Hochschild polytopes

The (m, n)-multiplihedron is a polytope whose faces correspond to m-painted n-trees, and whose oriented skeleton is the Hasse diagram of the rotation lattice on binary m-painted n-trees. Deleting certain inequalities from the facet description of the (m, n)-multiplihedron, we construct the (m, n)-Hochschild polytope whose faces correspond to m-lighted n-shades, and whose oriented skeleton is the Hasse diagram of the rotation lattice on unary m-lighted n-shades. Moreover, there is a natural shadow map from m-painted n-trees to m-lighted n-shades, which turns out to define a meet semilattice morphism of rotation lattices. In particular, when m=1, our Hochschild polytope is a deformed permutahedron whose oriented skeleton is the Hasse diagram of the Hochschild lattice.