The Hamaker-Reiner Conjecture on ASM Weak Order Intervals
Statement
Hamaker and Reiner conjectured that the order complex of an open interval in the ASM weak order is contractible unless is the long element of a standard parabolic subgroup, in which case it is homotopy equivalent to a sphere. False: there is an interval in the ASM weak order on whose order complex is not contractible even though has no such form, detected by a nonzero Mobius function value.
Record
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construction · #1
ChatGPT 5.4 Pro, with Colin DefantThe record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.
Arrived as a refusal to do what it was asked. Having polished the author's notes, the model was asked what else was worth exploring around weak order on MacNeille completions and suggested proving the Hamaker-Reiner conjecture. The author asked it to prove that conjecture; instead it returned the counterexample that appears as Figure 1 of the paper. The author describes both this and the Escobar-Klein-Weigandt proof as obtained autonomously.
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