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Let $P$ be the Bruhat order of a Coxeter group, and let $s<t$ in $P$. The set $\Delta(s,t)$ of all chains of the open interval $(s,t)$ (called the order complex of $(s,t)$) is a simplicial complex. By the work of Björner and Wachs, the geometric realization of $\Delta(s,t)$ is homeomorphic to a sphere. What is known about when $\Delta(s,t)$ is polytopal, i.e., is the boundary complex of a simplicial (convex) polytope?

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  • $\begingroup$ Related question: Bjorner-Wachs show that [s,t] is the face lattice of a regular cellular ball. Is that cell complex ever polytopal? $\endgroup$ – Sam Hopkins Feb 20 at 22:49
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    $\begingroup$ @SamHopkins: there is some information at arxiv.org/pdf/2001.05011.pdf. In particular, $[s,t]$ is polytopal if and only if it is a lattice if and only if it is a boolean algebra. $\endgroup$ – Richard Stanley Feb 21 at 0:27

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