# Weak generalized lifting property, Bruhat intervals and Coxeter matroids

## TL;DR

This paper introduces a weaker form of the generalized lifting property applicable to all finite Coxeter groups and demonstrates that parabolic Bruhat intervals form Coxeter matroids, revealing new combinatorial polytope properties.

## Contribution

It establishes a weaker lifting property for finite Coxeter groups and proves that all parabolic Bruhat intervals are Coxeter matroids, expanding understanding of their combinatorial structure.

## Key findings

- Weaker generalized lifting property for all finite Coxeter groups.
- Parabolic Bruhat intervals are Coxeter matroids.
- New combinatorial properties of associated polytopes.

## Abstract

We provide a weaker version of the generalized lifting property which holds in complete generality for all finite Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.