Hochschild polytopes

TL;DR

This paper introduces the Hochschild polytope, a new polytope derived from the multiplihedron by deleting inequalities, with faces corresponding to lighted shades and a lattice structure related to painted trees.

Contribution

It constructs the Hochschild polytope from the multiplihedron, establishing its face structure, lattice relations, and a shadow map linking painted trees to lighted shades.

Findings

Hochschild polytope faces correspond to lighted shades.

The polytope's skeleton matches the rotation lattice.

A natural shadow map forms a lattice morphism.

Abstract

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.