Cohen–Macaulayness criteria for alternating-sign-matrix varieties

For an alternating sign matrix A, do gradedness of [e,A], the corresponding equal-length condition and equidimensionality of every ASM variety X_B in that interval also imply Cohen–Macaulayness of all those varieties? The combinatorial and equidimensional conditions were already known to be equivalent.

Why it matters and what counts as progress

Why it matters: The missing implication would turn equivalent combinatorial conditions into a complete algebraic characterization of these ASM varieties.

What counts as progress: Prove the missing Cohen–Macaulay implication or identify a variety satisfying the equivalent conditions where it fails.

How the problem developed

26 Feb 2025 · Prior equivalences [1]
Laura Escobar, Karola Mészáros Klein and Melissa Weigandt Established earlier implications and equivalent combinatorial conditions related to Cohen–Macaulay alternating-sign-matrix varieties; their Conjecture 3.21 remained open.
8 May 2026 · AI-assisted criterion [2]
Colin Defant Reported an equivalence between gradedness, equal-length and equidimensionality conditions on an interval of alternating-sign-matrix varieties and Cohen–Macaulayness throughout that interval.

AI contribution

Weak Order on the MacNeille Completion of Bruhat Order

Colin Defant

Defant’s manuscript claims that, for every alternating sign matrix A, gradedness of [e,A], the corresponding equal-length condition and equidimensionality of all X_B in that interval are equivalent to Cohen–Macaulayness of all those ASM varieties. The new direction establishes Escobar–Klein–Weigandt Conjecture 3.21.

AI involvementAI-autonomous

ImportanceNotable

Technical verificationReported

Human understandingFull