Stanley's conjecture, cover depth and extremal simplicial complexes

  • Benjamin Nill Freie Universitat Berlin
  • Kathrin Vorwerk KTH
Keywords: Monomial ideals, Stanley decomposition, Partitionable simplicial complexes


A famous conjecture by R. Stanley relates the depth of a module, an algebraic invariant, with the so-called Stanley depth, a geometric one. We describe two related geometric notions, the cover depth and the greedy depth, and we study their relations with the Stanley depth for Stanley-Reisner rings of simplicial complexes. This leads to a quest for the existence of extremely non-partitionable simplicial complexes. We include several open problems and questions.
This paper is a report about a research project suggested by J. Herzog at the summer school P.R.A.G.MAT.I.C. 2008 at the University of Catania. In particular, the paper describes a direction where we expect that possible counterexamples can be found at least for a weaker version of Stanley’s conjecture.

Author Biographies

Benjamin Nill, Freie Universitat Berlin
Research Group Lattice Polytopes,
Arnimallee 3, 14195 Berlin, Germany
Kathrin Vorwerk, KTH
Department of Mathematics
KTH, 10044 Stockholm, Sweden