Orthogonality of subspaces in metric-projective geometry

Prażmowski, Krzysztof; Żynel, Mariusz

doi:10.1515/advgeom.2010.041

Cited by 10 publications

(13 citation statements)

References 12 publications

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Section: Introductionmentioning

confidence: 61%

“…There is an analogue of our result for the set of all non-isotropic k-dimensional subspaces of a sesquilinear form under the assumption that the dimension of the associated vector space is not equal to 2k [11]. The method exploited in [11] (a description of maximal cliques and their intersections) does not work for the case when the graph is formed by subspaces whose dimension is half of dim H (Section 3.3) and thus we use completely different reasonings. Our proof is based on the comparison of geodesics of length two in ortho-Grassmann graphs (Section 4) and a characterisation of compatibility (commutativity) in terms of geodesics in Grassmann and ortho-Grassmann graphs (Theorem 10).…”

Section: Introductionmentioning

confidence: 61%

“…In [11], an analogue of Theorem 3 is proved for the set of all non-isotropic k-dimensional subspaces of a sesquilinear form under the assumption that the dimension of the associated vector space is distinct from 2k. Methods used to prove Lemma 12 as well as methods exploited in [11] are based on some properties of maximal cliques in ortho-Grassmann graphs.…”

Section: Remarksmentioning

confidence: 99%

“…This observation is a crucial tool in the proof of Lemma 12. Similarly, transformations considered in [11] send ortho-stars to ortho-stars and ortho-tops to ortho-tops.…”

Section: Remarksmentioning

confidence: 99%

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Automorphisms and Some Geodesic Properties of Ortho-Grassmann Graphs

Pankov¹,

Petelczyc²,

Żynel³

2021

Electron. J. Combin.

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Let $H$ be a complex Hilbert space. Consider the ortho-Grassmann graph $\Gamma^{\perp}_{k}(H)$ whose vertices are $k$-dimensional subspaces of $H$ (projections of rank $k$) and two subspaces are connected by an edge in this graph if they are compatible and adjacent (the corresponding rank-$k$ projections commute and their difference is an operator of rank $2$). Our main result is the following: if $\dim H\ne 2k$, then every automorphism of $\Gamma^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator; if $\dim H=2k\ge 6$, then every automorphism of $\Gamma^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator or it is the composition of such an automorphism and the orthocomplementary map. For the case when $\dim H=2k=4$ the statement fails. To prove this statement we compare geodesics of length two in ortho-Grassmann graphs and characterise compatibility (commutativity) in terms of geodesics in Grassmann and ortho-Grassmann graphs. At the end, we extend this result on generalised ortho-Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators.

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Section: Introductionmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 61%

Section: Remarksmentioning

confidence: 99%

“…This observation is a crucial tool in the proof of Lemma 12. Similarly, transformations considered in [11] send ortho-stars to ortho-stars and ortho-tops to ortho-tops.…”

Section: Remarksmentioning

confidence: 99%

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Automorphisms and Some Geodesic Properties of Ortho-Grassmann Graphs

Pankov¹,

Petelczyc²,

Żynel³

2021

Electron. J. Combin.

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“…It has been proved in [8,11] for elliptic spaces, in [12] for symplectic spaces, and in [19][20][21][22]24] for hyperbolic spaces that transformations which preserve ortho-adjacency of lines are induced by collineations that preserve orthogonality, unless the underlying projective space has three dimensions. These results were generalized for k-subspaces in metric-projective settings in [28]. Transformations of lines that preserve ortho-adjacency in Euclidean spaces were investigated in [2][3][4][5] as well as in [17,31].…”

Section: Introductionmentioning

confidence: 99%

Coplanarity of lines in projective and polar Grassmann spaces

Petelczyc

Żynel

2015

Aequat. Math.

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Grassmann spaces of regular subspaces

2010

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The underlying metric affine geometry, or metric projective geometry, can be recovered from Grassmann spaces associated with the family of regular subspaces of respective space. In other words, automorphisms of such Grassmann spaces are collineations witch preserve orthogonality of the respective underlying space. This generalizes results of Prażmowska et al. (Linear Algebra Appl 430:3066-3079, 2009) and Prażmowska andŻynel (Adv Geom, to appear). Mathematics Subject Classification (2010). 51F15, 51A45.A quadric Q in a projective space (or, in a more "abstract" and general language, a polar space embedded into a projective space) may determine quite various incidence geometries. One can consider the geometry on Q; the points and lines of this geometry are projective points and lines which lie on Q. This is a primary model of polar geometry. One can take Q and the lines which are intersections of Q and secants of Q (cf.[10]). One can consider as points the projective points not on Q, and as the lines projective lines which are tangent to Q (cf. [9]) or which are projective nonsingular (nonisotropic) or hyperbolic (cf. [12,13]).In this huge variety of incidence systems related to quadrics a very important subspaces are those radical free, called regular subspaces. Motivations to consider them come from reflection geometry, a quite natural language in which a geometry is characterized in terms of its admissible reflections (cf. [2,3,20]). Axes of such reflections are exactly those subspaces which we call regular. The concept of regular subspaces can be adopted in metric projective as well as in metric affine geometry and this is what we are doing here.

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Orthogonality of subspaces in metric-projective geometry

Cited by 10 publications

References 12 publications

Automorphisms and Some Geodesic Properties of Ortho-Grassmann Graphs

Automorphisms and Some Geodesic Properties of Ortho-Grassmann Graphs

Coplanarity of lines in projective and polar Grassmann spaces

Grassmann spaces of regular subspaces

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