Knutson and Miller showed that Schubert polynomials are multidegrees of matrix Schubert varieties, affine varieties introduced by Fulton, which are closely related to Schubert varieties. Many roads to studying Schubert polynomials pass through unions and intersections of matrix Schubert varieties. The third author showed that the natural indexing objects of arbitrary intersections of matrix Schubert varieties are alternating sign matrices (ASMs), which had previously enjoyed rich study in enumerative combinatorics and, as the six-vertex ice model, in statistical mechanics. Intersections of matrix Schubert varieties are now called ASM varieties. Every ASM variety is expressible as a union of matrix Schubert varieties.
Many fundamental algebro-geometric invariants (e.g., Castelnuovo-Mumford regularity, codimension, degree) are well understood combinatorially for matrix Schubert varieties, substantially via the combinatorics of strong Bruhat order on Sn. The extension of strong order to ASM(n), the set of n × n ASMs, has so far not borne as much algebro-geometric fruit for ASM varieties.
Hamaker and Reiner proposed an extension of weak Bruhat order
from Sn to ASM(n), which they studied from a
combinatorial perspective. In the present paper, we place this work
on algebro-geometric footing. We use weak order on ASMs to give a
characterization of codimension of ASM varieties. We also show that
weak order operators commute with K-theoretic divided difference
operators.
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