Linear extensions and shelling orders

TL;DR

This paper establishes that linear extensions of the Bruhat order form shelling orders for matroids and explores their properties as Coxeter matroids, also applying combinatorial promotion and evacuation techniques to shelling orders.

Contribution

It proves that linear extensions of the Bruhat order are shelling orders for matroids and connects barycentric subdivisions to Coxeter matroids, introducing new combinatorial methods.

Findings

Linear extensions of Bruhat order are shelling orders for matroids.

Barycentric subdivisions of matroids are Coxeter matroids.

Promotion and evacuation techniques are applied to shelling orders.

Abstract

We prove that linear extensions of the Bruhat order of a matroid are shelling orders and that the barycentric subdivision of a matroid is a Coxeter matroid, viewing barycentric subdivisions as subsets of a parabolic quotient of a symmetric group. A similar result holds for order ideals in minuscule quotients of symmetric groups and in their barycentric subdivisions. Moreover, we apply promotion and evacuation for labeled graphs of Malvenuto and Reutenauer to dual graphs of simplicial complexes, providing promotion and evacuation of shelling orders.