2017/01/11 by Pankov, Mark
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1701.03054
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.