11 problems

Haase's face-subgroup conjecture. If $H\leq G$ and $P(H)\preceq P(G)$ is a face, then there is a partition

Let $a,b,c\geq 2$ be pairwise coprime integers, and let $P(a,b,c)$ be the permutation polytope associated with three disjoint cycles of lengths $ab$, $ac$, and $bc$. Facet-count co…

Let $G=\langle g\rangle$ be the cyclic permutation group of order $d$, acting with $t$ orbits, and for $I\subseteq [t]$ let $d_I$ denote the relevant order associated with the unio…

Stabilizer characterization conjecture. If $P(H)\preceq P(G)$ is a face, then

Wedge conjecture. The wedge over $F'$ can be combinatorially realized as a face of a permutation polytope if, or only if, the complement of $F'$ in $F$ is a face of $F$.

Free-sum conjecture. The free sum of centrally symmetric faces of permutation polytopes can be combinatorially realized as a face of a permutation polytope.

Affine-automorphism conjecture. If

Birkhoff-polytope uniqueness conjecture. If $P(G)$ is combinatorially equivalent to $B_n$ for some $n$, then $G$ is effectively equivalent to $S_n$.

Stable-degree conjecture. There exists a stably equivalent permutation representation $\rho'\colon G\to S_n$ such that

Strong embedding conjecture. There exists a permutation group $G\leq S_{2d}$ such that $P(G)$ is combinatorially equivalent, or more strongly lattice equivalent, to $P$.

Nonexistence conjecture. There are no four-dimensional faces of permutation polytopes having the combinatorial type of $Q_1$ or $Q_2$.