Apartments preserving transformations of Grassmannians of infinitedimensional vector spaces
Abstract
We define the Grassmannians of an infinitedimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 arXiv:
 arXiv:1701.03054
 Bibcode:
 2017arXiv170103054P
 Keywords:

 Mathematics  Combinatorics