Homotopy type of skeleta of the flag complex over a finite vector space and generalized Galois numbers

Abstract

Let $${\mathscr {F}}$$ stand for the flag complex associated to the lattice of proper subspaces of a finite-dimensional vector space V. This paper aims at giving a (discrete) Morse theoretical proof of the fact that the k-th skeleton of $${\mathscr {F}}$$ is homotopy equivalent to a wedge of spheres of dimension $$\min \{k,\dim ({\mathscr {F}})\}$$. The tight control provided by Morse theoretic methods (through an explicit discrete gradient field) allows us to give a formula for the number of spheres appearing in each of these wedge summands. As an application, we derive an explicit formula for the number of flags on V of a given dimension, i.e., the number of simplices in $${\mathscr {F}}$$ of the given dimension. Rather than depending on generalized Galois numbers, our formula for flags is given in terms of weighted inversion statistics of the symmetric group.