The permuto-associahedron revisited

TL;DR

This paper revisits the realization problem for a poset that combines features of permutohedra and associahedra, providing a new geometric construction of the permuto-associahedron.

Contribution

It offers a novel realization of Kapranov's poset by constructing its vertex set and normal fan simultaneously, expanding understanding of related polytopes.

Findings

Provided a new geometric realization of Kapranov's poset.

Constructed the vertex set and normal fan of the permuto-associahedron.

Connected algebraic and geometric combinatorics through explicit polytope construction.

Abstract

A classic problem connecting algebraic and geometric combinatorics is the realization problem: given a poset, determine whether there exists a polytope whose face lattice is the poset. In 1990s, Kapranov defined a poset as a hybrid between the face poset of a permutohedron and that of an associahedron, and he asked whether this poset is realizable. Shortly after his question was posed, Reiner and Ziegler provided a realization. Based on our previous work on the nested braid fan, we provide in this paper a different realization of Kapranov's poset by constructing the vertex set and the normal fan of a permuto-associahedron simultaneously.