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Antonio Pasini

University of Siena / Università di Siena, Dipartmento Ingegneria Dell'Informatione, Emeritus

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Angela Pasquale

University of Bern

University of Perugia

Insubria University

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Papers by Antonio Pasini

Note di Matematica 21

Some old and new constructions of the tilde geometry (the flag transitive connected triple cover ... more Some old and new constructions of the tilde geometry (the flag transitive connected triple cover of the unique generalized quadrangle W(2) of order (2, 2)) are discussed. Using them, we prove some properties of that geometry. In particular, we compute its generating rank, we give an explicit description of its universal projective embedding and we determine its homogeneous embeddings.

On certain submodules of Weyl modules

For k = 1, 2, ..., n−1 let V k = V (λ k) be the Weyl module for the special orthogonal group G = ... more For k = 1, 2, ..., n−1 let V k = V (λ k) be the Weyl module for the special orthogonal group G = SO(2n + 1, F) with respect to the k-th fundamental dominant weight λ k of the root system of type Bn and put Vn = V (2λn). It is well known that all of these modules are irreducible when char(F) = 2 while when char(F) = 2 they admit many proper submodules. In this paper, assuming that char(F) = 2, we prove that V k admits a chain of submodules V k = M k ⊃ M k−1 ⊃ ... ⊃ M1 ⊃ M0 ⊃ M−1 = 0 where Mi ∼ = Vi for 1, ..., k−1 and M0 is the trivial 1-dimensional module. We also show that for i = 1, 2, ..., k the quotient Mi/Mi−2 is isomorphic to the so called i-th Grassmann module for G. Resting on this fact we can give a geometric description of Mi−1/Mi−2 as a submodule of the i-th Grassmann module. When F is perfect G ∼ = Sp(2n, F) and Mi/Mi−1 is isomorphic to the Weyl module for Sp(2n, F) relative to the i-th fundamental dominant weight of the root system of type Cn. All irreducible sections of the latter modules are known. Thus, when F is perfect, all irreducible sections of V k are known as well.

Veronesean embeddings of dual polar spaces of

Given a point-line geometry Γ and a pappian projective space S, a veronesean embedding of Γ in S ... more Given a point-line geometry Γ and a pappian projective space S, a veronesean embedding of Γ in S is an injective map e from the point-set of Γ to the set of points of S mapping the lines of Γ onto non-singular conics of S and such that e(Γ) spans S. In this paper we study veronesean embeddings of the dual polar space ∆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 ε gr n which maps a maximal singular subspace v1, ..., vn of V (namely a point of ∆n) to the point ∧ n i=1 vi of PG(n V), the composition ε vs n := ν2n • ε spin n of the spin (projective) embedding ε spin n of ∆n in PG(2 n − 1, F) with the quadric veronesean map ν2n : V (2 n , F) → V (2 n +1 2 , F), and a third embeddingεn defined algebraically in the Weyl module V (2λn), where λn is the fundamental dominant weight associated to the n-th simple root of the root system of type Bn. We shall prove thatεn and ε vs n are isomorphic. If char(F) = 2 then V (2λn) is irreducible andεn is isomorphic to ε gr n while if char(F) = 2 then ε gr n is a proper quotient ofεn. 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 ε sv 2 is relatively universal. On the contrary, if char(F) = 2 then ε vs 2 is not universal. We also prove that if F is a perfect field of characteristic 2 then ε vs n is not universal, for any n ≥ 2.

Sets of generators and chains of subspaces

arXiv: Combinatorics, 2019

The rank of a point-line geometry G is usually defined as the generating rank of G, namely the mi... more 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 (possib...

A G ] 1 0 A pr 2 01 8 On transparent embeddings of point-line geometries

We introduce the class of transparent embeddings for a point-line geometry Γ = (P ,L) as the clas... more We introduce the class of transparent embeddings for a point-line geometry Γ = (P ,L) as the class of full projective embeddings ε of Γ such that the preimage of any projective line fully contained in ε(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.

Generation of $J$-Grassmannians of buildings of type $A_n$ and $D_n$ with $J$ a non-connected set of types

Let $X_n(K)$ be a building of Coxeter type $X_n = A_n$ or $X_n = D_n$ defined over a given divisi... more 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 $\Gamma(K) = Gr_J(X_n(K))$ be the $J$-Grassmannian of $X_n(K)$. We prove that $\Gamma(K)$ cannot be generated over any proper sub-division ring $K_0$ of $K$. As a consequence, the generating rank of $\Gamma(K)$ 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 $(n+1)^2$.

The generating rank of a polar Grassmannian

Advances in Geometry, 2021

In this paper we compute the generating rank of k-polar Grassmannians defined over commutative di... more 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, 𝔽 q ) for q = 4, 8, 9 and 2n ≤ N ≤ 2n + 2. We prove that for N > 6 and anisotropic defect (polar corank) d ≠ 2 they can be generated over the prime subfield, thus determining their generating rank.

The graphs of projective codes

Finite Fields and Their Applications, 2018

Consider the Grassmann graph formed by k-dimensional subspaces of an n-dimensional vector space o... more 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 ≥ 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.

On two nonbuilding but simply connected compact Tits geometries of type C3

Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial, 2019

A classification of homogeneous compact Tits geometries of irreducible spherical type, with conne... more A classification of homogeneous compact Tits geometries of irreducible spherical type, with connected panels and admitting a compact flagtransitive automorphism group acting continuously on the geometry, has been obtained by Kramer and Lytchak [5] and [6]. 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 [11]. The proof by Schillewaert and Struyve is of topological nature and relies on the main result of [5] and [6]. In this paper we provide a combinatorial proof of that claim, independent of [5] and [6].

Grassmann embeddings of polar Grassmannians

Journal of Combinatorial Theory, Series A, 2020

In this paper we compute the dimension of the Grassmann embeddings of the polar Grassmannians ass... more 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 nonmaximal 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 [4]) and prove that the Grassmann embedding is a quotient of this generalized 'Weyl-like' embedding. We also estimate the dimension of the latter.

Flag-transitive extensions of dual projective spaces

Bulletin of the Belgian Mathematical Society - Simon Stevin, 1998

We classify the flag-transitive circular extensions of line-point systems of finite projective ge... more

On natural representations of the symplectic group

Bulletin of the Belgian Mathematical Society - Simon Stevin, 2011

Let V k be the Weyl module of dimension (2n k) − (2n k−2) for the group G = Sp(2n, F) arising fro... more Let V k be the Weyl module of dimension (2n k) − (2n k−2) for the group G = Sp(2n, F) arising from the k-th fundamental weight of the Lie algebra of G. Thus, V k affords the grassmann embedding of the k-th symplectic polar grassmannian of the building associated to G. When char(F) = p > 0 and n is sufficiently large compared with the difference n − k, the G-module V k is reducible. In this paper we are mainly interested in the first appearance of reducibility for a given h := n − k. It is known that, for given h and p, there exists an integer n(h, p) such that V k is reducible if and only if n ≥ n(h, p). Moreover, let n ≥ n(h, p) and R k the largest proper non-trivial submodule of V k. Then dim(R k) = 1 if n = n(h, p) while dim(R k) > 1 if n > n(h, p). In this paper we will show how this result can be obtained by an investigation of a certain chain of G-submodules of the exterior power W k := ∧ k V, where V = V(2n, F).

A geometric approach to alternating k-linear forms

Journal of Algebraic Combinatorics, 2016

Denote by G k (V) the Grassmannian of the k-subspaces of a vector space V over a field K. There i... more Denote by G k (V) the Grassmannian of the k-subspaces of a vector space V over a field K. There is a natural correspondence between hyperplanes H of G k (V) and alternating k-linear forms on V defined up to a scalar multiple. Given a hyperplane H of G k (V), we define a subspace R ↑ (H) of G k−1 (V) whose elements are the (k − 1)-subspaces A such that all k-spaces containing A belong to H. When n − k is even, R ↑ (H) might be empty; when n − k is odd, each element of G k−2 (V) is contained in at least one element of R ↑ (H). In the present paper we investigate several properties of R ↑ (H), settle some open problems and propose a conjecture.

Of finite geometries of type with thick lines

Note Di Matematica, 1986

Several facts suggest the conjecture that every finite geometry of type with thick lines is eithe... more Several facts suggest the conjecture that every finite geometry of type with thick lines is either a bulding or flat ( see [4] , [5] , [8] and [9]). That conjecture plays a central role in the problem of classifying finite thick geometries of type and ( see [4] , [7] and [8] ).In this paper we find very simple conditions that hold in a finite geometry of type with thick lines if and only if the geometry is either a building or flat. Then the previous conjecture can be restated so: prove that the conditions discussed in this paper are theorems in the case of finite geometries with thick lines.

Flag-transitive L_h.L*-geometries

Le Matematiche, 2006

Locally partial geometries with different types of residues

Annals of Discrete Mathematics, 1992

Abstract A class of locally partial geometries (L. pGs for short) is constructed where both linea... more Abstract A class of locally partial geometries (L. pGs for short) is constructed where both linear spaces and generalized quadrangles occur as point-residues. We conjecture that this class gives all possible example of L. pGs with that anomaly. We prove this conjecture when plane-residues are projective planes (see section 5). In the general case, we are able to prove the conjecture when there is at least one point with a linear residue, satisfying an additional assumption.

On a series of modules for the symplectic group in characteristic 2

Contemporary Mathematics, 2012

ABSTRACT Let V be a 2n-dimensional vector space defined over an arbitrary field F and G the sympl... more ABSTRACT Let V be a 2n-dimensional vector space defined over an arbitrary field F and G the symplectic group Sp (2n,F) stabilizing a non-degenerate alternating form α(·,·) of V. Let G k be the k-th Grassmannian of PG (V) and Δ k the k-Grassmannian of the C n -building Δ associated to G. Put W k :=∧ k V and let l k :G k →W k be the natural embedding of G k , sending a k-subspace (x 1 ,...,x k ) of V to the 1-subspace (x 1 ∧⋯∧x k ) of W k . Let ε k :Δ k →V k be the embedding of Δ k induced by l k , where V k is the subspace of W k spanned by the l k -images of the totally α-isotropic k-space of V. In [Bull. Belg. Math. Soc. - Simon Stevin 18, No. 1, 1-29 (2011; Zbl 1264.20039)], exploiting the fact that the embedding ε k-2i is universal when char (F)≠2, R. J. Blok and the authors of this paper have proved that if char (F)≠2 then V k-2i (k) /V k-2i+2 (k) and V k-2i are isomorphic as G-modules, for every i=1,...,[k 2]. In the present paper they prove that the same holds true when char (F)=2.

Flag-transitive Extensions of Dual A ne Spaces

Lax Projective Embeddings of Polar Spaces

Milan Journal of Mathematics, 2004

Let Γ be a non-degenerate polar space of rank n ≥ 3 where all of its lines have at least three po... more Let Γ be a non-degenerate polar space of rank n ≥ 3 where all of its lines have at least three points. We prove that, if Γ admits a lax embedding e : Γ → Σ in a projective space Σ defined over a skewfield K, then Γ is a classical and defined over a sub-skewfield K 0 of K. Accordingly, Γ admits a full embedding e 0 in a K 0-projective space Σ 0. We also prove that, under suitable hypotheses on e and e 0 , there exists an embeddingê : Σ 0 → Σ such thatêe 0 = e andê preserves dimensions.

On some flag-transitive extensions of the dual petersen graph

Journal of Combinatorial Theory, Series A, 1995

We consider flag-transitive extensions of the dual Petersen graph satisfying one of the following... more

Note di Matematica 21

Some old and new constructions of the tilde geometry (the flag transitive connected triple cover ... more Some old and new constructions of the tilde geometry (the flag transitive connected triple cover of the unique generalized quadrangle W(2) of order (2, 2)) are discussed. Using them, we prove some properties of that geometry. In particular, we compute its generating rank, we give an explicit description of its universal projective embedding and we determine its homogeneous embeddings.

On certain submodules of Weyl modules

For k = 1, 2, ..., n−1 let V k = V (λ k) be the Weyl module for the special orthogonal group G = ... more For k = 1, 2, ..., n−1 let V k = V (λ k) be the Weyl module for the special orthogonal group G = SO(2n + 1, F) with respect to the k-th fundamental dominant weight λ k of the root system of type Bn and put Vn = V (2λn). It is well known that all of these modules are irreducible when char(F) = 2 while when char(F) = 2 they admit many proper submodules. In this paper, assuming that char(F) = 2, we prove that V k admits a chain of submodules V k = M k ⊃ M k−1 ⊃ ... ⊃ M1 ⊃ M0 ⊃ M−1 = 0 where Mi ∼ = Vi for 1, ..., k−1 and M0 is the trivial 1-dimensional module. We also show that for i = 1, 2, ..., k the quotient Mi/Mi−2 is isomorphic to the so called i-th Grassmann module for G. Resting on this fact we can give a geometric description of Mi−1/Mi−2 as a submodule of the i-th Grassmann module. When F is perfect G ∼ = Sp(2n, F) and Mi/Mi−1 is isomorphic to the Weyl module for Sp(2n, F) relative to the i-th fundamental dominant weight of the root system of type Cn. All irreducible sections of the latter modules are known. Thus, when F is perfect, all irreducible sections of V k are known as well.

Veronesean embeddings of dual polar spaces of

Given a point-line geometry Γ and a pappian projective space S, a veronesean embedding of Γ in S ... more Given a point-line geometry Γ and a pappian projective space S, a veronesean embedding of Γ in S is an injective map e from the point-set of Γ to the set of points of S mapping the lines of Γ onto non-singular conics of S and such that e(Γ) spans S. In this paper we study veronesean embeddings of the dual polar space ∆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 ε gr n which maps a maximal singular subspace v1, ..., vn of V (namely a point of ∆n) to the point ∧ n i=1 vi of PG(n V), the composition ε vs n := ν2n • ε spin n of the spin (projective) embedding ε spin n of ∆n in PG(2 n − 1, F) with the quadric veronesean map ν2n : V (2 n , F) → V (2 n +1 2 , F), and a third embeddingεn defined algebraically in the Weyl module V (2λn), where λn is the fundamental dominant weight associated to the n-th simple root of the root system of type Bn. We shall prove thatεn and ε vs n are isomorphic. If char(F) = 2 then V (2λn) is irreducible andεn is isomorphic to ε gr n while if char(F) = 2 then ε gr n is a proper quotient ofεn. 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 ε sv 2 is relatively universal. On the contrary, if char(F) = 2 then ε vs 2 is not universal. We also prove that if F is a perfect field of characteristic 2 then ε vs n is not universal, for any n ≥ 2.

Sets of generators and chains of subspaces

arXiv: Combinatorics, 2019

The rank of a point-line geometry G is usually defined as the generating rank of G, namely the mi... more 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 (possib...

A G ] 1 0 A pr 2 01 8 On transparent embeddings of point-line geometries

We introduce the class of transparent embeddings for a point-line geometry Γ = (P ,L) as the clas... more We introduce the class of transparent embeddings for a point-line geometry Γ = (P ,L) as the class of full projective embeddings ε of Γ such that the preimage of any projective line fully contained in ε(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.

Generation of $J$-Grassmannians of buildings of type $A_n$ and $D_n$ with $J$ a non-connected set of types

Let $X_n(K)$ be a building of Coxeter type $X_n = A_n$ or $X_n = D_n$ defined over a given divisi... more 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 $\Gamma(K) = Gr_J(X_n(K))$ be the $J$-Grassmannian of $X_n(K)$. We prove that $\Gamma(K)$ cannot be generated over any proper sub-division ring $K_0$ of $K$. As a consequence, the generating rank of $\Gamma(K)$ 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 $(n+1)^2$.

The generating rank of a polar Grassmannian

Advances in Geometry, 2021

In this paper we compute the generating rank of k-polar Grassmannians defined over commutative di... more 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, 𝔽 q ) for q = 4, 8, 9 and 2n ≤ N ≤ 2n + 2. We prove that for N > 6 and anisotropic defect (polar corank) d ≠ 2 they can be generated over the prime subfield, thus determining their generating rank.

The graphs of projective codes

Finite Fields and Their Applications, 2018

Consider the Grassmann graph formed by k-dimensional subspaces of an n-dimensional vector space o... more 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 ≥ 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.

On two nonbuilding but simply connected compact Tits geometries of type C3

Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial, 2019

A classification of homogeneous compact Tits geometries of irreducible spherical type, with conne... more A classification of homogeneous compact Tits geometries of irreducible spherical type, with connected panels and admitting a compact flagtransitive automorphism group acting continuously on the geometry, has been obtained by Kramer and Lytchak [5] and [6]. 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 [11]. The proof by Schillewaert and Struyve is of topological nature and relies on the main result of [5] and [6]. In this paper we provide a combinatorial proof of that claim, independent of [5] and [6].

Grassmann embeddings of polar Grassmannians

Journal of Combinatorial Theory, Series A, 2020

In this paper we compute the dimension of the Grassmann embeddings of the polar Grassmannians ass... more 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 nonmaximal 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 [4]) and prove that the Grassmann embedding is a quotient of this generalized 'Weyl-like' embedding. We also estimate the dimension of the latter.

Flag-transitive extensions of dual projective spaces

Bulletin of the Belgian Mathematical Society - Simon Stevin, 1998

We classify the flag-transitive circular extensions of line-point systems of finite projective ge... more

On natural representations of the symplectic group

Bulletin of the Belgian Mathematical Society - Simon Stevin, 2011

Let V k be the Weyl module of dimension (2n k) − (2n k−2) for the group G = Sp(2n, F) arising fro... more Let V k be the Weyl module of dimension (2n k) − (2n k−2) for the group G = Sp(2n, F) arising from the k-th fundamental weight of the Lie algebra of G. Thus, V k affords the grassmann embedding of the k-th symplectic polar grassmannian of the building associated to G. When char(F) = p > 0 and n is sufficiently large compared with the difference n − k, the G-module V k is reducible. In this paper we are mainly interested in the first appearance of reducibility for a given h := n − k. It is known that, for given h and p, there exists an integer n(h, p) such that V k is reducible if and only if n ≥ n(h, p). Moreover, let n ≥ n(h, p) and R k the largest proper non-trivial submodule of V k. Then dim(R k) = 1 if n = n(h, p) while dim(R k) > 1 if n > n(h, p). In this paper we will show how this result can be obtained by an investigation of a certain chain of G-submodules of the exterior power W k := ∧ k V, where V = V(2n, F).

A geometric approach to alternating k-linear forms

Journal of Algebraic Combinatorics, 2016

Denote by G k (V) the Grassmannian of the k-subspaces of a vector space V over a field K. There i... more Denote by G k (V) the Grassmannian of the k-subspaces of a vector space V over a field K. There is a natural correspondence between hyperplanes H of G k (V) and alternating k-linear forms on V defined up to a scalar multiple. Given a hyperplane H of G k (V), we define a subspace R ↑ (H) of G k−1 (V) whose elements are the (k − 1)-subspaces A such that all k-spaces containing A belong to H. When n − k is even, R ↑ (H) might be empty; when n − k is odd, each element of G k−2 (V) is contained in at least one element of R ↑ (H). In the present paper we investigate several properties of R ↑ (H), settle some open problems and propose a conjecture.

Of finite geometries of type with thick lines

Note Di Matematica, 1986

Several facts suggest the conjecture that every finite geometry of type with thick lines is eithe... more Several facts suggest the conjecture that every finite geometry of type with thick lines is either a bulding or flat ( see [4] , [5] , [8] and [9]). That conjecture plays a central role in the problem of classifying finite thick geometries of type and ( see [4] , [7] and [8] ).In this paper we find very simple conditions that hold in a finite geometry of type with thick lines if and only if the geometry is either a building or flat. Then the previous conjecture can be restated so: prove that the conditions discussed in this paper are theorems in the case of finite geometries with thick lines.

Flag-transitive L_h.L*-geometries

Le Matematiche, 2006

Locally partial geometries with different types of residues

Annals of Discrete Mathematics, 1992

Abstract A class of locally partial geometries (L. pGs for short) is constructed where both linea... more Abstract A class of locally partial geometries (L. pGs for short) is constructed where both linear spaces and generalized quadrangles occur as point-residues. We conjecture that this class gives all possible example of L. pGs with that anomaly. We prove this conjecture when plane-residues are projective planes (see section 5). In the general case, we are able to prove the conjecture when there is at least one point with a linear residue, satisfying an additional assumption.

On a series of modules for the symplectic group in characteristic 2

Contemporary Mathematics, 2012

ABSTRACT Let V be a 2n-dimensional vector space defined over an arbitrary field F and G the sympl... more ABSTRACT Let V be a 2n-dimensional vector space defined over an arbitrary field F and G the symplectic group Sp (2n,F) stabilizing a non-degenerate alternating form α(·,·) of V. Let G k be the k-th Grassmannian of PG (V) and Δ k the k-Grassmannian of the C n -building Δ associated to G. Put W k :=∧ k V and let l k :G k →W k be the natural embedding of G k , sending a k-subspace (x 1 ,...,x k ) of V to the 1-subspace (x 1 ∧⋯∧x k ) of W k . Let ε k :Δ k →V k be the embedding of Δ k induced by l k , where V k is the subspace of W k spanned by the l k -images of the totally α-isotropic k-space of V. In [Bull. Belg. Math. Soc. - Simon Stevin 18, No. 1, 1-29 (2011; Zbl 1264.20039)], exploiting the fact that the embedding ε k-2i is universal when char (F)≠2, R. J. Blok and the authors of this paper have proved that if char (F)≠2 then V k-2i (k) /V k-2i+2 (k) and V k-2i are isomorphic as G-modules, for every i=1,...,[k 2]. In the present paper they prove that the same holds true when char (F)=2.

Flag-transitive Extensions of Dual A ne Spaces

Lax Projective Embeddings of Polar Spaces

Milan Journal of Mathematics, 2004

Let Γ be a non-degenerate polar space of rank n ≥ 3 where all of its lines have at least three po... more Let Γ be a non-degenerate polar space of rank n ≥ 3 where all of its lines have at least three points. We prove that, if Γ admits a lax embedding e : Γ → Σ in a projective space Σ defined over a skewfield K, then Γ is a classical and defined over a sub-skewfield K 0 of K. Accordingly, Γ admits a full embedding e 0 in a K 0-projective space Σ 0. We also prove that, under suitable hypotheses on e and e 0 , there exists an embeddingê : Σ 0 → Σ such thatêe 0 = e andê preserves dimensions.

On some flag-transitive extensions of the dual petersen graph

Journal of Combinatorial Theory, Series A, 1995

We consider flag-transitive extensions of the dual Petersen graph satisfying one of the following... more