Adjacency Preserving Transformations of Grassmann Spaces

Wei, Huang

doi:10.1007/bf02942551

Cited by 32 publications

(26 citation statements)

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“…Disregarding the trivial cases A; = 1 and k = η-1, every A-transformation of Gk is induced by a semilinear transformation V -* V or (only when k = 2n) by a semilinear transformation of V onto its dual space V*. There is a wealth of related results, and we refer to [2], [6], and [9] for further references.…”

Section: Introductionmentioning

confidence: 99%

Transformations on the Product of Grassmann Spaces

Havlicek¹,

Pankov²

2005

Demonstratio Mathematica

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TRANSFORMATIONS ON THE PRODUCTOF GRASSMANN SPACES IntroductionLet Gk denote the set of all fc-dimensional subspaces of an n-dimensional vector space. We recall that two elements of Gk are called adjacent if their intersection has dimension k -1. The set Gk is point set of a partial linear space, namely a Grassmann space for 1 < k < η -1 (see Section 5) and a projective space for k 6 {1 , π -1}. Two adjacent subspaces are-in the language of partial linear spaces-two distinct collinear points.W.L. Chow [4] determined all bijections of Gk that preserve adjacency in both directions in the year 1949. In this paper we call such a mapping, for short, an A-transformation. Disregarding the trivial cases A; = 1 and k = η-1, every A-transformation of Gk is induced by a semilinear transformation V -* V or (only when k = 2n) by a semilinear transformation of V onto its dual space V*. There is a wealth of related results, and we refer to [2], [6], and [9] for further references.In the present note, we aim at generalizing Chow's result to products of Grassmann spaces. However, we consider only products of the form Gk x Gn-ki where Gk and G n -k stem from the same vector space V. Furthermore, for a fixed k we restrict our attention to a certain subset of Gk

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Section: Introductionmentioning

confidence: 99%

Transformations on the Product of Grassmann Spaces

Havlicek¹,

Pankov²

2005

Demonstratio Mathematica

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“…Interesting results closely related with Chow's theorem can be found in [5,6,7]. Now suppose that V is infinite-dimensional.…”

Section: Introductionmentioning

confidence: 99%

Order preserving transformations of the Hilbert Grassmannian

Pankov

2007

Arch. Math.

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Let H be a separable real Hilbert space. Denote by G∞(H) the Grassmannian consisting of closed subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We show that every order preserving transformation of G∞(H) can be extended to an automorphism of the lattice of closed subspaces of H. It follows from Mackey's result [8] that automorphisms of this lattice are induced by invertible bounded linear operators. Mathematics Subject Classification (2000). 46C05, 14M15.Keywords. Hilbert Grassmannian, invertible bounded operator. Introduction.Let V be a left vector space over a division ring R. If dim V = n is finite then we write G k (V ) for the Grassmannian consisting of k-dimensional subspaces of V , two elements of G k (V ) are called adjacent if their intersection is (k − 1)-dimensional. Chow's theorem [4] states that every adjacency preserving transformation of G k (V ) (1 < k < n − 1) is induced by a semi-linear isomorphism of V to itself or to the dual space V * , and the second possibility can be realized only for the case when n = 2k. Recall that a mapping l : V → V is semi-linear if it is additive and there exists an automorphism σ : R → R such that

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“…For example the 3-dimensional elliptic spaces are the only ones with Plücker transformations not induced by collineations and dualities [10]. More examples of Plücker spaces can be found in [3,5,13,15].…”

Section: Introductionmentioning

confidence: 99%