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Let me start with some background I want to use as analogy.

Consider a (convex) polytope $P$ and its set of triangulations. Among all the triangulations, a well behaved subset are the regular ones: roughly, those which come from the projection of a lifting of $P$ to one higher dimension using a generic linear functional in the last coordinate. The regular triangulations are "linear" in nature and indeed fit into a linear structure: they correspond to the vertices of the secondary polytope of $P$.

The secondary polytope describes relationships between the regular triangulations (e.g. vertices that are adjacent correspond to triangulations connected by a "flip"). We might want to enlarge the secondary polytope and fit all the triangulations into some structure together, and this is roughly what the Baues poset of $P$ doees. The minimal elements of the Baues poset are the triangulations, and the larger elements are the decompositions which are not complete triangulations.

We could hope that the Baues poset is not "too much bigger" than the secondary polytope: e.g., that it is homotopy equivalent. Some conjectures along these lines were formulated in the 90s, I believe, and I think that in general they were shown to be false. But nevertheless understanding the topology of the Baues poset in various contexts (including contexts beyond what I'm describing here) is still an interesting problem.

Now let me get to the actual setting of my question.

Suppose now $P$ is a simplicial polyotpe (still convex). Then it is shellable and we can look at its set of shellings. Among all the shellings, a well behaved subset are the line shellings: roughly, these correspond to orderings of the vertices of the dual (simple) polytope induced by a generic linear functional. The line shellings are, like regular triangulations, "linear" in nature and indeed it is not hard to see that they correspond to the regions of a certain hyperplane arrangement. I think this hyperplane arrangement has been studied, for instance, in the paper "Valid orderings of real hyperplane arrangements" by Richard Stanley.

Question: Has anyone tried to put a "Baues poset" like structure on the set of all shellings of $P$? We'd have to define "incomplete" shellings which correspond to the larger elements of the poset (I guess these could be 'shellings' where we add many facets at the same time).

Any references to work along these lines would be very much appreciated! Otherwise I think it might be a nice student project to look at some concrete examples...

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    $\begingroup$ I reached out to Jose Samper, who pointed out that sweeps are dual to line shellings. From this perspective, the following paper is relevant: arxiv.org/abs/2102.06134. $\endgroup$
    – Zach H
    Mar 4, 2021 at 16:09
  • $\begingroup$ Thanks, Zach, that paper indeed looks very relevant! $\endgroup$ Mar 4, 2021 at 16:12
  • $\begingroup$ In fact, I think that paper explains a complete answer to my question: the "poset of pseudo-sweeps" is the correct Baues poset for this context, and its topology is as good as one could hope for! $\endgroup$ Mar 4, 2021 at 16:18
  • $\begingroup$ Whoops, in Remark 6.3 the authors explain that only "broken line shellings" come from pseudo-sweeps, so this might not be a direct answer to my question (although it is very close...) $\endgroup$ Mar 4, 2021 at 16:47

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