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arXiv · 1701.03054

Apartments preserving transformations of Grassmannians of infinite-dimensional vector spaces

Mark Pankov
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Abstract

We define the Grassmannians of an infinite-dimensional vector space $V$ as the orbits of the action of the general linear group ${\rm GL}(V)$ on the set of all subspaces. Let ${\mathcal G}$ be one of these Grassmannians. An apartment in ${\mathcal G}$ is the set of all elements of ${\mathcal G}$ spanned by subsets of a certain basis of $V$. We show that every bijective transformation $f$ of ${\mathcal G}$ such that $f$ and $f^{-1}$ send apartments to apartments is induced by a semilinear automorphism of $V$. In the case when ${\mathcal G}$ consists of subspaces whose dimension and codimension both are infinite, a such kind result will be proved also for the connected components of the associated Grassmann graph.

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Mark Pankov. 2017-01-11. Apartments preserving transformations of Grassmannians of infinite-dimensional vector spaces. https://arxiv.org/abs/1701.03054

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