V)$ be the universal embedding of $Γ$. Let $\cal S$ be a subspace of $Γ$ and suppose that $\cal S$, regarded as a polar space, has non-degenerate rank at least $2$. We shall prove that $\cal S$ is the $\varepsilon$-preimage of a projective subspace of $\mathrm{PG}(V)$.

math.RT↗

arXiv2019

Sets of generators and chains of subspaces

The rank of a point-line geometry G is usually defined as the generating rank of G, namely the minimal cardinality of a generating set. However, when the subspace lattice of G satisfies the Exchange Property we can also try a different definition: consider all chains of subspaces of G and take the least upper bound of their lengths as the rank of G. If G is finitely generated then these two definitions yield the same number. On the other hand, as we shall show in this paper, if infinitely many points are needed to generate G then the rank as defined in the latter way is often (perhaps always) larger than the generating rank. So, if we like to keep the first definition we should accordingly discard the second one or modify it. We can modify it as follows: consider only well ordered chains instead of arbitrary chains. As we shall prove, the least upper bound of the lengths of well ordered chains of subspaces is indeed equal to the generating rank. According to this result, the (possibly infinite) rank of a polar space can be characterized as the least upper bound of the lengths of well ordered chains of singular subspaces; referring to arbitrary chains would be an error.

math.CO↗

arXiv2019

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 $2$ 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.

math.AG↗

arXiv2019

The generating rank of a polar Grassmannian

In this paper we compute the generating rank of $k$-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of $k$-Grassmannians arising from Hermitian forms of Witt index $n$ defined over vector spaces of dimension $N > 2n$. We also study generating sets for the $2$-Grassmannians arising from quadratic forms of Witt index $n$ defined over $V(N,{\mathbb F}_q)$ for $q=4,8,9$ and $2n \leq N \leq 2n+2$. We prove that for $N >6$ they can be generated over the prime subfield, thus determining their generating rank.

math.RT↗

arXiv2018

On two non-building but simply connected compact Tits geometries of type C3

A classification of homogeneous compact Tits geometries of irreducible spherical type, with connected panels and admitting a compact flag-transitive automorphism group acting continuously on the geometry, has been obtained by Kramer and Lytchak (Homogeneous compact geometries, Transform. Groups 19 (2016), 43-58 and Erratum to: Homogeneous compact geometries, Transform. Groups, to appear). According to their main result, all such geometries but two are quotients of buildings. The two exceptions are flat geometries of type C3 and arise from polar actions on the Cayley plane over the division algebra of real octonions. The classification obtained by Kramer and Lytchak does not contain the claim that those two exceptional geometries are simply connected, but this holds true, as proved by Schillewaert and Struyve (On exceptional homogeneous compact geometries of type C3, Groups Geome. Dyn. 11 (2017), 1377-1399). The proof by Schillewaert and Struyve is of topological nature and relies on the main result of Kramer and Lytchak. In this paper we provide a combinatorial proof of that claim, independent of Kramer and Lytchak's result.

math.GT↗

arXiv2018

On transparent embeddings of point-line geometries

We introduce the class of transparent embeddings for a point-line geometry $Γ= ({\mathcal P},{\mathcal L})$ as the class of full projective embeddings $\varepsilon$ of $Γ$ such that the preimage of any projective line fully contained in $\varepsilon({\mathcal P})$ is a line of $Γ$. We will then investigate the transparency of Plücker embeddings of projective and polar grassmannians and spin embeddings of half-spin geometries and dual polar spaces of orthogonal type. As an application of our results on transparency, we will derive several Chow-like theorems for polar grassmannians and half-spin geometries.

math.AG↗

arXiv2017

The graphs of projective codes

Consider the Grassmann graph formed by $k$-dimensional subspaces of an $n$-dimensional vector space over the field of $q$ elements ($1<k<n-1$) and denote by $Π(n,k)_q$ the restriction of this graph to the set of projective $[n,k]_q$ codes. In the case when $q\ge \binom{n}{2}$, we show that the graph $Π(n,k)_q$ is connected, its diameter is equal to the diameter of the Grassmann graph and the distance between any two vertices coincides with the distance between these vertices in the Grassmann graph. Also, we give some observations concerning the graphs of simplex codes. For example, binary simplex codes of dimension $3$ are precisely maximal singular subspaces of a non-degenerate quadratic form.

math.CO↗

arXiv2016

A geometric approach to alternating $k$-linear forms

Given an $n$-dimensional vector space $V$ over a field $\mathbb K$, let $2\leq k < n$. There is a natural correspondence between the alternating $k$-linear forms $φ$ of $V$ and the linear functionals $f$ of $\bigwedge^kV$. Let $\varepsilon_k:{\mathcal G}_k(V)\rightarrow {\mathrm{PG}}(\bigwedge^kV)$ be the Plucker embedding of the $k$-Grassmannian ${\mathcal G}_k(V)$ of $V$. Then $\varepsilon_k^{-1}(\ker(f)\cap\varepsilon_k(\mathcal{G}_k(V)))$ is a hyperplane of the point-line geometry ${\mathcal G}_k(V)$. All hyperplanes of ${\mathcal G}_k(V)$ can be obtained in this way. For a hyperplane $H$ of ${\mathcal G}_k(V)$, let $R^\uparrow(H)$ be the subspace of ${\mathcal G}_{k-1}(V)$ formed by the $(k-1)$-subspaces $A\subset V$ such that $H$ contains all $k$-subspaces that contain $A$. In other words, if $φ$ is the (unique modulo a scalar) alternating $k$-linear form defining $H$, then the elements of $R^\uparrow(H)$ are the $(k-1)$-subspaces $A = \langle a_1,\ldots, a_{k-1}\rangle$ of $V$ such that $φ(a_1,\ldots, a_{k-1},x) = 0$ for all $x\in V$. When $n-k$ is even it might be that $R^\uparrow(H) = \emptyset$. When $n-k$ is odd, then $R^\uparrow(H) \neq \emptyset$, since every $(k-2)$-subspace of $V$ is contained in at least one member of $R^\uparrow(H)$. If every $(k-2)$-subspace of $V$ is contained in precisely one member of $R^\uparrow(H)$ we say that $R^\uparrow(H)$ is spread-like. In this paper we obtain some results on $R^\uparrow(H)$ which answer some open questions from the literature and suggest the conjecture that, if $n-k$ is even and at least $4$, then $R^\uparrow(H) \neq \emptyset$ but for one exception with ${\mathbb K}\leq{\mathbb R}$ and $(n,k) = (7,3)$, while if $n-k$ is odd and at least $5$ then $R^\uparrow(H)$ is never spread-like.

math.AG↗

arXiv2015

Equations for polar grassmannians

Given an $N$-dimensional vector space $V$ over a field $\mathbb{F}$ and a trace-valued $(σ,\varepsilon)$-sesquilinear form $f:V\times V\rightarrow \mathbb{F}$, with $\varepsilon = \pm 1$ and $σ^2 = \mathrm{id}_{\mathbb{F}}$, let ${\cal S}$ be the polar space of totally $f$-isotropic subspaces of $V$ and let $n$ be the rank of ${\cal S}$. Assuming $n \geq 2$, let $2 \leq k \leq n$, let ${\cal G}_k$ the $k$-grassmannian of $\mathrm{PG}(V)$, embedded in $\mathrm{PG}(\wedge^kV)$ as a projective variety and ${\cal S}_k$ the $k$-grassmannian of $\cal S$. In this paper we find one simple equation that, jointly with the equations of ${\cal G}_k$, describe ${\cal S}_k$ as a subset of $\mathrm{PG}(\wedge^kV)$.

math.AG↗