MacNeille completions of parabolic quotients

TL;DR

This paper describes the Dedekind-MacNeille completion of Bruhat orders on parabolic quotients of the symmetric group, relating it to alternating sign matrices and their geometric properties.

Contribution

It explicitly characterizes the completion of these orders, revealing new lattice structures and their connections to ASM varieties.

Findings

The completion forms a subposet of alternating sign matrices.

Meet and join operations correspond to unions and intersections of ASM varieties.

Special cases of the construction are discussed in detail.

Abstract

Alternating sign matrices (ASMs) arise as the Dedekind-MacNeille completion of the Bruhat order on the symmetric group. They enjoy fruitful combinatorial and geometric properties, with a particularly rich history on enumerations and bijections. In this paper, we explicitly describe the Dedekind-MacNeille completion of the Bruhat order on any parabolic quotients of the symmetric group. It is naturally a subposet of the alternating sign matrices, with different lattice operations. Moreover, we demonstrate the relations between the meet and join operations in this lattice with taking unions and intersections of the corresponding ASM varieties, respectively. Finally, we conclude with a more detailed discussion of special cases.