Boolean intersection ideals of permutations in the Bruhat order

TL;DR

This paper characterizes when the intersection of two principal order ideals in the Bruhat order of permutations forms a boolean algebra, using reduced words, permutation patterns, and support, linking pattern presence to ideal structure.

Contribution

It provides a comprehensive characterization of boolean intersections in Bruhat order through multiple perspectives, connecting permutation patterns with algebraic properties.

Findings

Intersection is boolean iff certain permutation patterns appear

Characterization via reduced words, patterns, and support

Equivalence of properties established through pattern analysis

Abstract

Motivated by recent work with Mazorchuk, we characterize the conditions under which the intersection of two principal order ideals in the Bruhat order is boolean. That characterization is presented in three versions: in terms of reduced words, in terms of permutation patterns, and in terms of permutation support. The equivalence of these properties follows from an analysis of what it means to have a specific letter repeated in a permutation's reduced words; namely, that a specific 321-pattern appears.