Embeddings of Affine Grassmann Spaces

Dentice, Eva Ferrara; Re, Pia Maria Lo

doi:10.1007/s00022-009-2059-y

Cited by 3 publications

(3 citation statements)

References 4 publications

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“…For projective Grassmannians the following is known or could be easily obtained (cf. [4,14]). (12) and under a lax embedding the image of any other projective Grassmannian in…”

Section: Complement Of a Grassmann Substructurementioning

confidence: 99%

Complements of Grassmann substructures in projective Grassmannians

Żynel

2013

Aequat. Math.

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Abstract. We prove that a projective Grassmannian can be recovered from the complement of one of its Grassmann substructures. Even more, the underlying projective space with the interval of its distinguished subspaces can be recovered. Mathematics Subject Classification (2010). 51A15, 51A45.Keywords. Slit space, Grassmannian, projective space, affine space. IntroductionThe idea of deleting a hyperplane from a geometric structure is not new. It has been applied in various contexts starting from an affine space as a result of deleting a hyperplane from a projective space. In [3] the complement of a geometric hyperplane, i.e. a proper subspace that either meets each line in a unique point or contains that line, in a Grassmann space, also known as a space of pencils, is investigated.Instead of a hyperplane any subspace can be deleted to obtain a more interesting structure, which in the case of projective geometry is called a slit space (cf. [5,6]). Just to mention [11] which deals with the so called partial geometry that consists of points of a finite projective space P not contained in a fixed secundum (a subspace of codimension 2) W of P and lines of P which do not intersect W. In [7] configurations that arise from finite affine planes by removing a pencil of lines are investigated. The authors generalise the result of [13] that the complement of a line in a finite affine plane can be embedded into a projective plane of the same order. The next example is the affine space of rectangular matrices, introduced in [12, Ch. 3], which resembles the structure of subspaces in a projective space that are complementary to a distinguished subspace. Two different approaches to this structure are delivered by [1] and [10].In both papers it is shown that the structure of complements is embeddable into an affine space and not all lines of that affine space arise as lines of this structure.Another good reason (frankly the real reason) to play with complements or slit spaces is as follows. Consider a finite dimensional projective space P coordinatized by an even characteristic field and endowed with a non-symplectic projective polarity ⊥, a pseudopolarity. Let W be the set of all self-conjugate points of P w.r.t. ⊥. Then W is known to be a proper subspace of P. If p is a point in P we write p ⊥ for the set of all points conjugate to p; for a subspace U we have then U ⊥ := p∈U p ⊥ . We say that a subspace U is regular when its radical U ∩ U ⊥ is void. Note that U ∩ U ⊥ ⊂ W and those subspaces that are skew (disjoint) to W are regular. So, our present result moves us closer to solving a similar problem of recovering the ambient projective Grassmannian from the Grassmannian of regular subspaces w.r.t. a pseudopolarity.In spaces of pencils interval subspaces, i.e. those induced by intervals in the lattice of subspaces of the underlying space, form a very important class. Each of them carries the structure of a space of pencils and any subspace with this property is an interval subspace (see [15]). This is the reason to call such subsp...

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“…For projective Grassmannians the following is known or could be easily obtained (cf. [4,14]). (12) and under a lax embedding the image of any other projective Grassmannian in…”

Section: Complement Of a Grassmann Substructurementioning

confidence: 99%

Complements of Grassmann substructures in projective Grassmannians

Żynel

2013

Aequat. Math.

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“…One could note a similarity of the above construction to the, analogous, construction of the space of pencils P k (A) associated with an affine space A (cf. definitions in [17], [2], or, in a more modern paper [7]).…”

Section: E For Every a ∈ T(b) There Is Exactly Onementioning

confidence: 99%

“…Deleting H from Q that set becomes either p(A 1 ∩ A 2 , B), or p * (A 1 , B) depending on whether C ⊂ H or C ⊂ H respectively. To prove (7) note that the right hand side of it means that A 3 is ∼ + -adjacent to every element of all the cliques A 1 , A 2 belong to. In particular, A 3 belongs to each of the maximal ∼ + -cliques that contains A 1 , A 2 .…”

Section: Let Us Point Out Some Deviations For Extreme Values Of K and Mmentioning

confidence: 99%

Affine polar spaces derived from polar spaces and Grassmann structures defined on them

Prażmowski¹,

Żynel²

2012

Preprint

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Affine Tallini Sets and Grassmannians

Dentice¹,

2010

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Abstract. In 1982/1983 A. Bichara and F. Mazzocca characterized the Grassmann space Gr(h, A) of index h of an affine space A of dimension at least 3 over a skew-field K by means of the intersection properties of the three disjoint families of maximal singular subspaces of Gr(h, A) and, till now, their result represents the only known characterization of Gr(h, A). If K is a commutative field and A has finite dimension m, then the image Gr(h, A)℘ under the well known Plücker morphism ℘ is a proper subset A m,h,K of P G(M, K), the affine Grassmannian of the h-subspaces of A. The aim of this paper is to introduce the notion of Affine Tallini Set and provide a natural and intrinsic characterization of A m,h,K from the point-line geometry point of view. More precisely, we prove that if a desarguesian projective space over a skew-field K contains an Affine Tallini Set satisfying suitable axioms on "perp" of lines, then the skew-field K is forced to be a commutative field and the incidence structure is an affine Grassmannian, up to projections. Furthermore, several result concerning Affine Tallini Sets are stated and proved.2000 Mathematics Subject Classification: 51A45, 51A30 .

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Embeddings of Affine Grassmann Spaces

Cited by 3 publications

References 4 publications

Complements of Grassmann substructures in projective Grassmannians

Complements of Grassmann substructures in projective Grassmannians

Affine polar spaces derived from polar spaces and Grassmann structures defined on them

Affine Tallini Sets and Grassmannians

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