MacNeille completions of parabolic quotients

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.