Intervals and factors in the Bruhat order

TL;DR

This paper investigates specific intervals within the Bruhat order of the symmetric group, focusing on their structure and properties of permutations, especially the longest permutation and decomposable permutations.

Contribution

It characterizes when certain properties of reduced words hold in Bruhat intervals, highlighting differences between decomposable and longest permutations.

Findings

Property holds for the longest permutation

Property does not hold for decomposable permutations

Characterization of intervals isomorphic to principal order ideals

Abstract

In this paper we study those generic intervals in the Bruhat order of the symmetric group that are isomorphic to the principal order ideal of a permutation w, and consider when the minimum and maximum elements of those intervals are related by a certain property of their reduced words. We show that the property does not hold when w is a decomposable permutation, and that the property always holds when w is the longest permutation.