Inversion and Cubic Vectors for Permutrees

TL;DR

This paper introduces inversion and cubic vectors for permutrees, providing new algebraic and geometric insights, including lattice properties and cubical embeddings, generalizing known structures like permutahedra and associahedra.

Contribution

It presents two novel vector generalizations for permutrees, enabling explicit lattice operations and geometric realizations, advancing the understanding of permutree structures.

Findings

Defined inversion vectors for permutrees

Established a new constructive proof of the lattice property

Constructed a cubical embedding of permutreehedra

Abstract

We introduce two generalizations of bracket vectors from binary trees to permutrees. These new vectors help describe algebraic and geometric properties of the rotation lattice of permutrees defined by Pilaud and Pons. The first generalization serves the role of an inversion vector for permutrees allowing us to define an explicit meet operation and provide a new constructive proof of the lattice property for permutree rotation lattices. The second generalization, which we call cubic vectors, allows for the construction of a cubic realization of these lattices which is proven to form a cubical embedding of the corresponding permutreehedra. These results specialize to those known about permutahedra and associahedra.