Weak order: Alternating sign matrices, monotone triangles, and bumpless pipe dreams

Abstract

In 2018, Hamaker and Reiner introduced weak order for monotone triangles, which extended the usual notion of weak order on the symmetric group. Monotone triangles on $\{1, \ldots, n\}$ are well-known to be in bijection with the set ASM$(n)$ of $n \times n$ alternating sign matrices. Hamaker and Reiner defined weak order on ASM$(n)$ to be induced from weak order on monotone triangles via the standard bijection. Recently, the present authors used an a priori different definition of weak order on ASM$(n)$ to give a combinatorial characterization of the codimension of ASM varieties and to show that the natural K-theoretic representatives of these varieties satisfy a divided difference recurrence. In the present work, we establish compatibility of these definitions of weak order on ASM$(n)$. Additionally, we give three different explicit means of computing weak order covering relations on ASM$(n)$: on ASMs themselves, on monotone triangles in a manner different from that given by Hamaker and Reiner, and on bumpless pipe dreams, which are a newer family of combinatorial objects also in correspondence with ASMs. Finally, using the language of bumpless pipe dreams, we characterize the fibers of the weak order operators, each of which forms a sublattice of the strong Bruhat order on ASM$(n)$.