Lu–Wang weak polymatroidality conjecture
Let be a pure vertex-decomposable simplicial complex on the vertex set , and let denote the Stanley–Reisner ideal of its Alexander dual. The Lu–Wang conjecture asks whether is weakly polymatroidal for every such .
References
Primary source
Additional references
- Powers of Alexander Duals of Stanley-Reisner Ideals of Chessboard Complexes — arXiv — Arka Ghosh, S. Selvaraja
Progress summary
A 2026 preprint reports that the conjecture fails for infinitely many chessboard examples, but the broader question remains open.
The Lu–Wang conjecture predicts weak polymatroidality for the relevant Alexander-dual Stanley–Reisner ideals of pure vertex-decomposable complexes. The reported counterexamples concern only an infinite family of chessboard-complex ideals, so they do not settle the full statement.
Known related counterexample
A separate example shows that a vertex-splittable Alexander-dual ideal need not be weakly polymatroidal, for ideals generated in degrees greater than ; the source does not identify it as a resolution of the Lu–Wang conjecture.
September 2026 chessboard-complex counterexamples
A September 2026 report citing Ghosh and Selvaraja’s preprint says weak polymatroidality fails for an infinite family of chessboard-complex ideals. This is substantive claimed progress, but the broader conjecture for all pure vertex-decomposable complexes remains unresolved.
Current status (as of September 2026): Weak polymatroidality is claimed to fail for an infinite chessboard-complex family, while the general pure vertex-decomposable case remains open.
Solutions 0
No solutions have been posted yet.