Topology of weak-order intervals in alternating sign matrices

Hamaker and Reiner proposed a sphere-or-contractible classification for open intervals in the weak order on alternating sign matrices: intervals of specified parabolic forms have sphere homotopy type, and the remaining intervals are contractible.

Why it matters and what counts as progress

Why it matters: The sphere-or-contractible proposal predicts the topology of every open interval from its parabolic form, so a finite exception changes the classification.

What counts as progress: Prove the classification or find an interval outside the specified forms with noncontractible order complex.

How the problem developed

2020 · Conjecture [1]
Zachary Hamaker and Victor Reiner Proposed a dichotomy for the poset topology of intervals in alternating-sign-matrix weak order.
6 Sept 2026 · Counterexample [2]
Colin Defant Exhibited a counterexample to the Hamaker–Reiner interval-topology conjecture within the type-A alternating-sign-matrix setting, alongside broader results on MacNeille completions.

AI contribution

Weak Order on the MacNeille Completion of Bruhat Order

Colin Defant

Defant’s manuscript displays an interval in the weak order on six-by-six alternating sign matrices whose Möbius value is 1, although it has none of the specified parabolic forms. Its open-interval order complex is therefore not contractible, contrary to the Hamaker–Reiner dichotomy.

AI involvementAI-autonomous

ImportanceNotable

Technical verificationReported

Human understandingFull