A simplicial poset is a finite poset $P$ with minimial element $\hat{0}$ such that every interval $[\hat{0},x]$ is isomorphic to a Boolean lattice. Simplicial posets are generalizations of simplicial complexes (see, e.g., http://math.mit.edu/~rstan/pubs/pubfiles/82.pdf). Now, one way to define a matroid is as a simplicial complex for which the restriction to any subset of vertices is a pure complex. We could then naively define a simplicial poset $P$ to be matroidal if $P\setminus F$ is always graded, where $F$ is the order filter generated by any subset of atoms of $P$.

Have these matroidal simplicial posets been studied at all? Are there conjectures (e.g., about face numbers) for them?

  • $\begingroup$ Are they Cohen-Macaulay posets? Is the ring $A_P$ or $\tilde{A}_P$ of your reference a level algebra? $\endgroup$ – Richard Stanley Apr 5 '19 at 15:58
  • $\begingroup$ @RichardStanley: It might make sense to restrict to the Cohen-Macaulay case. I was in particular thinking there might be an analog of your matroid $h$-vector conjecture here. $\endgroup$ – Sam Hopkins Apr 5 '19 at 16:05
  • $\begingroup$ (It might even make sense to restrict to Gorenstein* $P$.) $\endgroup$ – Sam Hopkins Apr 5 '19 at 16:29
  • $\begingroup$ Are there "interesting" Gorenstein* matroidal simplicial posets? The only Gorenstein* geometric lattices are boolean algebras. $\endgroup$ – Richard Stanley Apr 5 '19 at 21:37

If X is a compact, connected d-dimensional PL-manifold, then it has a simplicial poset triangulation with d+1 vertices. This is one of the foundational results of theory of crystallizations of manifolds. See, for instance, "A graph theoretical representation of PL-manifolds- a survey on crystallizations", Ferri, Gagliardi and Grasselli, Aequationes Mathematicae, 31 (1986), 121-141, for a survey. Such a simplicial poset would be an example of a matrioidal simplicial poset and includes lots of topological types including interesting Gorenstein* topological types. I do not know if the PL is needed for the proof, or is just to make sure there exists at least one triangulation.

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  • $\begingroup$ I looked up the reference to the crystallization theory and it is very nice. But I don't understand immediately why these posets will satisfy the "matroidal" condition. Is that easy to see? $\endgroup$ – Sam Hopkins Aug 29 '19 at 0:33

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