# Apartments preserving transformations of Grassmannians of infinite-dimensional vector spaces

## TL;DR

This paper characterizes transformations of infinite-dimensional Grassmannians that preserve apartments, showing they are induced by semilinear automorphisms, extending classical finite-dimensional results to infinite dimensions.

## Contribution

It proves that bijections of infinite-dimensional Grassmannians preserving apartments are induced by semilinear automorphisms, including cases with infinite dimension and codimension.

## Key findings

- Transformations preserving apartments are induced by semilinear automorphisms.
- Results extend finite-dimensional Grassmannian automorphism characterizations.
- Connected components of Grassmann graphs also exhibit similar automorphism properties.

## 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.