Definition. Incidence positions and facet moves [marchetti2025rubik, Definitions 5--6] [frap-000L]
Definition. Incidence positions and facet moves [marchetti2025rubik, Definitions 5--6] [frap-000L]
Let \(P\) be a finite regular abstract polytope of rank \(n\geq 3\), with least and greatest faces \(F_{-1},F_n\). Write \(P^\circ =P\setminus \{F_{-1},F_n\}\). The incidence positions are
\[\mathcal R(P)=\{(F,G)\in P^\circ \times P^\circ :F\leq G\}.\]Choose a facet \(H\) and an element \(\alpha \) of the rotation subgroup \(\Gamma ^+(H/F_{-1})\). The facet extension lemma gives its unique extension \(\widehat \alpha \in \Gamma (P)\) fixing \(H\). The corresponding facet move is
\[\mu _{H,\alpha }(F,G)= \begin {cases} (\widehat \alpha (F),\widehat \alpha (G)),&F\leq H,\\ (F,G),&F\not \leq H. \end {cases}\]The first entry decides where the shape automorphism is applied; the second entry is transported with it. The facet is part of the move's name: one shape automorphism may stabilize several facets and give different local moves. Each move is a permutation of \(\mathcal R(P)\).