A characterization of the geometry of large maximal cliques of the alternating forms graph

Abstract

We prove that the geometry of vertices, edges and $q^n$-cliques of the graph $\mathsf{Alt}\left(n+1,q\right)$ of $\left(n+1\right)$-dimensional alternating forms over $GF\left(q\right)$, $n\ge 4$, is the unique flag-transitive geometry of rank 3 where planes are isomorphic to the point-line system of $\mathsf{AG}\left(n,q\right)$ and the star of a point is dually isomorphic to a projective space.