# P-flag spaces and incidence stratifications

## TL;DR

This paper introduces a new class of homogeneous spaces derived from posets, generalizing classical flag varieties and providing novel stratifications related to incidence groups and parking functions.

## Contribution

It defines P-flag spaces as quotients of the general linear group by incidence groups of posets, extending classical geometric structures and stratifications.

## Key findings

- Recover classical Schubert cell decompositions for Grassmannians and flag varieties.
- Establish new stratifications by parking functions for trivial posets.
- Unify various geometric stratifications within a general poset framework.

## Abstract

For any finite poset P we introduce a homogeneous space as a quotient of the general linear group with the incidence group of P. When P is a chain this quotient is a flag variety; for the trivial poset our construction gives a variety recently introduced in [20]. Moreover we provide decompositions for any set in a projective space, induced by the action of the incidence group of a suitable poset. In the classical cases of Grassmannians and flag varieties we recover, depending on the choice of the poset, the partition into Schubert cells and the matroid strata. Our general framework produces, for the homogeneous spaces corresponding to the trivial posets, a stratification by parking functions.